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0 such that (4.3)
II Vx112 %611X11,
holds for all x E ,0+ (,W, ). The number
it, (V) (II V112+µ+(V))1/2 + 11 VII
can be taken as S. Let X E .:P + (.Yt', ),
11 x 111 = 1. From the strictness of the plus-operator V
and Corollary 4.5 we conclude that the form [ (V ` V - µ+ (V )I, )x, y], is non-negative in .Y°,, and therefore we obtain from the Cauchy-Bunyakovski inequality for any y E YP, with 11 y I1, = 1 that
I [(V`V-h+(V)I,)x,Y]i1z max[O;µ-(V));
d)
there is a vo > 0 such that V ` V - POI,
0.
a) - b). If V is collinear with a uniformly (Jii, J2)-expansive operator, then V is a strict plus-operator (see Corollary 4.5) and there are numbers a > 0 and 6>0 such that [ Vx, Vx]2 '>a [x, x], + S jj x, j12. Therefore when x E ? + (JY,) we have X112 >
[Vx,
II VxlI2,
VI
i.e., V is a focusing plus-operator. b) c). Let V be a focusing strict plus-operator:
(xE e+(
[VX,Vx]2>, y I t'xlI2 I
i))
If a-( V) < 0, then the implication b) - c) follows from the definition of the strictness of a plus-operator V: µ+ (V) > 0.
Now suppose µ_ (V) > 0. We consider operators
Ve = Sf2 V, where
_ (1 - e)P2 + P2-. For sufficiently small e > 0 and x E taking Definition 4.22 into account, Sf2)
(.Y,) we have,
[ Vex, Vex)2 = [(V- ePi V)x, (V- ePi V)x]2 Vx, Vx]2 + (e2 - 2e) [P2 Vx, P2 Vx]2
(y+e2-2e)II VxIiz '> (y+e2-2c)II VexII2, and therefore Ve is a focusing plus-operator. Since [ Vex, Vex]
[ Vx, Vx]2 we
have µ-(V) 0 such that V`V- PI, : 0. Let the positive numbers v,, v2, v3 satisfy the following inequalities: µ_ (V) < v1 < P2 < P3 < µ+(V). Then (see Corollary 1.1.36) [ Vx, Vx]2 > v; [x, x] 1 (i = 1, 2, 3). From the premise about V we have that there is a sequence of vectors [xn) C -WI with II xn II = 1 such that [IX,,, VXn]2 - v2[xn, xn]1 = [(V`V- P211)X,,, Xn] I - 0 as n - oo.
Since the operator V*H2 V - v2 J1 is non-negative, we have
(V*J2V-v2J1)xn--'0.
(4.15)
Let (x) be a sub-sequence of the sequence (xn) consisting of semi-definite vectors, for definiteness, let us say of non-negative vectors. Then [(V`V- v211)Xn,Xn71 > [(V`V- v31l)X.,X,]I, and therefore
If
C .?
0.
(4.16)
(V*J2V- vIJ1)x,,-'B.
(4.17)
then similarly we obtain
Comparing the relation (4.15) with the relation (4.16) or (4.17) we obtain
x B-a contradiction. The implication d) > a) follows immediately from the fact that 1/ vo V is a uniformly (Ji, J2)-expansive operator.
C Remark 4.25: In proving the implication c) = d) it was established, essentially, that if V is a strict plus-operator and µ+ (V) > v > max [0; A- (V)J, then there is a S, > 0 such that [ Vx, Vx]2 v [x, x] 1 + S, x 11 ;. Moreover, the converse proposition is true: if [ Vx, Vx] 2 v [x, x+ 1 + x 11 1 for all x E .01 ,
v > 0, then µ+(V) > v > max(0;µ_(V)). For, µ+(v) >, v > µ_(V) by Corollary 1.1.36. We verify that v cannot coincide with µ±(V). Suppose, for example, that µ+ (V) = P. Then, by the definition of the number µ+ (V), there would be a sequence (x) C + ( 1) such that [xn ,xn] 1 = 1 and which i.e., [ Vxn, Vxn]2 - . ( V), [ Vxn, VXn]2 - µ+ (V) [Xn, Xn] 1 -- 0, contradicts the inequality [ VXn, VXn]2 - µ+ ( V) [X,,, xn] 1 , Sv x 1 3 sv [xn, I = 0, > 0. Similarly it can be verified that µ_ (V) # P. Thus for any plus-operator V (including even non-strict ones) 11
11
V, V- v11:0 (max10;µ_(V)) < v µ_ (V), and therefore we have, from Corollary 4.20, µ+ ( V`) > 1 > (V`). It remains only to apply Remark 4.25 again. We formulate next a theorem which characterizes the `power' of the set of uniformly (J,, J2)-expansive operators. Theorem 4.27: The set of uniformly (J,, J2)-expansive operators ((J,, J2)bi-expansive operators) is open in the uniform operator topology in the set of
all continuous operators acting from ., into .12, and its closure in this topology coincides with the set of all (J,, J2)-non-contractive operators ((J,, J2)-bi-non-contractive operators). Let V be a uniformly (J,, J2)-expansive operator [Vx, Vx]2 [x, x], + V I I - I V'- V112>0. Then
S11x11i(S>0),andlet V'besuch that 6-211 V'- VII
I
[V'x, V'x]2 = [(V+ V'- V)x, (V+ V'- V)x]2 [Vx,Vx]2-(211 V'- VII 11 VII+11 V'- VII2)11x11 [x,x]I + (s -211 V' - VII II VII + II V' - VII2)I1 x11 2 , i.e., V' is a uniformly (J,, J2)-expansive operator. If, in addition, V were a uniformly (J,, J2)-bi-expansive operator, then by Theorem 4.17 0 E p (V ), and therefore for V' from a sufficiently small neighbourhood of V we have 0 E p (V(, ). It then follows from Theorem 4.17 and Proposition 4.26 that V' is a uniformly (J,, J2)-bi-expansive operator. Since the uniform limit of (J,, J2)-non-contractive operators is a (J,, J2)-
non-contractive operator, the closure of the set of uniformly (J,, J2)expansive operators ((J,, J2)-bi-expansive operators) is embedded in the set of
(J,, J2 )-non-contractive operator (J,, J2)-bi-non-contractive operators). It remains to verify that if V is a (J,, J2)-non-contractive operator ((J,, J2)-binon-contractive operator), then it can be approximated in norm by uniformly (J,, J2)-expansive operators ((J,, J2)-bi-expansive operators). To do this we bring into consideration the operators
IV)=,1+eP; +,1-cP1.
(4.20)
It can be verified immediately that when 0 < e x]2 Vie')
[Ie')x, If')x] _ [x, x] + e II x 11 i,
are uniformly (J,, J2)-expansive operators ((J,, J2)-bi-expansive
§4 Plus-operators
129
operators) which as c --+ 0 approximate in norm the (J,, J2 )-non-contractive ((J1, J2)-bi-non-contractive) operator V. From Theorems 4.24 and 4.27 follows immediately
The set of focusing strict plus-operators (focusing doubly strict plus-operators) is open in the set of all continuous operators and its closure coincides with set of all strict plus-operators (doubly strict plusCorollary 4.28:
operators). We close this paragraph with the Remark 4.29: Plus-operator and sub-classes of plus-operators acting from the anti-space to . i into the anti-space to k2 will be called (as operators acting from W1 into .W2) minus-operators, and the names of sub-classes will be changed correspondingly (for example, (- Jl, - J2 )-non-contractive operators will be called (J,, J2)-non-expansive). All the propositions given above for plus-operators can be reformulated in a natural way for minusoperators. We leave the reader to do this, and later, in using such propositions, we shall refer back to this Remark 4.29.
4
If V is a continuous plus-operator acting from a Jl-space .i, into a
J2-space 2, with ci v = .1, then the operator V` V acts in 01, and therefore it is proper to ask about the description of its spectrum.
Theorem 4.30: Let V and V` be continuous plus-operators. Then
a(V`V)>0. From (4.18) and Theorem 3.24 it follows that a(V`V) C IR. Let 1, such - a E a(V`V), a > 0. Then there is a sequence C or,, 11 x that
(n-- co).
(4.21)
Without loss of generality we may suppose that the sequences [ [ x,,, x"] 1) [ [ Vx,,, 1)I' converge and have the limits 0, y, i and [ [ V` Vx, V`
and S respectively. From (4.21) we obtain (by mulitplying (V ` V+aI, )x scalarly by J, x,, and taking the limit as n oo) y + a/3 = 0 and (multiplying (V ` V + aI, )x scalarly by J, V` Vx and taking the limit as n -p oo) S + Cry = 0.
The first equality implies -y > 0; for, if R > 0, then y > 0 because V is a plus-operator, and if a < 0, then y > 0 because a > 0 by hypothesis and y = - 0. Now from S + ay = 0 and the fact that V is a plus-operator we conclude that S > 0, i.e., S = -y = 0 and therefore 0 = 0 also. Hence, lim
[(V`V-µ+(V)Il)x,,,x],=y-µ+(V)0=0,
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
130
and therefore (cf. proof of Theorem 3.24) (V c V - µ+ (V )Il )xn -p 0 as n oo. Comparing the last relation with (4.21), we obtain xn - 0-a contradiction. We now suppose that V is a continuous operator acting in a J-space W' with
v=.y'. Theorem 4.31: If V is a uniformly J-expansive operator, then its field of regularity contains the unit circle T = (i 11 i; I = 1). Moreover, V is a uniformly J-bi-expansive operator if and only if T C p(V). Suppose E E 7 and that E is not a point of regular type of the operator V.
Then there is a sequence (x,) C . when n S > 0,
with 11 xn 11 = I such that (V - EI )xn
B
oo. Since V is a uniformly J-expansive operator, we have, for some
((Vxn, Vxn] - (xn, xn]
(xn, (V - SI )xn] + S ((V - U )X., xn]
+ ((V-EI)xn, (V-EI)xn] >s 11 xnlli, and therefore xn - 0 when n oo; a contradiction. Now let V be a uniformly J-bi-expansive operator. From what has just been proved and Theorem 1.16, we conclude that IT C p (V ). Conversely, let V be a uniformly J-expansive operator and let I C p (V ). Since
VV'- I= (V- I)(V` -I)-, (Vcv- 1)(V- I)-'(V` -I), so
V`V- I,>0 Corollary 4.32:
implies W-60.
Let V be a focusing strict plus-operator in a J-space .Y. Then
all E such that max (0, µ _ (V)) < I E < µ+ (V) are points of regular type of the operator V, and they are regular if and only if V is a doubly strict focusing plus-operator.
This assertion follows directly from Theorem 4.24, (4.19), and Theorem 4.31.
Exercises and problems 1
Prove that for every strict plus-operator V there is on c = c( V) > 0 such that for
all plus-operators from an e-neighbourhood (in norm) of the operator V the plus-deficiency is the same (M. Krein and Shmul'yan [2] ). 2
Prove that if
V1, VZ are strict plus-operators and S+ (V,) = S. (V2) (M. Krein and Shmul'yan [2] ).
V, - V2 E v'm, then
§4 Plus-operators
131
3
Verify that if V is a strict plus-operator acting from a J,-space .W'i into a J2-space .W2, then Pz V also is a strict plus-operator. Hint: Use the fact [Pz Vx, Pz Vx]z > [Vx, Vx]z.
4
Prove that if V is a J-non-contractive continuous operator acting in a Krein space .Ye and 1P v =.Ye, then the disc 1 = (X II X II < 1) belongs to the field of regularity of
the operator V,,, D C if and only if V is a J-bi-non-contractive operator (M. Krein and Shmul'yan [21). Hint: Use (4.8) and Theorem 4.17. 5
Prove that the conditions a)-e) in Theorem 4.17 are equivalent to the condition f ) 00ap(V*,). Hint: Use Exercise 4.
6
Let .;V = Y+ O+ .JP- be a J-space, dim Y = dim .-W+, and let V be a semi-unitary
operator mapping .# into .N". Verify that V is a strict, but not a doubly strict, plus-operator. Hint: Use Theorem 4.17. 7
Prove that a continuous (J,, J2)-non-contractive operator V is (J,, Jz)-bi-noncontractive if and only if its graph Fv is a maximal non-negative subspace in the Jf-space ,Yr (4.5) (cf. Rintsner [4] ). Hint: Compare (4.14) with Theorem 1.8.11.
8 A plus-operator V is said to be stable if all operators from a certain neighbourhood of it are plus-operators. Prove that assertions a)-d) of Theorem 4.24 are equivalent to the stability of the plus-operator V (M. Krein and Shmul'yan [5] ). 9
Let V= II V;;II?;=1 be a continuous uniformly (J,, Jz)-expansive operator, with 9 v = .YPj. Prove that the operator will be uniformly (J,, J2)-bi-expansive if and only if 0¢ap(V1*1 ).
Hint. cf. Exercise 5 and Proposition 4.26. 10
Prove that the set of all continuous strict plus-operators acting in a J-space J ' forms a subgroup and that µ+(T, T2) > µ+(TI)A+ (T2) > µ_ (T, T2) (M. Krein and Shmul'yan [5] ).
11
Prove that if T, and T2 are strict plus-operators in a J-space .,Y, then 6+ (Ti T2) = 6+ (T,) + 6+ (T2), where, we recall, 6+ (T) is the plus-deficiency of the
operator T (see Theorem 4.15) (M. Krein and Shmul'yan [2] ). 12
Prove that every strict plus-operator in the space II. is doubly strict (Ginzburg [2)).
Hint: Cf. Exercises 4 and 5, using the fact that II, is finite-dimensional. 13
Let V be a (J,, Jz)-non-contractive operator acting from a Pontryagin space H into a Pontryagin space II, , with x = x', and It v = Ih. Then V is a continuous operator (cf. I. lokhvidov [17]). Hint: Verify that the operator V satisfies the conditions of Corollary 4.8.
14
Give an example showing that the condition x = x' in Exercise 13 is essential (Azizov).
15
Let V be a J-non-contractive operator, let .'+ C .,P+, 20+ = M'+
nY[li,
and
dim S'0 < oo. Then V1'+ = Y'+ implies V!'0 = T°+ (Azizov).
Hint: First verify that f°+ C VP°++, and then use the fact that t'0+
is finite-
dimensional. 16
Let .Yy = .Yr+ (D .%(- be a J-space,
-Y.
being separable infinite-dimensional spaces
132
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric with orthonormalized bases (e, )m. (e; )mm respectively. Verify that the linear operator V defined on the basis as follows
Vei =e+ ,, i=0, ± 1,...i
Ve; =e+i,
i= -1, -2,...i
Veo =B
is a J-bi-non-contractive operator and
Y+=CLin[(e,'+ei )0-
(earn E,u+
V'++2'+,
but nevertheless VYO+ ie 2'0++, where Y0+ = 2+ n 2+11 (cf. Problem 15) (Azizov). 17
Prove that if V is a (Ji, J2)-non-contractive operator with J v = .#, and if [Vxo, Vxo] = [xo, xo] for some xo, then [ Vxo, Vy] = [xo, y] for all y E 9 v, and therefore Vxo E V p and V c Vx0 = xo, i.e., xo E Ker(V ` V - I,) (Azizov).
Hint: Use the fact that the graph r v = ( ( Vx, X) I x E / v) is non-negative in the Jr-metric (4.11) and that the vector ( Vxo, xo) (EI'v) is isotropic in it. 18
If V is a J-bi-contractive operator, then a,(V) fl T = 0 (cf. Corollary 2.17) (E. lokhvidov [1]). Hint: Use the result of Exercise 17 applied to the operator V`.
19
Let V be a J-non-contractive operator acting in a J-space Y with / v = W. Prove that Ker (V - EI) C Ker(V` - 41) when E E T; in particular, if V is a J-bi-noncontractive operator and t E T, then Ker (V - EI) = Ker( V` - EI ), and therefore 4 v- t, = . when E o ap( V) (Azizov). Hint: Use Exercise 17.
20
Let V be a J-bi-non-contractive operator, Yo the neutral subspace, and Vxo = Yo. Prove that the operator V induced by the operator V in the )-space .h° _ YY 1 /2'o is J-bi-non-contractive (Azizov). Hint: Use Exercise 17 and Theorem 1.17.
21
Let V be a J-non-contractive operator with f/ v = . , and let Y C Ax(V) fl Ker (V` V- I) and V2' c .T. Prove that Y [1] Y (V) when kµ ;d I (Azizov).
Hint: As in Exercise 8 on § 1, use induction with respect to the parameter p + q, where p, q are the least non-negative integers for which the equalities
(V- XI)Px=B, (V-µI)Qy=B hold for xE', yEY,,(V). 22
Prove that if for an arbitrary J-non-contractive operator V we have Vx = Xx,
Vy=µy and IXI=IAI=1, kite, then [y,x]=0. Hint: Use the result of Exercise 17. 23
Prove that if V is a W-non-contractive operator ( W-non-expansive operator) and Vxo = kxo, Vx, = kx, + x0 when I X = 1, then x0 [1] Ker( V - XI) (cf. Exercise 11 on §2). Hint: Use the fact that [Vx, Vx] - [x, x] is non-negative (non-positive) on l v.
24
Prove that for a a-non-contractive operator V all the root subspaces 2',,(V) (I X I = 1), with the exception of not more than x of them, are negative eigensubspaces (cf. Corollary 2.28) (Azizov). Hint: Use the results of Exercise 22 and 23.
25 A a-non-contractive operator V in a separable space H. can have no more than a countable set of different eigenvalues on the circle T (Azizov, cf. I. lokhvidov [1] ). Hint: Use the results of Exercises 22 and 24. 26
Let ,Yi (i = 1,2) be Ji-spaces. A continuous operator V.-,W - . Z with s v = 01 is called a B-plus-operator if [x, x], > 0, x ;d 6 - [ Vx, VX] > 0. It is clear that V is a
plus-operator. Prove that in the case when .W' = II, every B-plus-operator is a strict plus-operator, but the converse assertion if false ([XVI] ).
§5 Isometric, semi-unitary, and unitary operators
133
27
Verify that in the first assertion of Exercise 26 the condition .YY1 =1I. is essential. Hint: In a J-space which is not a Pontryagin space consider an orthoprojector on to an improper maximal positive subspace and use Proposition 4.14 and Theorem 1.8.11.
28
It is clear that every uniformly (J,, J2)-expansive operator is a B-plus operator. Construct an example showing that even if YP1 = YP2 = rik the converse of this assertion is false.
29
Suppose that V is a B-plus-operator and V` a plus-operator. Show that V` is also a B-plus-operator. In particular (cf. Exercise 12), when W, =.'Y2 = II,,, for every B-plus-operator V the operator V` is also a B-plus-operator.
Hint: Consider an arbitrary FE .,//+ (.)l"2); prove that the subspace (VF)1' negative, and use Theorem I 1.19 (cf. [XVI] ). 30
is
Prove that for a plus-operator V with V('O + (.Y'1) n iv) n p °wY2) ;e 0 it is always true that 31 v C ?+ (.W'2) (Brodskiy [ 1 ] ).
31
Prove that under the conditions of Theorems 4.6 it follows from
-P+,(, r,) n Ker V = 0, 9 v J Wi and the fact that the operator V (see (4.4)) is bounded and V11Jrf t =,,Y2' that V is bounded (Brodskiy [1], cf. I. Iokhvidov [17]). 32
Verify that in a J-space .# = X+ Q+ Y' with dim .R'+ = co the operator V11
V= 11
0
01 1
1-
I
is a strict plus-operator, where V is a semi-unitary operator with V,. + ;d but V` is not even a plus-operator (I. Iokhvidov [XVII]). , - .02 be a strict minus-operator. Then Ker Pi VP; I .Jt"i = [0), and Let V: when 9 v = Ye, the equality .*p2 VP, _ Ye2 is equivalent to the minus-operator V being doubly strict (cf. Ginzburg [2] ). Hint: In proving the first assertion, use the results of Exercise 3 and the equality (4.3), and in proving the second, use Exercise 7 on §1 and Theorem 4.17. The Remark 4.29 has also to be taken into account. Ye+,
33
34
Let
V. .Y 1 -y .02 be a (J,, J2)-bi-non-expansive operator, let 2';(C J';) be
uniformly positive subspaces, W; be the Gram operators of the subspaces TI-L" and let P; be the J;-orthogonal projectors on to Y1J (i = 1,2). Then P2 VP1 I-V[l" is a (W,, W2)-bi-non-expansive operator (Azizov). Hint: Without loss of generality assume that Y; C . Y, (i = 1, 2). Write Vin matrix form relative to the decompositions .0, _ Y, [+] Y[", 2 = M'2 [+] Y?] and calculate the matrices V*J2V(,< J,), VJ, V* (,< J2).
§5. Isometric, semi-unitary, and unitary operators Definition 5.1: A linear operator U acting from a W,-space lei into a W2-space '2 is said to be
I
1) (W,, W2)-isometric if [Ux, Ux]2 = [x, x], when x E iu; 2) (W,, W2)-semi-unitary if it is (W,, W2)-isometric and 9u=7r,; 3) (W,, W2)-unitary if it is (W,, W2)-semi-unitary and Mu = .)2.
In particular, if W, = I, and W2 = 12, then 5.1 is the definition of (ordinary) isometric, semi-unitary, and unitary operators.
134
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
From Definition 5.1 it can also be seen that an arbitrary operator U mapping
a neutral lineal cu into a neutral lineal 11u can serve as an example of a (WI, W2)-isometric operator. Therefore, in contrast to (Hilbert) isometric operators, (WI, W2)-isometric operators in infinite-dimensional spaces can be
unbounded and can have a non-trivial kernel. It is clear that U is a (WI, W2)-isometric operator (respectively a (J1, J2)-unitary operator) if and
only if it is simultaneously a (WI, W2)-non-contractive operator and a (WI, W2 )-non-expansive operator (respectively a (JI, J2)-bi-non-contractive operator and a (JI, J2)-bi-non-expansive operator), and for such operators the corresponding propositions in §4 hold. 5.2 In order that a linear operator V with an indefinite domain of definition should be collinear with some (WI, W2)-isometric operator U, i.e., V= XU (X * 0), it is necessary and sufficient that V should be a plus-operator and a strict minus-operator (or a minus-operator and a strict plus operator) simultaneously.
The necessity is trivial. Now suppose, for example, that V is a plusoperator and a strict minus-operator. Then (see Theorem 4.3) there are constants a > 0 and 0 > 0 such that [ Vx, Vx]2 > a [x, x] I and - [Vx, Vx]2 > -/3[x, x]I when xE qv, and therefore (a -0)[x, x] I < 0. Since _q v is indefinite (by hypothesis), a = (3 > 0, and U = (1 /f) V is a (WI, W2)-isometric operator. Later we shall use the following simple proposition, which follows immediately from the polarization formula (see Exercise 1 on 1 §1). 5.3 For an operator U to be ( W1, W2)-isometric it is necessary and sufficient that [ Ux, Uy12 = [x, y] I for all w, y E c1u.
Remark 5.4: It follows form Proposition 5.3 that if x, y E 9u, then [ x [1] y] a [ Ux [1] Uy], and therefore a lineal Y(C 9u) is isotropic in CAu if and only if U2 is isotropic in 9?u. The possibility was mentioned above of a (WI, W2)-isometric operator with
a neutral domain of definition being unbounded. It turns out that the last condition is not essential; there are even (JI, J2)-semi-unitary unbounded operators.
Example 5.5: Let e _ O+ 0 .e- be an infinite-dimensional J-space with infinite-dimensional let be a maximal dual pair of semi-definite subspaces of the classes h + respectively (see Exercise 4 on § 1.10), let go = + n 9- with dim Yo = 1, and let 9± _ Yo + 2+, where 2'+ are
definite lineals dense in 9+. Then relative to the scalar product ± [x, y] (x, y E `+) the lineals 2+ are Hilbert spaces (see Definition 1.5.9 and Propositions 1.1.23). Therefore there are isometric operators U+ mapping on to r+. We now define an operator U on elements x = x+ + x-, x` E ./%-, by the formula U(x+ + x-) = U+x+ + U-x-. It is easy to see that .W+
this is a J-semi-unitary operator, and that the image of the subspace V+ is the
§5 Isometric, semi-unitary, and unitary operators
135
lineal '+ (not closed in ,Y). From Proposition 5.2 and Corollary 4.13 we conclude that U is an unbounded operator. In this example a sufficient condition for the operator U to be unbounded turned out to be that U.+ was not closed. It turns out that a condition of this sort is also necessary. Theorem 5.6: Let U be a (J1, J2)-semi-unitary operator. Then the following conditions are equivalent.
a) U is a continuous operator; b) U.W' are subspaces; c) 9?u is a subspace.
a) - b) follows immediately from Proposition 5.2 and Corollary 4.13. b) - c) we verify that UM it are uniformly definite subspaces. To do this it
is sufficient (see Proposition 1.5.6) to prove that the subspace U.W' are complete relative to the intrinsic norm I [ Ux, Ux]z 11,2 (x E . i ). But the latter follows immediately from the facts that .ei are Hilbert spaces with the scalar products ± [x, y] 1, and ± [ Ux, Uy] z in . Wl , and U I . i are isometric
operators. Thus U!Pi are uniformly definite subspaces. It only remains to apply Theorem 1.5.7.
c) - a). Since [ Ux, Uy]z = [x, y] 1 for all x, y E M, and M1 is not degenerate, Ker U= (0). Therefore the operator U-1 exists. It follows from the same relation that the fact that U is a (J1, J2)-semi-unitary operator is equivalent to the inclusion. (5.1) U-' C U` Therefore U-1 admits closure and it is defined on the subspace Mu, i.e., U-1
is a continuous operator. Hence if follows by Banach's theorem that U is continuous. Remark 5.7. Essentially it is proved in the implications a) - b) - c) that U.1± and U.Y( are regular subspaces. Corollary 5.8:
Every (J1, J2)-unitary operator is bounded.
Remark 5.9: If . i are Gi-spaces, i.e. 0 ¢ op(Gi) (i = 1, 2), then the relation (5.1) also holds for (G1, G2)-semi-unitary operators, and therefore if Ru is a subspace, then U-1 and U are bounded operators. In particular, all (G1, G2)unitary operators are bounded, and the condition that Ube a (G1, G2)-unitary operator can be rewritten in the form
U-' = U` (= Gi 1U*G2),
Iu= .01,
u=.(z.
(5.2)
2E Now let ,Yi be Ji-spaces, i = 1, 2, and let U = II Ui; II;; - 1 be the matrix representation of a (J1, J2)-semi-unitary (J1, J2)-bi-non-contractive operator.
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
136
By Remark 4.21 and Theorem 4.17 the operator U11 is continuously invertible on the whole of ez, and U21 U11' (=- r) is the angular operator of the uniformly positive subspace U.Yi , and therefore II IF < 1. We bring into consideration the operator z
U(r) =11 U(r )ii 11 i.i=1 = I l r
(I2 -
r*r) - v2
(I2 -
r* (12+ -
r*r) - 1/2
rr*)- 1/2
(12 - rr*)-1/2
I I
(5.3)
operating in the space .J'2 = M2+ c Mj . Using Formula (5.2) it can be verified immediately that U(F) is a J2-unitary
operator, and it follows from 3.21 after a straightforward verification that U(I') is a uniformly positive operator. By the construction of U(F) we have that V = U(r) -' U is a (J1, J2)-semi-unitary (J,, J2)-bi-non-contractive operator, mapping i into Jez , and therefore (see Remark 5.4) also ,fj into z
So V has the matrix representation V = II Vii II?I=1, where V,,: is unitary, Vzz: I -i is a semi-unitary operator, and V12 = 0,
V21 = 0.
Conversely, let Y+ be an arbitrary maximal uniformly positive subspace in -Y2 and let r be its angular operator. We introduce the J2-unitary operator U(F) in accordance with formula (5.3). Let dim Yi = dim Mz , dim I < dim . z , and V,,: e, M2 be unitary, V22: . I Wz a semi-unitary operator, and V21 = 0, V12 = 0. Then V= Vii 11 i J=1 is simultaneously a semi-unitary operator and a (J1, J2)-semi-unitary operator, and U= U(P) V is a (J,, J2)-semi-unitary (J,, J2)-bi-non-contractive operator, mapping l into the given space T+ C'2. We summarize the above argument:
Theorem 5.10:
A one-to-one correspondence has been established between
all triples of operators (r, VI1, V22), where r: w2 contraction, Vi,:. ; - .02+ is a unitary operator, V22:
z
is a uniform
i - .)"
a semi-
unitary operator, and all (J1, J2)-semi-unitary (J,, J2)-bi-non-contractive operators: (r, VI1, V22)
U= U(r)V,
where U(F) is constructed according to formula (5.3), and V= II Viil1?i=,,
V21 =0, V,2=0;
U- (r; V,,, V22), where r = Uz, UI,', V ,
V22 = Pz U(r) ' UPI I . I .
Corollary 5.11: Under the conditions of Theorems 5.10 the operator U is (J,, J2)-unitary if and only if V22 is a unitary operator. Corollary 5.12: If U= II Uii 1I?i=I is a (J,, J2)-semi-unitary (JI, J2)-bi-noncontractive operator, then U21 E .9 - U,2 E 99.
§5 Isometric, semi-unitary, and unitary operators
137
This follows from the implication
9- 0 rE9.4*r*E.91'.- U21EY.. If U = U(r) V is a J-unitary operator and if r E .91,,, then
Corollary 5.13:
C\T C AM. Since
Iz -(12
rE,St ,
-r*r)-ins
IZ - (Iz - rr*)- 112 E .gym,
and therefore U- V E 9,,. Since V is a unitary operator, it only remains to use Theorem 2.11, taking into account that the spectrum of a unitary operator lies on the unit circle.
Remark 5.14: In §§4 and 5 we have not so far investigated the spectral properties of J-non-contractive and J-isometric operators, since later, in §6, we shall establish, by means of the Cayley-Nayman transformation, a connection
between the classes of operators and the classes of J-dissipative and Jsymmetric operators respectively, and, in so doing, a connection between their spectral properties.
3
In this paragraph we introduce and describe a special class of J-unitary
operators, namely, stable J-unitary operators. This description is obtained as a consequence of a more general result.
Definition 5.15: A J-unitary operator U acting in a J-space W is said to be stable if 11 U" 11
1. Prove that then a2 = (ai) ' is also a spectral set for U, that the projector Pa,uo, is J-self-adjoint, and therefore d'=PQ,ua, is a projectionally complete space. Moreover, if of (I ai I > 1) and a2'= (a(`)-' is a second pair of such spectral sets for U, and a, n a' , = 0, and 2' = Po; u0J, then 2' [1] Y'. Hint: Use Exercise 14. For the rest the proof is entirely similar to that of Corollary 3.12.
16
§6 1
V be a J-isometric operator with k, µ E o ( V) and 1\µ * 1. (cf, [XIV]). 2',,(V) [1] Hint: cf. Exercises 21 and 22 on §4. Let
Then
The Cayley-Neyman transformation We shall find it convenient to introduce this transformation not merely for
linear operators but also in a more general setting-for linear relations (see Definition 1.2) be a given Hilbert space. We consider the set 2' of all linear relations Let T, i.e., of all possible lineals of the space ' x _V1. Thus T = (<x, y)) is the
§6 The Cayley-Neyman transformation
143
linear set of ordered pairs (x, y) (x, y E .Ye), and linear operations in ,Ye x Yt' (and therefore also in T) are defined in the natural way (component-wise). We recall (see § 1.1) that a linear relation T is the graph PA of some linear operator M. If and only if it follows from (w, y) E T and x = B that y = 0; so A: that T = rA = ((x, Ax) ).1 E VA (CAA C Jr'). Such linear relations T are called single-valued linear relations. The sets
VT= (xE.YP there is an yE.J'P such that (x, y) E T) 1 T = (y E ,-,Y I there is an x E .) such that (x, y) E T) J
(6.1)
'
are called respectively the domain of definition and the range of values of the linear relation T. For all T E Y we also introduce the following definitions and notation: Ker T= Ix E.WI (x,0>ET),
Ind T= (yE.Y°j(0,y)ET), (6.2)
-T=(-1)T, XT=((x,Xy)I (x, y)ET)(XEC), T-t=((Y,x)I<x,Y)ET).
(6.3)
(6.4)
Ker T and Ind T are called respectively (see Definition 1.1) the kernel and the indefiniteness of the linar relation T. It is clear that X T, T -' E Y. We also introduce the identical linear relation I= ((x, x) ) XE.,y and the sum of two linear relations T1, T2 E ?:
T1 +T2=(<x,yi+Y2) I<x,Yk)ETk, k= 1,21.
(6.5)
It is clear that Tt + T2 E Y and 9T,+ T2 = cI T, n cT,. In particular, we consider the linear relation
Tx=T- XI
(TE22,XEC)
(6.6)
and by means of it we introduce for a linear relation T its resolvent set
p(T)_ (XEC
I
Ker Tx = (0), 3rr_
),
(6.7)
the spectrum
u(T) = C\p(T)
(6.8)
and, in particular, the point spectrum ap(T) = (XECIKer T>,;e (0)).
(6.9)
Points X E p(T) are called regular points, but 9? E ap(T) are eigenvalues of the linear relation T. For eigenvalues X the lineal Ker Ta = (x E Y' I < x, Xx) E T) is called the eigenlineal, and its non-zero vectors are called eigenvectors of the linear relation T corresponding to the eigenvalues X)'.
We can introduce a further classification of points X(E C) of the point spectrum op (T) of an arbitrary linear relation T by dividing these points into '
It will be convenient later, when Ind T ;d 101, also to suppose that, by definition, oo E oo(T)
(Ca(T)).
144
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
two sub-classes ap, (T) and u,2(T) (cf. Theorem 1.16)
up,,(T)= (XEap(T)I.9?r, ;6 W),
(6.10)
(6.11) ap,2(T)_(XEap(T)I_JiT The continuous spectrum ac(T) and the residual spectrum ar(T) of a linear relation T are defined by the formulae
ac(T) = (X E a(T)\ap(T) 191r, # 4rA =
),
ar(T)_ (XEa(T)\ap(T)I4T,*W)
(6.12) (6.13)
respectively
From (6.7)-(6.9), and (6.12), (6.13) it now follows that a(T) is the union of four mutually non-intersecting parts:
a(T) = up,,(T) U ap,2(T) U ac(T) U ar(T).
The field of regularity r(T) of a linear relation T is the set
r(T) = IX E C\ap(T) I T, =.;?T,)
(6.14)
It is clear that p(T) C r(T) (since e_,), and that r(T)\p(T) C arw(T). It is easy to convince onself that the definitions introduced of the sets p(T), a (T ), ap,, (T), ap,2 (T ), ac(T) and ar(T) for a linear relation T generalize the
definitions of the corresponding sets for closed linear operators A. When T= I A the two sets of definitions are simply identical.
2
For a linear relation T E Y we now introduce the direct (Kr) and the inverse
Cayley-Neyman transformations defined for a non-real parameter To do this we first introduce the corresponding transformations element-
wise, i.e., for all pairs (x, y), (u, v) E ' x, kt(x, y) _ (y - ('z, y - ('x),
(v-u,(v-('u)
(6.15) (6.16)
O A direct calculation will verify the proposition. 6.1 The .Ye x.
transformations kt and kt ' are mutually inverse bijections of
onto. xM.
Now for any (k;4)
E C and S, T E 2 we put
Kr(T)= (ks(x,y))<x,y)ET, Kt '(T)= (kt '(u,v))Es.
(6.17) (6.18)
6.2 The transformation Kt((' # (") maps ? on to Y bijectively, and Kt carries out the inverse transformation.
§6 The Cayley-Neyman transformation
145
This assertion follows directly from 6.1. 6.3
Let TE 9 and S= Kt(T) ((' ;d ) or, equivalently, T= KF t(S). Then Ker S = Ker Tt, Ker T = Ker (CS - CI),
Ind S = Ker Tr, Ind T = Ker St,
(6.19)
!2T= R,,
(6.21)
'Ws ='RT"
V, = RT"
(6.20)
The formulae (6.19) and (6.20) follow directly from a comparison of the definitions (6.2), (6.3), (6.5) with (6.17). The same applies to (6.21) for the verification of which the definitions (6.1) must be brought in again. We now investigate how the spectrum of a linear relation T(E') and its
components are transformed on transition from the linear relation T to another linear relation S = Kr(T). To do this we introduce, for (' # (", a mapping 4;t: C
C in the extended complex plane t = C U (co 1:
(X - (' )(X - .)-t, X ;4 (', X * O°, 00,
(6.22)
= oo.
1,
The function µ = ct(X) maps
Theorem 6.4:
ap,1(T), ap,2(T), ac (T), ac(T), r(S), and p(T) bijectively on to ap.t(S), ap,2(S), ar(S), a,(S), r(S), and p(S)
respectively.
When X ;4 (', X ;d oo, we have by (6.17), (6.15) and (6.22)
S,,=S-µI _ <X,y)ETl
(y-('X) [(Y - rX,
-
II
(y-XX)) <X,y>ET}
from which it follows that when
)X E C
Ker S, = Ker Ta,
(6.23) ,s = ?1Ta. We note that, taking (6.22) and Proposition 6.3 into account, the relations (6.23) remain valid for all X E C if, for any linear relation T, we put
Ker T. = Ind T by definition. 1 We point that, on this understanding, the Definitions (6.10) and (6.11) make sense also for a = oo E ap(T) (see the Footnote on p. 143), and so the assertions of Theorem 6.4 remain vald in this case. This is also true when X =,r E up(T).
146
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
In accordance with this we shall also suppose (cf. (6.7))' that
X=ooEp(T)* (Yr=de, Ind T= (0]]. All the assertions of the Theorem now follow directly from a comparison of the formulae (6.23) with the definitions (6.7)-(6.14). Corollary 6.5:
The function
(µ -
1)1,
00,
µ
l,µ # 00,
µ=1,
(6.24)
µ = 00
effects for the linear relations S and T = KF ' (S) the mapping inverse to the mapping ct (see Theorem 6.4).
3 We return to the most important case for us when the Hilbert space W generating the set 22 (C ,,Y x W) of linear relations is a W-space, i.e., it is equipped with an indefinite W-metric [x, y] = (Wx, y)(x, y E) (see 1.§6.6).
Definition 6.6: A linear relation T(E 2') is said to be W-non-expansive (respectively, W-non-contractive, W-isometric) in W if for all (x, y) E T we have [y, y] 5 [x, x] (respectively, [y, y] > [x, x], [y, y] = [x, x]). It is clear that Definition 6.6 generalizes the corresponding definitions for linear operators V (see Definitions 4.2 and 5.1). The latter definitions are obtained from Definition 6.6 when T = F v is the graph of the operator V. In the case when 91 is a G-space, i.e. [x, y] = (Gx, y) (0 g up(G) we introduce Definition 6.7. For a TE 2' the linear relation
T'= ((u, v) E
X W [y, u] = [x, v] for all (x, y) E 71
is called its G-adjoint. Remark 6.8: It can be seen from this definition that T`(E2) exists for any linear relation T. However, even in the case when T= GA is the graph of a
linear operator A ('A, WA C W) the linear relation T` is not necessarily a graph (i.e., a single-valued linear relation)-see Exercise 1 below. However, comparison of the Definitions 6.7 and 1.1 shows that, when T= FA and VA = .W', these definitions are equivalent, i.e., T` = FA'
Definition 6.9: A linear relation T is said to be G-symmetric if T C T`, and to be G-selfadjoint if T` = T. We note that the condition T C T` is equivalent We recall that the condition Ind T= LO is equivalent to T being the graph of a linear operator. But then by Banach's theorem, in the case when this operator is closed, the condition VT implies that the operator is bounded, and this gives meaning to the notation, oo E p(T) in this case.
§6 The Cayley-Neyman transformation
147
in an obvious way to the requirement that [y, x] = [x, y] for all (x, y) E T, and in this form we generalize it to an arbitrary W-space, and call such T W-symmetric.
When T= FA with CIA = ,e the Definition 6.9 goes over into the definitions (of the graph) of a G-symmetric and a G-selfadjoint operator A respectively (cf, with the Definitions 3.1 and 3.2 respectively). Returning to the general case of a W-space we introduce
Definition 6.10: A linear relation is said to be W-dissipative if Im [y, x] > 0 for all (x, y) E T. When T = PA, this condition is equivalent to the operator A being W-dissipative (cf. Definition 2.1). We examine how the Cayley-Neyman transformation affects one or other of the properties of a linear relation T given in the formulae (6.6)-(6.9). The identity (6.25) [Y - ("x, Y - N - [Y - x, y - x] = 4 Im (' Im [ y, x] is established by a simple calculation. From it follows immediately the
proposition. 6.11 Let the linear relations S, T(E.') be connected by the mutually inverse transformations (cf. (6.15)-(6.18))
S=Kt(T) =[ 0). Hint: cf. Theorem 6.13 and Proposition 6.11. 4
Let V be a W-non-contractive operator in .", let a be its bounded spectral set, and
let P. be the corresponding Riesz projector. Then when I a I > 1 (respectively invariant relative to V is non-negative (nonI a I < 1) the sub space 2 = is positive). In particular, for a W-isometric operator the subspace 2 = P neutral (cf. Theorem 2.21). Hint: To the graph of the restriction V 1 21 (for any f 1) apply Proposition 6.11, Corollary 6.5 and formula (6.31), and then Proposition 6.17 and Theorem 2.21. 5
Let the operator V be the same as in Exercise 4, and let A with I A I > 1 (I A I < 1)
be a certain set of its eigenvalues. Then C Lin(2'a(V))XEA is a non-negative (non-positive) subspace. In the case of a W-isometric V both these subspaces are neutral. Hint: Prove this by analogy with Corollary 2.22, basing the proof on the result of Exercise 4. 6
Let V be a bounded A-non-contractive operator with Cl v = n.. Then a( V) fl (X I I X I > I I consists of not more than x (taking algebraic multiplicity into
account) normal eigenvalues (cf. Corollary 2.23) (Brodskiy [1]). Hint: Use the results of Exercise 5 on §1, Exercise 22 on §4, then Remark 6.19, Proposition 6.17 and Corollary 2.23.
§6 The Cayley-Neyman transformation
153
7
Let n be the set of eigenvalues of a J-isometric operator V and let A fl (A *)-'10. Then C Lin(f,,(V))xEA is a neutral subspace (cf. Corollary 3.14) ([XIV]). Hint: Use the results of Exercise 5, and Exercise 16 on §5.
8
Let U be a ir-unitary operator in II.. Then its non-unitary spectrum consists of not
more than 2x (taking multiplicity into account) normal eigenvalues situated symmetrically relative to the unit circle T (cf. Corollary 3.15) ([XIV]). Hint: Use the results of Exercise 5 on §1, Exercise 22 on §4, Remark 6.20, and Corollaries 6.18 and 3.15. 9
Let U be a a-unitory operator; then a,(U) = 0 (cf. Corollary 3.16). Hint: Use the results of Exercise 8 on §4.
10
If V is a ir-non-contractive operator (1) v = II. ), if k E ap( V) and I k
1, then the
root lineal 'r( V) can be represented in the form ',,(V) =.il"a [+].,ff,, where dim .4", < oo, V,AI",, C,;l"x,, ff,, C Ker( V- XI), and fix is a non-degenerate subspace (in particular, it may happen that .,ff), = (0)). If d, (k), d2(X ), ... , d,, (X) are the orders of the elementary divisors of the operator V 1 SPX, then
Z 2 XEao(V),IXI=1 J=1
[d'(k)I 2
J
+IXj>1 Z
x
(Azizov [8]; cf. [XIV]).
Hint: Use the results of Exercise 5 on §1, Remark 6.20, Corollary 6.18 and Theorem 2.26. 11
Prove that a J-symmetric operator A is J-non-negative if and only if the satisfies the V= Kt(A) J-isometric operator (; ;4 (') Re[I'(J- V)x, x] > 0 for all xE Jv (Azizov, L. I. Sukhocheva).
condition
12
Prove that if V is a J-unitary operator and Re [('(I- V)x, x) > 0 for some f E C and for all x E ., then a(V) C T (Azizov, Sukhocheva).
13
Let A and V be the same as in Proposition 6.17, and let .9 be a finite-dimensional
subspace. Then (2 C 9A, AY C ?) a (V C 9, v, VV C i) (cf. [III] ). Hint: Use Corollary 6.18. 14
15
Let V be a W-non-contractive operator, and let Vxo = exo, Vx, = ex, + xo with e I = 1. Then xo is an isotropic vector in Ker( V - eI). Hint: Apply the transformation Kt ` (I' * (') to VI te(VV) and use the results of Exercise I1 on §4. Prove that if A C B, where A, B E 2, then Kr(A) C Kr(B) and Kt ' (A) C KF ' (B) when (' ;d f.
Remarks and bibliographical indications on chapter 2 In §§1,4,5 the exposition is carried out at first in the most general form-for operators acting from one space .W1 into another 2 (T:,W, .,Y2), and only later is it made concrete for the case of operators acting in a single space. This
is the first time, apparently, that this has been done (at any rate, so systematically; cf. the monograph [V] and our survey [IV]) and mainly in the
interests of the theory of extensions of operators (see Chapter 5). In this connection it should be borne in mind when reading these notes and especially the bibliographical indications that in them, with rare exceptions, no account
154
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
is taken of generalizations (as compared with the primary sources) made in
the text to the case of operators T : Y,2. §1.1. Adjoint operators (relative to an indefinite metric) were first considered in the space IIx by Pontryagin [1], in J-spaces by I. Iokhvidov [5], [6] and by Langer [2], in G-spaces by I. Iokhvidov [12] and next by Azizov and 1. Iokhvidov [1]. §1.2. The idea of using the indefinite metric (1.4) applied to graphs of linear operators for discovering the properties of the operators themselves was first put forward and used by Phillips [1]. Later Shmul'yan [4], [5] and others developed it. Our exposition here follows Ritsner's monograph [4]. §1.3. The formula (1.10) was established by Azizov. For all the rest of the material see [III]. § 1.4. The Propositions 1.11 and 1.12 are 'folk-lore'. Sources of Theorem
1.13 can be found already in Pontryagin's article [1]. The formulation and proof given in the text are due to Azizov who used an idea of Langer's in [XVI]. Normal points are studied in [X]. Theorem 1.16 represents a certain development of Langer's results [2]. §1.5. The device presented here of passing to the factor-space ,W/,W° was first applied in [XV]; Theorem 1.17 we find essentially in Langer [9]. §2.1. W-dissipative operators were introduced in the book [VI]. ir-dissipative operators were first introduced in a particular case by Kuzhel' [5], [6]; they were studied in detail independently by Azizov [1], [4], [5], [8], and also by M. Krein and Langer [ 3 ] ; J-dissipative operators by Azizov [ 5 ], [ 8 ] E. Iokhvidov [ 1 ], Azizov and E. Iokhvidov [ 1 ] in which the main theorem
of this paragraph was established. A geoemtrical proof of the well-known Lemma 2.8 (see [VII]) is due to Azizov. §§2.2, 2.3. The Definition 2.10 and Theorem 2.11 are taken from [X]. Corollaries 2.12 and 2.13 are due to Azizov, as is all the material in §2.3 (see Azizov [8] ).
§2.4. Theorem 2.20 has been borrowed from [VI] and [XXII]. The remaining material of this paragraph was obtained by Azizov. §2.5, §2.6. Theorem 2.26 due to Azizov generalizes the corresponding result of Pontryagin for 7r-Hermitian operators (see also [XIV], [XVI]). The other results of these paragraphs were also established by Azizov.
§3.1. In an abstract formulation (but in a different terminology) rHermitian and a-self-adjoint operators were first studied by Pontryagin [1], who mentions that his attention was drawn to them by S. L. Sobolev (see the
remark on Chapter 3 below). After Pontryagin they were studied by 1. Iokhvidov [1], [2] (in the 'finite-dimensional' case cf. Potapov [1] ), and in more detail see [XIV]. J-self-adjoint operators were considered by Ginzburg [2], I. Iokhvidov [6] and later (in great detail) by Langer [1]-[3]. G-symmetric and G-self-adjoint operators are encountered partially even in Langer [2], and they are considered in detail in [III], to which we refer the reader for details. Proposition 3.7 and its Corollaries 3.8, 3.9 are found in an article by Azizov
Remarks and bibliographical indications on Chapter 2
155
and E. Iokhvidov [1]. As regards Corollary 3.12 see I. Iokhvidov [6], Langer
[2]. Remark 3.13 is due to Azizov, Corollaries 3.14 and 3.15 go back to Pontryagin [1] (cf. [XIV]). §3.2. In proving Theorem 3.19 it would have been possible to use Proposition 3.7 and to refer to a well-known `definite' result. However, we decided to bring in what we think is a simple proof, due to Azizov. Lemmas 3.20 and 3.21 used in it have such a tangled pre-history that we are inclined to attribute them to 'folk-lore'. Lemma 3.22 is due to Azizov. Corollary 3.25 is found, essentially, in Potapov [1]. Theorem 3.27 in the case of a continuous operator A is due to Ginzburg [2], and in the form in which it is formulated-to Langer [8]. §3.3. All the examples, except Example 3.33, are due to Azizov. §4.1. Plus-operators Vin a real space II, (9) v = II,, V bounded) were first
considered by M. Krein (see M. Krein and Rutman [1], and later in an arbitrary (complex) II, by Brodskiy [1]). The latter, in contrast to our Definition 4.1, imposed the condition ' v = Ilk (we point out that from this condition v n t,+ + = 0 already follows-see Lemma 1.9.5). The definition and the name 'plus-operator' of a plus-operator V in a general J-space itself were introduced in the articles of M. Krein and Shmul'yan [1], [2]. In contrast to Brodskiy, with these authors 9 v = . and the operator V is bounded a priori. For such operators they formulated Theorem 4.3 and presented for it a proof which remains valid in the more general case (see [XVI] for the previous history of this theorem), and they established a classification (which had appeared earlier in a particular case in Brodskiy's [I] ) of plus-operators into strict and non-strict (this terminology itself is due to them).
J-non-contractive (J-non-expansive) operators in a Krein space were intro-
duced and studied by Ginzburg [1], [2], generalizing the corresponding considerations in Potapov's [1] for finite-dimensional spaces. Earlier M. Krein [4] (see also [XIV]) had considered in IIk the so-called non-decreasing linear
operators V: L4 v= Ilk, V bounded and [Vx, Vx] > [x, x] for XE
.?+. We
point out that this inequality means, even in the general conditions of Definition 4.1, that V is a strict plus-operator with µ+(V) > 1 (cf. [XVI]). For plus-operators V in IIk with v = II, Brodskiy [ 1 ] discovered that they are either finite-dimensional (and then, possibly, unbounded), or they are continuous (cf. with our Corollary 4.8). As I. Iokhvidov [17] pointed out, the same Brodskiy article contains essentially all the ideas used in the proof of Theorem 4.6 and its corollaries. For these results in a rather fuller and explicit form, see I. Iokhvidov [17]. T. Ya. Azizov pointed out that the requirement that , v n ,Y+ * 0 imposed in these papers can be omitted in Theorem 4.6. Regarding Corollary 4.7 see I. Iokhvidov [17]. A curious generalization of plus-operators V was recently proposed and investigated in [XVI]: [ Vx, Vx] > µ [x, x] for some µ E fR and all x E H. For such V finite µ±(V) again exist and µ_ (V) 5 µ < µ+(V), and a number of facts were established, many of which probably remain true in the more
156
2 Fundamental Classes of Operators in Spaces with an Indefinite Metric
general situation when V:. '1 -.02 (in the spirit of our Definitions 4.1 and 4.2). §4.2. The concept of a doubly strict plus-operator and the basic facts about such operators in J-spaces were established in the articles of M. Krein and
Shmul'yan [2), [3]; there is a later bibliography of this topic in [IV]. The proof of Proposition 4.10 in the text was given by I. Iokhvidov [17] (there the priority of Yu. P. Ginzburg on this question is pointed out). J-bi-non-contractive (J-bi-non-expansive) operators were introduced and studied by Ginzburg [2]. Theorem 4.19 and its Corollary 4.20 are due to M. Krein and Shmul'yan [2], but the proof in the text is Azizov's (cf. Ritsner [4] ). It embodies most of the earlier known criteria for J-non-contractive operators V to be J-bi-non-contractive (see Ginzburg [1], [2], I. Iokhvidov [14], [17],
M. Krein and Shmul'yan [1], [2]; in connection with the rejection of the a priori requirement for V to be continuous, see I. Iokhvidov [ 17]. See Ritzer [4], E. Iokhvidov [8] for the generalization of the concept of a J-bi-non-expansive operator (cf. Shmul'yan [4] ). §4.3 Focusing plus-operators were first examined by Krasnosel'skry and A. Sobolev [1], later by A. Sobolev and Khatsevich [1], [2], and in detail by Khatskevich [6], [7], [10], [11], [15]; uniformly J-expansive operators-in the book [VI]. Theorem 4.24 was established by Azizov [10] (see Azizov and Khoroshavin [1]). Some of its assertions were obtained independently by M. Krein and Shmul'yan [5]. Theorem 4.27 is contained essentially in [VI]. §4.4. Theorem 4.30 in the 'finite-dimensional' case was established by Potapov [1], in the general case-see Ginzburg [2], M. Krein and Shmul'yan [3]; the proof in the text is due to Azizov. Theorem 4.31 has been borrowed from [VI]; Corollary 4.32 is due to A. Sobolev and Khatskevich [1], [2] §4, Exercise 12. In connection with Remark 4.29 a warning must be given
against mechanical transfer of results of the type in Exercise 12 from plus-operators to minus-operators (respectively, from (J1, J2)-non-contractive to (J,, J2)-non-expansive operators); it must be remembered that the roles of
the subspaces .; ands (respectively, of the projectors P; and P,.-), i = 1, 2, are interchanged. §4, Exercise 26. B-plus-operators in space H were introduced by Brodskiy [1] (see [XVI] ), together with the name `B-plus-operator' itself. In conclusion we point out to the reader that much more information about the operators acting in II spaces that are mentioned in §4 can be found in the monograph [XVI].
§5.1. Isometric (in particular, unitary) operators in infinite-dimensional spaces with an indefinite metric were first considered by M. Krein (see M. Krein
and Rutman [1]), I. Iokhvidov [1], [2], I. Iokhvidov and M. Krein [XIV], [XV]. Proposition 5.2, see M. Krein and Shmul'yan [2]. Example 5.5 in the text was given by Azizov; for the other examples, see I. Iokhvidov [12]. Theorem 5.6 is due to I. Iokhvidov [12]. §5.2. In connexion with Theorem 5.10 see [XIV], M. Krein and Shmul'yan [3], and also cf. Azizov [11]. Corollary 5.13 is found in M. Krein [5].
Remarks and bibliographical indications on Chapter 2
157
§5.3. Theorem 5.18 (and its corollaries) in the case of a commutative group is due to Phillips [3]. In the general case of amenable groups we are inclined to attribute it to 'folk-lore', since the proof given in the text in no way corresponds to that given in [VI], for example, for a single operator. This fact
was formally noted by Azizov and Shmul'yan; see also Exercise 7 on this section. For groups in IIj this result was proved by Shmul'yan without the requirement of amenability. Definitions 5.21 and 5.22 were given by M. Krein [XVII]. Theorem 5.23 is
due to M. Krein [XVII] (for details, see [VII), the proof given here was somewhat modified by Azizov. For similar results for stable and strongly stable J-self-adjoint operators, see Langer [11, McEnnis [1].
§5.4. All the results of this paragraph are in M. Krein and Shmul'yan [3]. §5, Exercises. The operators in Exercise 4 are called J-unitary operators by Shul'man. The condition in Exercise 9 was first considered by Masuda [1].
§6. I. Iokhvidov [1] was the first to apply Cayley-Neumann transformations to operators in spaces with an indefinite matric. Then these transformations were considered in detail in [XIV], [III], and were widely applied by many authors. In §§6.1-6.3 we mainly follow Ritsner [4].
INVARIANT SEMIDEFINITE SUBSPACES
3
In this chapter we shall set out results on one of the central problems in the
theory of operators in Krein spaces and, in particular, in Pontryagin spaces-the problem of the existence of maximal semi-definite invariant subspaces for operators and sets of operators acting in these spaces. It will be assumed that the J-non-contractive operators appearing here are bounded and defined on the whole space.
Statement of the problems
§1 1
We have already encountered the concept of an invariant subspace in
Proposition 2.1.11. We now go into it in detail. Definition 1.1: Let T.
-
be an operator densely defined in a Hilbert
Y. We shall say that the subspace 2' is invariant relative to T if Jr FY = 2' and T: 2' - Y. In particular, the subspace 2' = (0) is invariant
space
relative to any operator T; in this case we put by definition p(T j (0)) = C. We note that if Tis an operator defined everywhere in .0, then the condition is always satisfied and moreover Cr -Y) - It would be IT _ possible in Definition 1.1 to require, instead of the condition that clrfl2' be
I:
dense in 2', the inclusion 2' C 9r, but then the set of invariant subspaces would be impoverished. On the other hand, it would be possible to extend this
class by dropping the condition that C4T r) 2' = 2' and leaving only the condition that it fl 2' C .', but then this would lead to the situation, unnatural in our opinion, when any subspace which intersected )T only along the
vector 0 would be an invariant subspace for T. We therefore stay with the Definition 1.1.
In Pontryagin's foundation-laying work [1] it is proved that (in the terminology we have adopted) every 7r-selfadjoint operator A in IIX has a x-dimensional non-negative invariant subspace 2+, which can be chosen so 158
§1 Statement of the problems
159
that Im a(A 12+) >, 0 (Im a(A 12) 5 0). Further development of this result led to the following problems. Problem 1.2: Does every closed J-dissipative (and, in particular, every J-selfadjoint) operator A have an invariant subspace 2+ E _lt+? If it does, is there an .T+ such that Im a(A I Y+) 3 0 (and for a J-selfadjoint A is there also one such that Im a(A I Y+) S 0)?
Problem 1.3: Let d= (A) be the set of maximal J-dissipative operators A with p(A) n C+ ;e 0 whose resolvents commute in pairs, and let Y+ (C.+) be their common invariant subspace, and let p (A 12'+) n C+ ;'6 0 (A Ed).
have a common invariant subspace k+ E /u+ which Does the family contains Y+ and is such that p(A I - +) n C+ ;6 0 (A E d)? We notice at once that, in such a general formulation, Problem 1.2 has a negative answer even for J-selfadjoint J-positive operators (see Theorem 4.1.10 below).
Example 1.4:
Let W1 = Lin (e) (j W; be a Hilbert space, with II a II = 1,
dim .i = oo, and e 1 Wi We define relative to this decomposition a .
completely continuous selfadjoint operator G in el by means of the matrix G22=aG'22, G=IIGGj1I?i=i, where G11=0, G2'2 is a negative completely continuous operator in . I, f E Xi \RGI II f II = 1, and a > 0 is such that II G II < 1, and we introduce in Ml the G-metric [x, y] = (Gx, y). Since r1 is decomposed into the sum of the neutral one-dimensional subspace Lin (e) and the negative AeI, and since it follows from the inclusion f E.I \,G,, that 0 o ap(G), so Ye1 is a G(')-space with x = 1 (see 1.§9.6). Let P be the orthoprojector from
1 onto .01'. Then PG is
a completely continuous G-selfadjoint operator, which has, as is easily verified, not one non-negative eigenvector. In the space
,N' = 1
O+
2,
W2=
(1.1)
1,
we define a J-metric [(x1,X2),(Y1,Y2)1 _ [J(x1,X2),(y1,Y2)] by means of the operator J (see 1.(3.9)) '1=II JuII%i=1,
Jll=G, J12=(I-G2)1/2,
Let Al be the orthoprojector from .
J21=J2,J22=-G.
on to .,Y1 (C.W), and let the linear
A2 I .'2 = I G I -'. Then A 1 = A 1* > 0, operator A2 be such that Ker ..d2 = and A2 = Az > 0. Since the operator Al is bounded, it follows (see 2.Proposi-
tion 1.9 and 2.Corollary 3.8) that A = Al J+ JA2 = A`30. Since W1 C VA,
we can express A in matrix form relative to the decomposition (1.1): A = II Aii II Ii=1 It can be verified immediately that
A11=PG,
A22= -GI GI-'
A12=P(I-G2)1i2+(I-G2)"21GI-1,
A21=0,
3 Invariant Semi-definite Subspaces
160
is well-known that the selfadjointness of G implies iI G I = c, and
It
G. From the matrix representation of the operator A it follows that 0 O ap(A), i.e., A > 0. Since A 21 = 0 and G I G I -1 eRG = eG, so A 'A C 9A. From this the equality VA" = CAA (n = 1, 2, 3, ...) follows. Therefore if X'+ (E,,#' ) were an invariant subspace relative to A, it would also be invariant relative to therefore CAA = W1 Q+
A 2 = 11 (A 2)tj II a=,:
(A2),1 =(PG)2, (A2)21=0,
(A2)22= PGP(I- G2)1,12-(I- G2)1/2G-1,
(A2)22=I2.
Therefore
if x = (x,, x2) E f+ (xi E .wi, i = 1, 2), then also
A2x= 1), and a ( V I 2 )I 0 such that
[y,y] > [x,x]+6llxl12> III IIYIIZ, i.e. Y+ is a uniformly positive subspace; b) For an arbitrary x E 9?- the inequality - [x, x] > - [ Vx, Vx] + 6 II X II 2 > S 11 X II 2 holds (since Vx E : - ), and therefore Y- is a uniformly negative subspace.
Therefore Y` are Hilbert spaces relative to the scalar products ± [x, y] (x, y E Y±), and so in them V I Y+ is an expansive operator and V I 97 is contractive, which implies the inequalities I a( V I p7+) I> 1 and I a( V Y-) 11. But since (see 2.Remark 4.25) any operator X V with 1
µ+(V)
<X
1 and I a(X M Y-) I< 1 will be uniformly inequalities and I a(V I 2+) I > 'µ+ (V) the J-bi-expansive
I a(VIY-) I ,µ+(V) and Ia(VI'_)I 0. We define the operator W I ie+ by putting Wx+ = Woox+ + W1ox+ + W2ox+, where x+ E)'e+ and
woo = Wii: J°+ -.Ye+; W10 = ((1 + a)I+ + W12 W2)U2U1o:
+_
"y+,
U,o is a partially isometric operator mapping .ye+ on to Ker W11 (C.W+) and Ker U1o = W20 = W22 W*12((1 + CO I+ + W12W1*2 )-1i2U1o: ,y+ _.O-.
It is immediately verifiable that, when 0 < a < S/(l + 11 W22 112), the operator will be uniformly J-bi-expansive. ,
Let T= 11 Ti; 11 ?;-, be the matrix representation of an operator T, defined everywhere, relative to the canonical decomposition -e = W+ O+ .-e- of the 2
3 Invariant Semi-definite Subspaces
170
J-space .. We bring into consideration the functions
Gi(K+)=K+ T,1+K+T12K+- T21- T22K+,
(2.3)
Gr (K-) = K_ T22 + K_ T21K_ - T21 - T,1K_ ,
(2.4)
whose domains of definition are respectively the operator balls X1 (see 1.8.19).
Lemma 2.2: A subspace Y+ E -tf± with angular operator K± is invariant reative to an operator T with )T= W if and only if the operator K+ is a solution of the equation Gf (K±) = 0 respectively.
Let TY+ C Y+ or, what is equivalent, suppose that for every x+ E W+ + which is a solution of the equation
there is a y+ E
T(x+ + K+x+) = y+ + K+y+), which, in its turn, is equivalent to the system TI2K+x+ = y+ T21x+ + T22K+x+ = K+y+}
(2.5)
We now substitute the value of y+ from the first equation of the system (2.5) into the second and, since x+ is arbitrary in ,.+, we obtain that Gi (K+) = 0. Conversely, suppose Gi (K+) = 0. Then we take as the required y+ the vector defined by the first equation of the system (2.5).
The assertion that TY- C £- a Gr (K_) = 0 is proved similarly.
Definition 2.3: We shall say that an operator T satisfies the condition A+ (A-) and we shall write T E A+ (TEA-) if there is an operator K+ E X +, II K+ II < 1 (K_ E .yl-, II K-1 I < 1) such that Gi (K+) (respectively,
Gr (K_) is a completely continuous operator. Remark 2.4: Using 1.Theorem 8.17 we can by a simple calculation satisfy ourselves that T E A+ (respectively, T E A_) if and only if there is a canonical decomposition ,Y = W' [ + ] ,- (respectively, .0 = R' [ + ],,912- ) such that the `corner' Pi TP1 I i (respectively, Pz TPz I ' ), where P;± are the J-orthogonal projectors on to (i = 1, 2) is completely continuous. Therefore the inclusions T E A+ or T E A_ do not depend on the actual decomposi-
tions of the space as might at first sight appear from the Definition 2.3. It follows from Theorem 2.1 that every uniformly J-bi-expansive operator satisfies the conditions A+ and A_. However, even for such operators there is not always a single decomposition for which both the `corners' are completely continuous simultaneously, or, what is equivalent, there is not a Ko EX' with II Ko II < 1 such that Gi (Ko) E .y'., and GT (Ko) E / O simultaneously (see Exercise 2 below). Nevertheless the following proposition holds:
§2 Invariant subspaces of a J-non-con tractive operator
If U is
2.5
a
J-bi-non-contractive J-semi-unitary
U E A_ ).
{ U E A+)
operator,
171
then
In particular, if U is a J-unitary operator, then
{UEA+) (UEA-). Let U21 E .y'.. It follows from 2.Corollary 5.12 that U12 E .Se., and hence
(UE A-). The assertion about J-unitary operators is proved similarly.
3
Let V be a J-bi-non-contractive operator. We introduce the notation
1+(V)=
V-T+=-V+,I a(VI1+)I %:µ+(V)),
(2.6) (2.7)
IVY- =Y, Ia(VIY )I 1)C fl(Y+I9+E/+ (V)) and VEA+>Lin(X (V)IXI [Inx, InX] > [X, x] + en II X II
2.
Since Vn11 = (1 + e,,) V,1, the operator Vn will be uniformly J-bi-expansive if
and only if V is a J-bi-non-contractive operator (see 2.Remark 4.21). It follows from Theorem 2.1 that the operator Vn has an invariant subspace Y' E ..//+. Let Kn denote its angular operator. By Lemma 2.2 it satisfies the equation Gj,(K,,) = 0. Since (see 1.Proposition 8.20) the ball X+ is bicompact in the weak operator topology, we can choose from the sequence (K',} (c,, 0 as n - co) a sub-sequence (K,,) which converges in this topology to a certain operator K o E X + as n oo. Since e,, 0 when n co, KaV,,11 = (I + p,,)K,,V11 , Ko Vii, V2122K,1= (1 - e,t) V22
V,121 = (1 + ea) V21 - V21,
K,1-s V22Ko.
Since it follows from Remark 2.4 that we can without loss of generality assume that V12 E .99., we can use Lemma 2.7 and obtain KnV,112K,, = (1 - c,)K,,V12K;, - KO V12Ko.
These relations enable us to conclude that Ko satisfies the equation Gv (Ko) = 0, i.e., by Lemma 2.2 the subspace SP+ = (x+ + Kox+ I x+ E W+1 (Ea/+) is invariant relative to the operator V.
a) Let V be a J-bi-non-contractive operator. We verify that then I a(V I Y+) I > µ+(V). It suffices (taking Theorem 2.1 and 2.Remark 4.25 into account) for us to deal with the case when µ+ (V) = 1. By Proposition 2.6 we have p(V I 2P+) = p(Vo), where Vo = V11 + V1z_Ko. Since (see 2.Remark 4.21) 0 E p(V11) and V_T+ = Y+, so 0 E p(V, 1) n p(Vo). Since (see Exercise 4 on 2.§4, I = ( X II X I < 1) C p(V11) and V12 E .1 ,, it follows from 2.Theorem 2.11 that D C p(Vo). We _ shall prove that Y+ Ei+(V), i.e.,
ao = l n a(Vo) = 0. Let X E p(Vo) n OD. The operators V. =_ 17,111 + V,112K,, converge strongly, by Lemma 2.7, to the operator Vo, and by Theorem 2.1 and Proposition 2.6 we have X E p(V,,). We now verify that supn(II (V,, - XI+)-' II < oo. To do this it is, by a well-known Banach-Steinhaus theorem, sufficient to show that the set ((V,,- XI')-'x+} is bounded for every x+ E +. Let (V,, - XI+)-'x+ = xi We rewrite this equality in the equivalent form
[I+-e,1[(V0-XI +)-'V1zMI Ko)-I+1(170-XI +)
V1V12K,T)
I+)-'V12(Kn- Ko)]2}x,i _ [I+ - (V0- XI+)-'V12(K,1-Ko)](Vo- XI+)-'x+.
(2.8)
Since
II [(Vo-XI+) V12(Kn-Ko)]2II
52II
(Vo-XI+)-`V,z(K,,-Ko)(V0-XI+)
`V12I,
§2 Invariant subspaces of a J-non-contractive operator 0, and V12 E .v'm and by Lemma 2.7 (Vo - )'I+) -' V12 (K,r - KO)
K,1- K,,
173
(S) i 0,
so (see, for example, [XVIII] )
II(Vo- XI+)-'V12(Ka-Ko)(Vo-XI+)-'V2II -0. Sax+, where II T,, 11 - 0, and (S,1) Therefore (2.8) takes the form (I+ + TT)x,i = is a uniformly bounded sequence. Hence for sufficiently large values of >i we obtain that x i = I+ + T a ) ' S,,x+ and I xi) is a bounded sequence, and therefore (V# < Co. Now let Xo E co. Since ao is an isolated point of the spectrum of the operator I
Vo, there is an open disc Do (C D) whose boundary I'o consists of regular points of this operator and is such that Do fl ao = (Xo). Since Fo is compact we conclude that the set ((V,j - XI+ )-' I Xo E Fo) is uniformly bounded with respect to X and n, and therefore the sequence of Riesz projectors
Pa = - 1
2ai converges strongly to the projector
Po= -tai 1
ro
(Va
- XI+)-' dX
r, (V'o- XI+)-' dX
on the root space 2?(Vo) (see 2.Theorem 2.20). But Pa = 0 for all ri, and therefore Po = 0, which implies that 2'x0(Vo) = 10). Thus I a(Vo) I > 1, i.e., f+ (V) ;.d 0. Now let VE A+. We shall suppose that V21 E 91. (see Remark 2.4). We bring into consideration, by 2.Formula (5.3), the operator U(F) with F = V22 Vi1' (E .gym ). The operator U = U-' (F) V is, together with V, J-binon-contractive and, as is easily verified, II U22 II S 1. Therefore U22+F(I+_P*r)-1/2U12+((I-
V22 =
1,r *)_1/2-I-)U22
is the sum of a compression and a completely continuous operator. This enables us, using the same scheme as we used in proving that V E A_ - 9+ (V) # 0, i.e., again starting from Proposition 2.6 (but this time its second part), to prove that V E A+ = f - (V) * 0. We also notice the following obvious implication:
VEA± a V`EA-. Therefore, if V E A_ (respectively, A+ ), then
(2.9)
V`) ;4 0 (respectively,
1+(V) * 0). So assertion a) has been proved. b) Let 2+ E1+ (V). Then A"- = X+ I-'] E ..!!- and V`. IV- C ./I"-. Since I .,l _ -)= (V`)22 = V22 and (V`)21 = - V,Z, so by Proposition 2.26, a(V` u(V2*2 - V2Q), where Q is the angular operator of the subspace .'!"-. From V 2 E .9'm we conclude (see 2. Theorem 2.11) that [ X I I X I > 1) C )5( V` I ,I'- ). Let I Xo I > 1 and V`xo = Xoxo, 0 * xo E ./V-. Since moreover xo E :1'+ (see
2-Exercise 5 on §6), so xo E Poo and therefore xo E 2+ fl .4'-, which implies (see
3 Invariant Semi-definite Subspaces
174
2.Exercise 17 on §4) VV`xo = xo, i.e., Vxo = (l/Xo)xo and 11/X I < 1-we
have obtained a contradiction of the fact that Y+ Eq+(V). Therefore A"- E,J-(V`). Similarly it can be verified that, if /V_ E7_(V`), then 99+ = T- (-L] E '+ (V), and the second implication in b) holds. c) Let X E ap(V) with I X I ie 1 and let xE Y),(V). Then the spectrum of the restriction of the operator V on to the finite-dimensional invariant subspace
Lin ((V - XI)'x(o consists of the one point (X). By 2.Theorem 1.13 we have x [1] A' for all A' E(+ (V`) if I X I < 1 or for all .4' E j- (V`) if i X I > 1. Using the proposition b) which has already been proved we obtain that xE fl (2- 12- E f-(V)) (I X I < 1) or, respectively, that xE fl (Y+ I Y+ E f+(V)( (I X I > 1). Therefore the inclusions indicated in
assertion c) are valid for every 'x(V) and are therefore also valid for their linear envelope.
Corollary 2.9:
If V is a a-non-contractive operator or a 7r-bi-non-expansive
operator, then f±(V) ;4 0 and the inclusions in assertions b) and c) of Theorem 2.8 hold for V In particular, ir-semi-unitary and 7r-unitary operators have these properties.
This follows from the fact that V21 and V12 are finite-dimensional continuous operators and therefore V E A+ fl A_ . Corollary 2.10:
If V(EA+) is a J-semi-unitary J-bi-non-contractive operator,
then f±(V) ?6 0 and assertions b) and c) in Theorem 2.8 hold for it. In particular, J-unitary operators U E A+ fl A- have these properties. It is sufficient to compare Proposition 2.5 with Theorem 2.8.
Remark 2.11: In accordance with 2.Remark 4.29 all the statements of problems and results in §§1 and 2, as also in §§3-5 later, for J-non-contractive and J-bi-non-contractive operators can be reformulated without difficulty in terms of J-non-expansive and J-bi-non-expansive operators, and this we leave the reader to do.
Examples and problems 1
Give an example of a uniformly I-expansive operator which does not have the property & (and even less, the property 4_) (Azizov). ./P+, J4" (O( and in it Hint: Consider a J-space N' _ .,Y® ( ./P-, dim N1 = dim an operator X U where I X I > I and U is a semi-unitary operator mapping .rY into
2
Let.r = .,Y+ Q+ .YP- be a J-space, .,Y+ and ./P- being two copies of one and the same infinite-dimensional Hilbert space. Verify that the operator V = 11 V1 11 zJ= 1, where VII=AN, V12 = (1/2)I, V21=0, V22=,Y/2I, is a uniformly J-bi-expansive
operator. Prove that there is no Ko E .W + with IlKo ll < 1 such that Gi (Ko) E s/' and Gv (Ko*) E Y. simultaneously (Azizov).
§2 Invariant subspaces of a J-non-contractive operator 3
175
Let V be a J-bi-non-contractive operator with V E A_ (A+ ), let co be the spectral set of
the operator V with ao c (X II X I > 1) (respectively, ao C ED), and let PaO be the
corresponding Riesz projector. Prove that then PPO.f c fl (7+ )+ E/+( V) (respectively, P,,Yf c fl (2- I J'- E f - ( V)) (Azizov). Hint: Use 2.Theorem 1.13 and Theorem 2.8.
Let V be a r-non-contractive operator, let I Xo I > 1, and let 2'o be the isotropic
4
part of the lineal 91a0 '(V) (the possibility that 2',,O(V) = J'x 1(V) = J'o = (B)) is not
excluded). Verify that then dim 22o < dim 22x0(V) and that the subspace 2'o + J'xO( V) is non-degenerate (Azizov).
Hint: Use Theorem 2.8, 2.Theorem 1.13, and 2.Exercise 17 on §4. 5
Let V be a r-non-contractive operator, and let 22i, 222 be arbitrary invariant subspaces of it from .,//+. Prove that then, if I X I > 1, we have
dim(221,(VI2',)+2'i,-'(VIY',))=dim(2,(VI222+. x-'(V IX2))=dim 4(V), and if I X I = 1, then dim 22X( V I I',) = dim 2'a (V 12'2) (Azizov). Hint: Use Exercise 4, Theorem 2.8, 2.Theorem 1.13, and 2.Exercise 17 on §4. 6
Let V be a r-semi-unitary operator in 11 let .J'+ E /+ (V) and to (V I d'+) I > 1, and let ap (V) fl a *-, (V 197+) = 0. Then the operator V' has a single x-dimensional positive, and a single x-dimensional neutral, invariant subspace. When x = 1 the operator V` has no other invariant subspaces from //+, and when x > 1 the power
of the set of them is either not greater than 2,-2 or is not less than that of the continuum (Azizov). Hint: Use Exercise 6 and the hint for it. 7
Let II1 =11+ (D rl_ with 11+ = Linle') and let (e, 11' be an orthonormal basis in r1-. Prove that the linear operator V defined on e+ U (e, )i as Ve+ = 2e+ + ei , Ve; = e;+, (i = 1, 2, ...) is bounded and that its closure is a r-semi-unitary operator which satisfies the conditions of Exercise 6 (Azizov).
8
Let A be a bounded uniformly J-dissipative operator with VA =.1y. Prove that the that J'+ are uniformly operator A has a single pair of invariant subspaces 9 E definite and Im a(A 122+) > 0, Im a(A I Y-) < 0 ([VI] ). Hint: Use Theorems 1.13 and 2.1.
9
Let A be a maximal J-dissipative operator, let A E (L), 1/A = ., and let A12 be an A22-completely continuous operator. Prove that the operator A has at least one invariant subspace 22+ E .,//+, 9,+ C c/A and Im a(A 19'+) > 0 (cf. Langer [2], M. Krein [5], Azizov and E. Iokhvidov [1]). Hint: Use Theorems 1.13 and 2.8. Let A be a maximal r-dissipative operator (in particular, a r-self-adjoint operator) in
10
H. with c/A = M. Prove that it has invariant subspaces 1'± E .. //± such that Im a(A 12'+) > 0, Im a(A I Y'-) < 0. Moreover, Lin(95,(A) I Im X > 01 C 99+ and C Lin(A'1,(A) I Im X < 01 C -Y'-1 (Pontryagin [1], Azizov [4], [8], M. Krein and Langer [3]. Hint: Use Theorem 1.13 and Corollary 2.9. 11
Under the conditions of Exercise 10 let OA < r (regarding the symbol OA, see 2.Exercise 6 on §2) and let the right-hand (respectively left-hand) boundary ray of the corresponding angle form with the positive (respectively negative) semi-axis an angle
,p, > 0 (respectively ,p2 > 0). Prove that the operator A then has no points of the spectrum within the angles (- roe, ,p,) and (- r + p1, r - a2) and if, moreover, Ker A is definite, then the operator A has a single pair of invariant subspaces 2'± E. // that J'` are definite and J'+ = Lin (21,(A) Ker A n.#,, I Im X > 01, and r- = (Lin (2',, (A `), Ker A fl .? - I Im it > 0)) `11 (Azizov [8] ). 1
3 Invariant Semi-definite Subspaces
176
Fixed points of linear-fractional transformations and invariant subspaces §3
1
Let Y = W+ O+ .e- be a J-space and V = 11
V;; 11
;- 1
be a J-bi-non-
contractive operator. It follows from 2.Theorem 4.17 that on the operator ball + _ ,X+( +, -) (see 1.Proposition 8.19) the Krein-Shamul'yan linear-
fractional transformation Fv: K - Fv(K) _ (V21 + V22K)(V11 + V12K) ',
(3.1)
or (in equivalent form) Fv (K) V11 + F+V (K) V12K - V21 - V22K = 0
(3.2)
is properly defined. 3.1 Let K be the angular operator of the subspace Y+ E 11+. Then Fv (K) is the angular operator of the subspace V P+ (E_lf+ ), and therefore the function
Fv maps the ball X+ into If x = x+ + Kx+, then
.yf+.
Vx= V11x+ + V21x+ + V12Kx+ + V22Kx+ = (Vii + V12K)x+ + (V21 + V22K)x+ = y+ + (V21 + V22K)(V + V12K)-'y+,
where y+ denotes the vector (V11 + V12K)x+. This proposition and the writing of the function F+V in the implicit form
(3.2) enables us to introduce the concept of a generalized linear fractional
transformation Fv defined by a bounded J-non-contractive operator V (V v = A e) as a mapping of elements of the ball Jy'+ into the set of subsets of this ball:
Fv(K)= (L' E.yl+I L'V11+L'V12K- V21 - V22K=0).
(3.3)
The following proposition is proved in the same way as 3.1 was: 3.2 &+
Let V be a J-non-contractive operator, let K be the angular operator of E. if+, and let L be the angular operator of the subspace V5F+. Then
F+V (K) = X+ (L), where .Yl+ (L) is defined by the formula in 1.8.9.
Definition 3.3: A subset J ( C .YC+ is said to be invariant relative to the generalized linear fractional transformation Flt if F+V (K) fl X ;4 0 for any K E N. Moreover, the restriction F+V I W is understood to mean the mapping Fv I J': K (E.X) -+ F+V (K) fl x. In particular, if J consists of a single operator KO, then KO is called a fixed point of the transformation Fv . From Proposition 3.2 it follows immediately that
3.4 A subspace 2' E If+ with the angular operator Ko is invariant relative to a J-non-contractive operator if and only if Ko E F+V (Ko), i.e., Ko is a fixed
point of Fv .
§3 Fixed points of linear fractional transformations
177
We note that this proposition coincides with Lemma 2.2, if in the latter we put T = V, a J-non-contractive operator. Proposition 3.4 shows another way of seeking the solution of the problem of invariant subspaces. This way consists in investigating when the function Fv has a fixed point. In order to apply this idea we need some topological concepts and results; we introduce the latter without proof. 2
Definition 3.5: Let E be a Hausdorff linear topological space, and Jt' a subset of it. A mapping F which carries points K E X into now-empty convex subsets
F(K) C E is said to be closed if the fact that generalized sequences (K6) and ( Fa) (F6 E F(Kb )) converge to KO and Fo respectively implies that Fo E F(Ko).
Theorem 3.6: (Glicksberg [IX] ). Let iC be a non-empty bi-compact convex subset in a locally convex Hausdorff topological space E, and let F be a
closed mapping of points K E J into non-void convex subsets F(K) C X. Then the function F has at least one fixed point in J', i.e., there is a point Ko EX such that Ko E F(Ko).
It is easy to see that under the conditions of Theorem 3.6 the following proposition holds: 3.7 The set of fixed points of the mapping F is closed In our case the role of E will be played by the space of linear continuous operators acting from one Hilbert space into another, and the role of ..W by the ball yl + of this space or its closed convex subsets. The topology is the weak operator topology. We now pass on to the key result of this section.
Theorem 3.8: Let ,' =+ O+ Y- be a J-space and V = I I Vii I I i;=, a J-non-contractive operator with V12 .99-; let Fv be a generalized linear-
fractional transformation generated by the operator V according to the formula (3.3), and let .X ( C X + ) be a non-empty convex subset, closed in the weak operator topology, which is invariant relative to F+V. Then the mapping
Fv I JY has at least one fixed point. The set of fixed points of the function Fit X is closed in the same topology. We verify that we are under the conditions of Theorem 3.6. Let KE X. Then it follows from a comparison of Proposition 3.2 with 1.Theorem 8.23 that the non-empty set Fv (K) is bi-compact and convex, and the same is true of the non-empty set Fv (K) fl X. Thus the function Fv I .X carries points from it' into its convex non-empty subsets. It remains to verify that Fv I it is a closed mapping. Let (Kb) be a generalized sequence of elements from .yl, and let F6 E Fv (K6) fl it' and K6 - Ko, F6 - Fo. From the Definition (3.3) we then have F6 V1 i +. F6 V12K6 - V21 - V22K6 = 0.
3 Invariant Semi-definite Subspaces
178
2.7 F6 V - Fo V11, V22K6 - V22Ko, by Lemma and F6 V12Ks - Fo V12Ko, we have Fo E Fv (Ko). Since i' is closed, Fo E -W, i.e.,
Since
Fo E Fv (Ko) n W. It now follows from Theorem 3.6 that Fv I i' has at least one fixed point, and from Proposition 3.7 that the set of all such points is closed.
3
As a corollary from Theorem 3.8 we obtain
Theorem 3.9: A J-non-contractive operator V (EA_) has the property
J-bi-non-contractive operator V (EA+) has the property
+; a
_; a J-unitary
operator U (EA+ n A _) has the property 4? Ill.
Let V be a J-non-contractive operator and V E A_ . Without loss of generality, by virtue of Remark 2.4, we assume that V12 E .9'm. If a nonnegative subspace 2'o with the angular operator Ko is completely invariant relative to V, then it follows from Proposition 3.2 that the weakly closed convex set V, (KO) is invariant (see 1.Theorem 8.23) relative to the generalized
linear-fractional transformation Fv. Therefore by Theorem 3.8 the function Fv I .W+ (Ko) has at least one fixed point Ko which, by Proposition 3.2, will be the angular operator of a maximal non-negative invariant subspace !'o of the operator V, and Yo E .SPo.
Now let V be a J-bi-non-contractive operator, VE A+. By Remark 2.4 we can suppose that V21 E Y., and therefore (V`)12 = - Vzl E Y- Consequently we conclude from Theorem 3.8 that the function Fv° has a fixed point in any convex weakly closed subset from X+ which is invariant relative to Fv'. In particular, if 3 o C Velo C Mo, and Qo is the angular operator of the subspace Mo, then it follows from the invariance of 9101) relative to V` (see 2. Proposition 1.11) and from Proposition 3.2 that the weakly closed convex subset X-* (Qo) (see 1.Theorem 8.23) is invariant relative to Ft'. Let 20 = (x+ + Kox+ I x+ E+ ), where KO is a fixed point of the function Fv , I X *(Qo). Then 20 is invariant relative to V` and is J-orthogonal to 91o
(see 1.Proposition 8.22). Therefore x[01] (Elf-) is a subspace invariant relative to V and containing 91o. The last assertion about a J-unitary operator U is proved similarly. Namely, because of Remark 2.4 we can suppose that U12 E Y,. If (2'+, Y_) is a dual pair with U T+ = -T+, and if K+ are the angular operators of the subspaces 91± respectively, then, as a convex weakly closed subset X in .y1 + which is invariant relative to Fu, we consider the non-empty subset .W = J'+ (K+) n .X_* (K_ )
(see 1.Theorem 8.23). By Theorem 3.8 Fu I.X has a fixed point K+, and therefore 9+ = (x+ + K+ x+ I x+ E ,Y+) is a subspace which is invariant relative to U, contains Y+, and is J-orthogonal to Y_. Therefore (9+,. '+1]) is a maximal dual pair which is invariant relative to U and contains (Y+, Y_).
§3 Fixed points of linear fractional transformations
179
Corollary 3.10: Let V be a 7r-non-contractive or a a-bi-non-contractive operator. Then it has the property If V is a a-unitary operator, then it has the property c Ill. Since V12 and V21 are finite-dimensional operators, VE A+ fl A_, and it only remains to apply Theorem 3.9.
Corollary 3.11: Let 11, =11+ 0 II_ be a Pontryagin space and U be a 7r-unitary operator. Then U is stable if and only if all its eigen-subspaces Ker(U- XI) are non-degenerate.
Z Let U be a stable operator. In accordance with 2.Corollary 5.20 we can without loss of generality suppose that U11± = I l+, and therefore
Ker(U- XI) = (Ker(U - xI) fl II+) [ (D] (Ker(U- xI) fl Ii-). This equality implies that Ker(U - XI) are non-degenerate.
Conversely, suppose that all the Ker(U - XI) are non-degenerate. Then X E aa(U) implies I X I = 1 (see Exercise 5 on 2.§6). Therefore (see Exercise 22
on 2.§5) Ker(U- XI) [1] Ker(U -µI) when X pe µ, and so there are precisely p (0 < p < x) different eigenvalues X1, X2, .. X, of the operator U to which correspond non-negative eigenvectors. Since the Ker(U- X,I) (i = 1, 2, . . ., p) are non-degenerate, it follows that
,
Ilk = Ker(U- X,I) [+]Ker(U- X2I) [+]
[+]Ker(U- X I) [+].-I
where
.il'= [Ker(U- X1I) [+]Ker(U- X21) [+] ... [+]Ker(U- X I)J and U,4' C 'U in accordance with 2. Proposition 1.11, and by construction U I A has no non-negative eigenvectors. By Corollary 3.10 /V is a negative subspace and therefore U I N is a unitary operator relative to the scalar product - [x, y] 1,/V. Consequently u is a unitary operator relative to the scalar product (which is equivalent to the original one)
(x,Y)1 = L (xi,y;)1=1
where
X= Z; x,+x.,, y= Z; y,+ y.,, i=1
x,,y,EKer(U-X,I) for i= 1,2,...,p,
,_1
and
x,, Y., E./l. From this it follows that the operator U is stable. Theorem 3.9 enables us to make Theorem 2.8 more precise for the case of a J-unitary operator.
3 Invariant Semi-definite Subspaces
180
Theorem 3.12: Let U (EA+ U A_) be a J-unitary operator, let A be its non-unitary spectrum, A = A, U A2, A, fl A2 = 0, and let Az-' (X-' I X E A2) = Ai. Then the operator U has invariant subspaces J e± E such that the non-unitory spectra of U 12+ and U 19_ coincide with A, and A2 respectively. Moreover, A C p(U), and if X E A, (respectively, X E A2), then
the root subspace .If (UII+) (respectively, .x(UI If_)) coincides with -SPa(U). El
A C p(U) by virtue of Remark 2.4 and 2.Corollary 5.13. Let
If+=CLin(.)JU)IXEA1),
2'_=CLin(.If,,(U)IXEA2).
Since Al- ' f1Ai = 0 by hypothesis, .± C 1° by virtue of Exercise 7 on 2.§6. By
construction Y+ are completely invariant subspaces of the operator U and as(UIY+)=F1,(U)( x(UI2_)=Yx(U))when XEAl (respectively, XEA2). By Theorem 3.9 there are subspaces -ie7± E J(± which are invariant relative to
U, which contain .± respectively. Carrying out an argument similar to that used in proving Theorem _2.8 we realize that the non-unitary spectra of the operators U 12+ and U I _ consist of normal eigenvalues. By construction A, C a(U I I+) and A2 C a(U 12_ ), and the corresponding root lineals satisfy the requirements of the theorem. It remains only to notice that the skewconnectedness of .,,(U) and 2'x - ,(U) (see 2.Corollary 3.12) implies that
a(UI.+)nA2=0anda(UI2_)f1A,=0. In this paragraph we investigate the question of the number of invariant subspaces possessed by J-bi-non-contractive operators which have at least one 4
invariant maximal uniformly positive subspace (from not on the sets of uniformly definite subspaces from . C and ./l(- will be denoted by t!o and ..llo respectively). But first we prove the following Lemma 3.13: Let .e = M+ O+ operator of the subspace .ii'- E assertion holds: if the subspace
.
- be a J-space and let Q be the angular Then W = .,Y+ + /V- and the following
Y_(x++Kx+x+Ee+,K: Xe+
-W-, IIKII µ,
§4 Invariant subspaces of a family of operators
187
to prove the neutrality of 2, and 2'2 it suffices to verify the neutrality of the subspace Ea.' when 21 < 1 and the subspace (I - E,,).' when µ > 1 respectively. From the spectral theory of self-adjoint operators it follows that or (U I E),.') C [ Xmin, X1, where Xmi,, = inf ((Ux, x) 11 x 11 = 1); moreover 0 E p (U)
because U is a J-unitary positive operator, and therefore Xmin > 0. The subspace E,,. is also invariant relative to the operator U` (= U-' ), and Since X 1 is verified similarly. The orthogonality of 2, and 22 follows from the orthogonality property of the
spectral function: Ex (I - E,) = 0 when X < and µ > 1. b) The orthogonality of Ker(U - I) to 2, and .?2 follows from the orthogonality of Ker(U - I) to EX.Y( when X < 1 and to (I - E,). when µ > 1.
We now verify that Ker(U - I) [1] 2, and Ker(U - I) [l] 2'2. Because of the continuity of the J-metric it suffices to verify that Ker(U - I) [1] Ea.' when X < 1 and that Ker(U - I) [1] (I - E,).W when µ > 1 respectively, but this in turn follows directly from 2.Theorem 1.13 when we take into account that a(U` I Ker(U - fl) = (1). We consider the subspace .' = Ker(U- I) [ S] (2, Q+ 22) which is invariant relative to U; we shall prove that W' = e. For otherwise we would have . _ .e' G) W ' 1 where ' 1 (7 0) is a subspace invariant relative to U, and U' = U I ,W" 1 is a positive operator. Let Ex' be its spectral function. It is easy to see that E,,.' 1 C E for all v E R. Therefore, since Ex.Y' 1 is orthogonal to E),.t (C 9?1 C Jr") when X < 1 and (I - E, ). Y' 1 is orthogonal to (I - E,, ).w' (C 22 C . ') when µ > 1, it follows that Ex ' 1 = (01 for X < 1
and (I- E,), ,Y' = (0) for µ > 1, and so a(V') _ (1). Hence we conclude that V' = I Ye ' 1 , which contradicts the orthogonality of W ' 1 to Ker(U - 1).
Consequently Ae' =. c) Taking into account 1.Formula (7.1), this follows from b). d) Let 2 be an invariant subspace of the operator U. It is well-known (see, e.g., [XXII] and cf. 4.Remark 1.8) that then ExS C 2. Since Eat is a neutral
subspace when X < 1, the definiteness of 2 implies Eat = (0). Similarly (I - E,)9? = (01 when µ > 1. Therefore X1(2, O+ 22), i.e., 2 C Ker(U - I). Corollary 4.3: Let the operator U satisfy the conditions of Lemma 4.2, and W defined on W let emu be the algebra of all continuous operators A: which commute with U. Then -Bu has a common non-trivial neutral invariant
subspace if and only if U;4 I. Each such subspace is J-orthogonal to Ker(U -
I).
If U = I, then Wu coincides with the algebra of all continuous operators
A: . -+ ., which, as is easy to see, has non-trivial (and including also neutral) invariant subspaces. But if U ;e I, then, for example the 2, and .?2 appearing in Lemma 4.2 are
188
3 Invariant Semi-definite Subspaces
neutral and invariant relative to -j6u, since each of the operators of this algebra commutes with Ex for all X E IR (see, e.g., [XXII] and cf 4Theoreml.5). Let 2 be any non-trivial neutral subspace invariant relative to .tiBu, and let P
be the J-orthogonal projector from ' on to Ker(U - I). Then, as is easily verified. P2' is an invariant subspace of the algebra u I Ker(U - I), which coincides with the algebra of all continuous operators acting in Ker(U- I).
Therefore either P2' = (0) or P2 = Ker(U - I). Since (P` = ) P E X u, so PY C 9 and therefore P-T is a neutral subspace. It follows from assertion b) in
Lemma 4.2 that Ker(U- I) is a projectionally complete subspace, and therefore PL = (0), i.e., 2'[±] Ker(U- I). The following lemma is of a general character and seems to be well-known.
Lemma 4.4: If Y = ( V) is a group consisting of normal operators and containing the conjugate V* whenever it contains V, then VU*U = U*UV for any U, V E Yl.
By hypothesis the operator W= V*VU*UE I' and therefore it is normal. Since W is a (U*U)-selfadjoint operator and the scalar product (U*U , ) is equivalent (in the sense of equivalence of the corresponding norms) to the original one, so a(W) C FR, which, taking the normality of W into account, is equivalent to its selfadjointness: W= W*, i.e., V*VU*U= U*UV*V. As is well-known (see. e.g., [XXII]) it follows from this that (V*V)12'2U*U= Let V= be the polar representation of the normal operator V; here S is a unitary operator commuting with U*U(V*V)1,12.
S(V*V)1,12
(V*V) 1/2. Since S(U*U)25-1 V*V= (V*V) 1/25(U*U)25- 1 (V* V) 112 = (VU*U)(U*UV)
= (U*UV*)(VU*U) = (U*U)(V*V)(U*U) = (U*U)2V*V,
so S(U*U)2 = (U*U)2S. We conclude, as above, that SU*U= U*US. Consequently VU*U= U*UV.
We turn now to the formulation and proof of the main result of this paragraph. Theorem 4.5: Let ,Y be a J-space and 4/ = { U) be a group consisting of
normal J-unitary operators and containing the operator U* whenever it contains U. Then =V! has the property
W.
Let (2'+, 2'-) be an invariant dual pair of the group ail, and let (2'+, ) be its extension into a maximal invariant dual pair. We use Exercise 6 on § 1 and we shall suppose that (2P+, 2'-) is a definite dual pair. By assertion d) of Lemma 4.2 we have 2'± C n{Ker(U*U- I) UE -V!). Moreover, the maximality of J+ implies the equality U*U = I for all U E W. For, if we had Uo*Uo ;d I for some Uo E -V/, then it would follow from Lemma 4.4 and
§4 Invariant subspaces of a family of operators
189
Corollary 4.3 that the group 4! has a common non-trivial neutral subspace
J-orthogonal to Ker(Uo Uo - I) and all the more to k+-so we have a contradiction. Thus U*U = I for all U E 4l, i.e., 4! is a group consisting of operators simultaneously J-unitary and unitary. By Theorem 4.1 it has the property 4) 1-1, and therefore 9± E -11:2:.
Corollary 4.6: If -V/ = ( U) is a commutative group consisting of normal J-unitary operators, then the minimal group containing 4l and W* has the property I']. The set 4!* = (U* I U E =R!) is also a commutative group consisting of
normal J-unitary operators. Moreover, if U, V E 4l, then UV= VU, and therefore by a well-known theorem of Fuglede U*V = VU*, i.e., the elements of the groups 4! and 41 commute with one another. Consequently, the group
FV generated by the union of 4l and 4l* is commutative and consists of normal J-unitary operators. Moreover, if UE 41, then U*E Ql. It only remains to use Theorem 4.5. The operator ball .yl+, as was shown in I. Proposition 8.20, is bicompact
3
in the weak operator topology, and therefore (see 1.Proposition 8.21) the centralized system of its closed subsets has a non-empty intersection-on this is based the proof of the following key proposition. Theorem 4.7:
Let Y' = ( V) be a family of J-bi-non-contractive operators, let F; = (Fv I V E Y ) be the corresponding linear fractional transformations of the ball JY (see (3.1)), and let .Ylb be the closed set of fixed points of the transformation F. If for each finite set (Vi) C 'F we have ni .X 0 and let P be the a-orthogonal projector on to +1J . Then, as is easily verified, the operators V, = PV 12i+] form a commutative family y
, = ( V,), and moreover the V1 are 7r-non-contractive operators in
2P+11 (see
the more general Lemma 5.9 below. Since 9+ is the maximal invariant non-negative subspace of the family Y, it follows, on the one hand, that Y', has no non-negative invariant subspace 2, (B}: for otherwise Lin(9+,2,} would be an extension of 9+ into a non-negative invariant subspace of the family Y' and on the other hand, by Corollary 2.9, each of the operators V, has in Yl+'l a x,-dimensional non-negative invariant subspace. Moreover, if V, xo = Xxo, with (B ;4)xo E JO+, then I X I = 1. For, if I X I > 1, then, by Exercise 5 on 2.§6, 21,(V,) is a non-negative subspace and it is invariant relative to Y-which is impossible. But if we had I X I < 1, then (see Exercise 5 on 2.§6) xo E -, i.e., xo 410, and the isotropic part of 2a(V1) would again be invariant relative to y, which once more is impossible. Therefore, I X I = 1 and Ker( V, - XI) is non-degenerate. Consequently (see Exercise 23 on 2.§4)
the operator V, has no associated vectors corresponding to this X. Let X1, X2, ..., X, be all such points from a,(V,) to which correspond non-negative eigenvectors. It is clear that', = Lin (Ker( V, - X,I)) ° is a certain Pontryagin
space H. From Corollary 2.9 we conclude that x = xl. Moreover, ,)'1 is invariant relative to 'Vi. We consider the family 111 _ V1 01 = ( V I
, ). It
again is commutative and consists of 7r-non-contractive operators one
of which is, as is easily seen, a stable 7r-unitary operator. By carrying out the procedure indicated as many times as there are operators in Y/, we obtain a family 7i' = ( V') consisting of a-unitary operators which are the restrictions
of the original operators on to an invariant subspace IIx, with x' = x,. According to Corollary 4.9 the family '//' has a x1-dimensional non-negative subspace-we have obtained a contradiction. We now verify that the family Y has the property 4_ . Let Je_ be the maximal invariant non-positive subspace of the family, containing the original one. Again by virtue of the result of Exercise 4 on § 1 we can suppose that 1- is a negative subspace. Then 9-I'] is a Pontryagin space fl,', with x' = x and it is
invariant relative to the family Y" = (V` I V E Y } consisting of it-noncontractive operators. In accordance with what we have proved above, Y ` has a x-dimensional non-negative invariant subspace Y+ in *U1; but then ii'L+)
(E,11-) is invariant relative to Y and Y+> D 9-, and this is possible only when Yl+11 coincides with_ .
If ! = (U) is a family of pairwise commutating ir-unitary operators, then it has the property 4fll. Corollary 4.12:
Let (9+,.9'-) be the maximal invariant dual pair of the family W, containing the original one. In accordance with Exercise 6 on §1 we can
3 Invariant Semi-definite Subspaces
192
suppose that (2'+, 2'-) is a definite pair, and therefore [9+ [+19, ] [ll is a Pontryagin space rI,,, invariant relative to 1l with x' = x - dim 2'+ By Theorem 4.11 when x' > 0 the family 4! has in I1x' a x'-dimensional non-negative invariant subspace 2, and a maximal non-positive invariant subspace .'z= P;11 f H. Consequently (Lin(9+,.2,), Lin(Je1_,2'Z)) is an extension of the dual pair (2'+, 2-) preserving invariance relative to ill, and this is possible only when 91, = 2'2 = [0), i.e., k± E J1
.
Let .e be a G(")-space, let ail= (U) be a commutative family of G(')-unitary operators, and let 2'_ be its maximal invariant non-positive subspace. Then 2- E U. Corollary 4.13:
We use the results of §3.5 and canonically embed the G(')-space W in II.,
and we also extend the family 41 by continuity into a commutative family %1l = (U) of 7r-unitary operators. In accordance with Corollary 4.12 Wl has the
property c -, and therefore there is an 2 E - (ILL) which contains 2'_ and is invariant relative to TV. Then (cf. 1.Proposition 8.18) 2'fl E mil- O, 22 fl W' 3 2'-, and 2' fl W' is invariant relative to JIV. Since 2'_ is maximal it
follows that 2'_ = k fl Y. The result obtained enables us to prove a series of propositions about the existence of invariant subspaces; one of them is
Theorem 4.14: Let 4! = (U) be a commutative family of J-unitary operators, let (2+, 2-) bean invariant dual pair of 4l with def 2'+ < oo or def ,'_ < cc, and let (2'+, 2'-) be its extension into a maximal invariant dual pair. Then 2+ E It+.
Without loss of generality we shall suppose that def 2'+ < co, and so def 2_+ < co also. In accordance with Exercise 6 on § 1 we can suppose that
(2'+,2'-) is a definite dual pair. Consequently 9+11 is a G(')-space with x = def i+, the W 191+'1 are G('')-unitary operators, and 2' is their maximal
invariant non-positive subspace. We conclude from Corollary 4.13 that 2'_ E tl- (2'1+-LI), and so (cf. I.Theorem 10.2) 2'_E ll ( ). Therefore (x!11, 2'_) is an invariant dual pair containing (2'+, 9-), and since the latter is maximal we obtain 9+ = E ,tl+
Exercises and problems I
Investigate whether in Theorem 4.5 the condition (U E N!) _ (U* E -Y!) can be omitted.
2
Let Y = (V) be a commutative family consisting of a-non-contractive and ir-bi-nonexpansive operators. Then it has the property 4i (Azizov (6)).
3
Generalize Corollary 4.13 to the case where i+! _ (U) is a commutative family of G`-non-contractive operators and VU = VU = N' (Azizov).
§5 Operators of the classes H and K(H)
193
4
Prove that if -VI = ( Ul is a commutative family of J-bi-non-contractive operators, if 1'+ (respectively, 2'_) is a maximal completely invariant (respectively, maximal invariant) non-negative (respectively, non-positive) subspace of the family -f/ and def
5
Let W = Lin (e, fl, 11 e 11 = 11 f 11 = 1, (e, f) = 0. We introduce into . ' a J-metric by means of the operator J: J(ae + /3f) = /3e + af. Prove that the group generated by the operators J and U: U(ae+Of)= Xae+X-'(if (I a ;4 1) is soluble, consists of
'+ < co (respectively, def 2'_ < oo), then Y+ E fl+ (respectively, Y_ E Lf-) (Azizov).
jr-unitary operators, and has no common non-trivial invariant subspaces (cf. 2.Theorem 5.18) (Azizov). 6
Let d be a commutative algebra of operators acting in a J-space and closed relative to
the operations of conjugation and J-conjugation, i.e., A E d - A * E d, A` E d. Prove that if (Y+, °-) is any maximal dual pair invariant relative to d, then T± E ./ff± (Phillips [3] ). Hint: Use Theorems 1.13 and 4.5.
§5
Operators of the classes H and K(H)
In 1.§5.4 we introduced the concepts of the classes h± with which we shall operate in this section. 1
Definition 5.1: We shall say that a bounded operator T belongs to the class H
(TE H) if it has at least one pair of invariant subspaces 2+ E ..Zf+ and Y_ E -&- and every maximal semi-definite subspace Y± invariant relative to T belongs to MI respectively. From this definition and 2. Proposition 1.11 the implication
TEHa T`EH
(5.1)
follows immediately. We now investigate a number of other properties of operators of the class H.
Theorem 5.2: If an operator T has an invariant subspace of the class -it, fl h+ (respectively, .,L(- fl h-), then T E A+ (respectively, T E A_ ). In particular, T E H= T E A+ n A_ . D
Let K+ be the angular operator of the invariant subspace 99+ E
of the
operator T. If 2+ E h+, then K+ can be expressed in the form of a sum K+ = K1 + K2, where 11 K1 11 < 1, and Kz is a finite-dimensional partially isometric operator. By Lemma 2.2 Gi (K+) = 0, and therefore
GT(Ki )(K+-K2 )T>>+(K+-K2 )T12(K+-K2 )T21-T22(K+-K2 ) _ -K2+T11-K2 T12K+-K+T12K2+-K2+T12K2+ +T22Kz E.V and by Definition 2.3 TE A+. Similarly one proves that if the operator T has an invariant subspace _T- E , ll- fl h-, then TE A_. From Definition 5.1 and what has been proved, it follows that TE H = TEA+ n A-.
3 Invariant Semi-definite Subspaces
194
It follows from this theorem that the propositions proved earlier (see §§2-3) for operators of the class A+ hold also for operators of the class H. In particular, from Theorem 3.9 and Exercise 8 on §3 we obtain
Corollary 5.3:
If T (E H) is a J-bi-non-contractive operator, (respectively, a
J-semi-unitary J-bi-non-contractive operator or, in particular, a J-unitary operator), then T has the property c (respectively, Ill ). Corollary 5.4: Let T (EH) be a J-bi-non-contractive operator. Then each of its completely invariant non-negative (respectively, invariant non-positive) subspaces belonging to the class h+ (respectively, h-). It is sufficient to use Corollary 5.3, the Definition 5.1, and the simple fact that a subspace of a subspace of the class h± belongs to h+-.
Let T be a J-bi-non-contractive operator of the class H. Then
Corollary 5.5:
there is a constant xT < oo such that the dimension of each of the neutral invariant subspaces of the operator T does not exceed xT. Let (2') be the set of all neutral invariant subspaces of the operator T. It follows from Corollary 5.4 that dim 2 < oo for all ' E (i). If we assume that there is no constant XT < oo bounding the dimension of the subspaces Y, then with dim 9?;, oo, among them, there could be found a sequence dim 2;, , = E',E) = E) i.e., we have
proved 1)-6).
4 Spectral Topics and Some Applications
214
Now let c > 0. Since e
e
J -IIAII-0
v dE,,= JB'/2C
'
J-IIAII-0
v dFFBtn = JB"2C_,F- eBvz = JB1/2
PBiiz,
and IIAII
IIAII
v dE, _ -
f
e
v d(I - E,,) = - JB'/2Cf '
v d(I - FF)B'/z
e
e
= JB'nCC
IAII
I
f
I
IAll
P FB' 2 = JB1/2 CC 'CfI - FF)Bvz e
= JB'12(I- FF)Bin = A - JB'nFeBvz, it follows that 11
11
P dE, = A - JB'1z(Fo - F_o)B'/z -IIAII-0 We put S = JB'iz(Fo - F_o)B1/2. Since Fo - F_o is a projector on to Ker C, we have
Sz = JBvz(Fo - F-o)Bvz JB1 z(F0 - F-o)B1/2 = JBvz(F0 - F-o)C(Fo - F_o)Bin = 0. Similarly,
SE),=ExS=0 when X,)
=(I-E),)S=0 when X>0. Corollary 1.6: A bounded J-non-negative operator A has a J-spectral function E with a single critical point X = 0. Moreover,
a(E)\(0) = a+(E) U a_ (E) and a+(E) = (0, oo) fl a(E), a- (E) = (- oo, 0) fl a(E).
As E(A), when 0 = (a, l], a 0, 0 96 0, we put E(i) = Ea - Ea. The fact that this determines a J-spectral function of the operator A with a single critical point X = 0 can be seen from the properties of the function Ex investigated in Theorem 1.5. In particular, it follows from properties 2) and a+(E) = (0, oo) fl a(E), a(E)\(0) = a+ (E) U a_(E), and 3) that
a- (E) = (- oo, 0) fl a(E). Remark 1.7: It follows from condition 5) of Theorem 1.5 that if T is a bounded operator which commutes with A, then it also commutes with S.
Remark 1.8: Since (see the proof of property 5) in Theorem 1.5) the
§1 The spectral function
215
operators E, (X ;d 0) are the strong limits of certain polynomials of A, every subspace 2 which is invariant relative to A is invariant relative to Ea and S. Moreover, ' = E,,/ [ + ] (I - EE,)!, E 2' and (I - E,)2' are invariant relative to A, and a(A I E),?) C (- oo, X], and a(A I (1- Ea).T] C [X, co). If, moreover, ? is invariant relative to a bounded operator T which commutes with A, then E),2', (I - EE,)Y and S9? are also invariant relative to this operator.
We note also that by virtue of condition 3) in Theorem 1.5 the integral m p dE,. In future, where it is f II - o p dE. can be written in the form I% necessary to do so, the operators E, and S corresponding to the operator A II
will be denoted by the symbols EEA) and SA.
Corollary 1.9:
If A and D are commutating J-non-negative operators, then
E> A)Ea°) = Ea°)EX(A); SAES;°) = Ea°DSA = O when X < 0 and SA(I (I_ E)(°)SA = O when X > 0; SASD = SDSA = O; DSA = SAD = 0.
The permutability of the operators ExA), EaD), SA and SD follows from El condition 5) in Theorem 1.5 and Remark 1.7. In accordance with Exercise 2 on
2.§2, WsA is a neutral lineal. On the other hand, we conclude from the commutativity of the operators SA and EaD) when X 71 0 that it is invariant relative to EX(D). Consequently Ea°)JsA is a negative lineal when X < 0, and (I- EX(°")?sA is a positive lineal when X > 0 (see Theorem 1.5 condition 2)).
Hence it follows that E!°I is, _ (B) when X < 0 and (I_ E)(°)) 3 s, _ (0) when X > 0, i.e., Ex°DSA = 0 when X < 0 and (I - Ea°))SA = 0 when X > 0. By virtue of condition S in Theorem 1.5 the operators D and SA commute, and therefore the lineal ?sA is invariant relative to D. Since Y"sA C -,ip °, it follows from the inequality I [DSAx, Y]12 < [DSAx, SAx] [Dy, y] = 0 that DSA=0. It remains to verify the equality SASD = 0. Since
VE,(°) = s-lim (DEYD + D(I - Ef' )), we have flo
We now use the fact DSA = 0 and SD = D - f
p dEv°)SA = 0.
p
Let ./ = (A) be a commutative family of bounded J-selfadjoint operators. We shall say that it has the property 'J if every invariant dual pair of this family admits extension into a maximal dual pair invariant relative to .,d.
3
Later in Exercise 8
it
will be proved that a family of definitizable
J-selfadjoint operators has the property (i(11 if this family is either finite, or if for all the operator entering into it the `corners' P+ AP- I W- are completely continuous. But here now we shall prove the validity of the following result. Theorem 1.10: If d = (A) is a commutative family of bounded J-nonnegative operators, then d has the property 4) [l1 .
4 Spectral Topics and Some Applications
216
Let (2'+, 9-) be an invariant dual pair of the family .1. We have to prove +
that there are Y+ E Jf such that Y+ D Y+, Y+ [1] 2-, and l2' C Y+. We first verify this assertion for the case when the Y+ are definite. Let
f+(A)=C Lin((I-EaA))l a>0}, f_(A)=C Lin{EaAt.I X (A) + such that Af' C f' and Xo E p(A I e,"). Then follows from Xo E p(A I f') that Xo E apl A2), where A2 = QA I (f' [+] f (A)I`), and Q is the
.W = Y.
.(A) + (f' [+] f (A)WWW ).
Since
Xo E ap(A1)
it
projector on to 6," [+] f (A) 1' parallel to 27x0 (A ). Let A2xo = koxo. Then (A - XoI)xo E 27>(A ), i.e., xo is a root-vector of the operator A corresponding to X0 and not lying in 27ao (A)-we have obtained a contradiction Let Jl be a certain complex of properties invariant relative to a bounded projection and an equivalent renormalization of the space, and let each bounded dissipative operator having this complex of properties have a complete system of root vectors. Then every J-dissipative operator A E K(H) with a non-degenerate f (A) also has a complete system of root vectors in M. Lemma 2.5:
We suppose the opposite. Since the properties in the complex .W are invariant relative to a bounded projection, they are in particular invariant also relative to a J-orthogonal projection. It follows from Lemma 2.4 that either
Lemma 2.5 is true, or there is a bounded J-dissipative operator A E K(H) having this complex of properties and such that ap(A) = 0. We suppose the latter holds. From 3.Theorem 5.1b we have that the operator A has at least ± ± one pair of invariant subspaces 27± E . i fl h . Since ap(A) = 0, the Y± are uniformly definite subspaces, and therefore .e = 2+ + 27-. Since the operators ±A I Y± have the properties of the complex ,* and are dissipative in relation to the scalar product ± [ , ] 127± which is equivalent to the original one, it follows by the hypothesis of the theorem that
f (A I 2±) _ 2'- and therefore up(A) - 0-we have obtained a contradiction.
Lemma 2.5 enables us to transfer to the case of bounded J-dissipative operators of the class K(H) a whole series of assertions about the completeness of the system of root vectors for ordinary dissipative operators since, as a rule,
the conditions in these assertions are invariant relative to the operations of bounded projection and equivalent renormalization of the space (see, e.g., [XI] ). We shall not cite all these assertions in the main text, but we shall introduce some of them later in the form of exercises. We show by the example of Theorem 2.6 how to carry out these exercises. We also prove Theorem 2.8;
this is interesting because it cannot be generalized, not even in the case of operators of the class H (see Example 3.3 below). But first we recall some familiar terms and introduce some new ones. .02 be a completely continuous operator, and let [s, (A )J be Let A :.W'1 the set of eigenvalues of the operator (A *A ) 1/2 (taking multiplicity into account), or in other words, [ s. (A )] is the s-number of the operator A ([XI]). We say that A E .y'p if Es. (A) < oo. In particular, if p = 1, then A is called
a nuclear operator, and if p = 2, A Hilbert-Schmidt operator. We remark that even if .,Y, is non-separable, all the s-numbers except perhaps for a count-
§2 Completeness and basicity of root vectors of J-dissipative operators 223
able set are equal to zero. It will therefore be convenient to us to suppose below that sk (A) * 0 when k = 1, 2, ... , v; v < oo. Usually the s-numbers
sk(A) are numbered in the order `biggest first', and then the formula min (II AK 11
that if equivalent
1 dim K < n) holds ([XI]). Hence, it follows in particular
are the s-numbers of the operator A in another norm
there is an m > 0 such that (n = 1, 2, ...). Let lea) be an orthonormalized basis, and { fa } be a J-orthonormalized Riesz basis of a J-space W. The number sp A = Ea(Aea, ea), where A E .9'1,
to the
first
one, then
is called the trace of the operator A. As is well-known (see, e.g.,
[XI] )
sp A = paXa, where (Xa) is the set of eigenvalues of the operator A taking multiplicity into account, and therefore sp A does not depend on the choice of the orthonormalized basis and the equivalent renormalization of the space. This enables us to write the formula for the trace in the equivalent form: sp A = Z. [Afa, fa] sign [fa, fa]. All three of these formulae will be used later.
Let A E K(H) be a completely continuous J-dissipative Theorem 2.6: with a nuclear J-imaginary component A,, and let operator
n (p; AR)Ip = 0, where AR is the J-real part of the operator A, (n = 1,,2.... ) n(p;AR) is the number of numbers of the form situated in the segment [0,p]. Then e(A)=,W if and only if Yo(A) is limp
non-degenerate.
The necessity for non-degeneracy follows from Corollary 2.2. Sufficiency: From Corollary 2.2 we obtain that if !Fo(A) is non-degenerate,
then f(A) is also non-degenerate. Let P be the J-orthoprojector on to d(A)HHH, and let A, = PA I f (A)1`1. Since (see, e.g., [XI], 2.2.1), we have limp-. n(p; (A,)R)/p = 0. Therefore if f (A) * .,Y, 11
we can by virtue of Lemma 2.4 suppose without loss of generality that the original operator A is a Volterra (i.e., A E Y. and a(A) = {0)) J-dissipative operator of the class K (H) with 0 ¢ up (A) and we shall prove that the equality limp - . n (p; AR)p = 0 is impossible. Indeed, it follows from 3.Theorem 5.16 that the operator A has invariant subspaces Y ± E, it ± fl h ± . They are
non-degenerate; for otherwise it would follow from 2.Lemma 2.19 that ap(A) 3;60. Consequently the Y' are uniformly definite and therefore .N'= M'+ + Yl-. Suppose, for example, that 2+ *- (0). We consider the It is a Volterra dissipative operator relative to the operator A+ = A (definite) scalar product [ , ] +, and limp-. n(p;AR )gyp = 0. Since the scalar product [ , ] I Y'+ is equivalent on 11 '+ to the original one, it 19+.
I
follows that limo-. n(p; AR )/p = 0, where fi(p,AR) is the number of numbers of the form are the s-numbers in the interval [0, p], and the of the operator AR relative to the scalar product [ , ] 191+. Let n+ (p; AR) be the number of numbers of the form 1/X (AR) in the interval [0, p], where the
lXn (AR)) are the positive eigenvalues of the selfadjoint operator A. Since
4 Spectral Topics and Some Applications
224
0 < n+ (p; AR) < n (p; AR +), we have limp - m n+ (p; AR )Ip = 0. It follows from
[XI] 4.7.2 in this case that limp-.. n+ (p; AR )Ip = (1 fir) sp Ai . Consequently
sp A; = 0. Since Al' is a non-negative operator, sp Al = 0 implies At = 0, i.e., A = AR is a completely continuous self-adjoint operator, and therefore ap(A+) 0-we have obtained a contradiction. We arrive similary at a contradiction if Y- ;4 (B). Later we shall more than once make use of the following simple proposition.
If A is a selfadjoint operator with a spectrum having no more than a countable set of points of condensation, then fo(A) =, and Lemma 2.7:
is an orthonormalized basis composed of the eigenvectors of the in Wthere ' operator A. Since Y) ,(A) = Ker(A - XI) and Ker(A - XI) 1 Kerr(A - µl) when X ; µ, there is in fo(A) an orthonormalized basis composed of eigenvectors of the operator A. It remains to verify that fo(A)1 = 101. Suppose this is not so.
Since a(A I fo(A)1) contains no eigenvalues, and the set of points of condensation of a(A I 6,,o(A)1) can be no more than countable, there is at least
one isolated point of the spectrum of the operator A(fo(A)1. This point must, as is well-known (see, e.g., [1]), be an eigenvalue-we obtain a contradiction.
A J-dissipative operator A will be called simple if there is no subspace invariant relative to A and A` on which these operators coincide. Theorem 2.8: If an operator A with a nuclear J-imaginary component Ai is a simple 7r-dissipative operator or if a(A) has no more than a countable set
of points of condensation, then f (A) = II, if and only if the lineal Lin(3),(A)l X E s(A)) is non-degenerate and Eq Im X. = sp Ar, where X. traverses the set of all eigenvalues of the operator A taking multiplicity into account.
Since A = AR + iA1 and the non-real spectrum of the ir-selfadjoint operator AR consists of normal eigenvalues, it follows from Ar E .9', that the non-real spectrum of the operator A consists of normal eigenvalues. Therefore
all the points of condensation of a(A) lie in R. Moreover, by virtue of 2.Theorem 2.26, a(A) _ . { "a [+] lla, where A.;{ ",, C .'{ x, dim . /I'x < oo, /la is non-degenerate, and therefore and //x C Ker(A - XI) and Lin(91>,(A)I XEs(A)) is a subspace.
Suppose that f (A)=H,. We then obtain from Lemma 2.1 that Lin ('),(A )l X E s(A )) is non-degenerate. We consider the subspace 9'= C Lin (..//a I X E a(A) fl IR). This is a non-degenerate subspace: for other-
wise it would follow from 2. Lemma 2.19 that the operator A I U' has an eigenvector xo ;4 0 isotropic in U', and by virtue of the fact that 9' is 7r-orthogonal to C Lin(. a, U',,(A)j X = X,µ * µ) it would follow that xo [l] f (A)-we obtain a contradiction. We consider the decomposition
§2 Completeness and basicity of root vectors of J-dissipative operators 225
fl = Y[+] o[l]. Since AY C M and A`Y C !, it follows that A11 C 'Ill and the operator A I -1' satisfies the same conditions as A but with this difference that all the root subspaces of the operator A I YI' are finitedimensional. Since Y C "(A) we shall suppose without loss of generality that ' = (B). Since (by construction) ./Vx Rd (B) only when X E s(A ), so aa(A) fl IR
has not more than a finite number of points, and therefore the operator has not more than a countable set of eigenvalues. Consequently, just as in [XI],
1.4.1, an orthonormalized basis (ek) can be constructed in H. such that (Aek, ek) = Xk and Xk runs through the set of eigenvalues of the operator A taking multiplicity into account. Hence, [(I/2i)(A - A*)ek, ek] = Im Xk. Since
A,E.q'1, and A,-(1/2i)(A-A*)=(1/2i)(-A`+A*)=(1/2i)(- JA*J+A*) is a finite-dimensional operator, and moreover
sp(I (JA*J-A*)I sp(21 J(A*-JA*J)J) =sp(I _
(A*-JA*J))
-sp2 (JA*J-A*)l
it follows that sp
(f (JA *J - A *) I = 0, 21 (A - A *) E J,
and sp A, = sp (2i
(A -A*)
Conversely, let Lin (Yx(A )I X E s(A )) be non-degenerate and let sp A, = Z. Im Xa. Again by virtue of Lemma 2.1.E (A) is non-degenerate. Again without loss of generality we suppose that 99 = C Lin(,/tt,, I X _ ) = (B). We con-
sider the operators A' = A E(A) and A, = PA I E(A)where P is the
7r-orthoprojector on to E (A) [lI . Since the operator A' satisfies the same conditions as the operator A, we have sp A,'= Ea Im Xa. Let (fk')) and (ffz)) be wr-orthonormalized based in 6(A) and E (A )Ill respectively. Then
sp A, = Z [A,fk'), fk')]sign [fk'), fk')] + k
= Lj k
[A,f«z),
f.(2) ]
sign [f.(2), f.(2)]
a
[Alfk'),
Im Xk +
fkl)]slgn[fkl), fk')] +
a
[(A 1),f(2), f.(2)] sign
[(A1)lfaz), 1«z)]sign
[f«z), fz)l
[f(2),1(2)]
k
which, by virtue of the equality sp A, = Ek IM Xk implies the equality [(A, )rf«z), f«z)] sign [f«z), f«z)] = 0.
Suppose for definiteness that H. is a Pontryagin space with x negative squares. Then f(A) is also a Pontryagin space with x negative squares, and therefore
4 Spectral Topics and Some Applications
226
f (A )t1J is a positive subspace. Hence it follows from Z [(At )rff2) f .M] sign
0
a
that (A,), = 0, i.e., A, = Ac. Since Ai = Ac I f (A)111, we conclude from 2. Theorem 2.15 that Af(A)111 Cf(A)[1] and AI 6(A)111 =A`I 6,(A)111 = A,. If A is a simple T-dissipative operator, it follows from this that f (A)111 = (0). If, however A is not simple but a(A) has no more than a countable set of points of condensation, then the operator A,, which is self-adjoint relative to the definite scalar product [ , ] I 6(A)[11, also has this property. Therefore, by virtue of Lemma 2.7, ap(A,) 0 if r(A)111 (0); f(A)111 = (0).
but,
by Lemma 2.1, ap(A) = 0
which
implies
that
Corollary 2.9: If A E K(H) is a nuclear J-dissipative operator, then nondegeneracy of 20 (A) is equivalent to the equality "(A) =,_W.
It follows from Corollary 2.2 that if 6(A)=, , then 2o(A) is nondegenerate. Conversely, let Yo (A) be non-degenerate. Then again from Corollary 2.2 we
obtain that 6,(A) is also non-degenerate. We make use of Lemma 2.1, by virtue of which it suffices to prove that when Y ;4 (0) there are no Volterra nuclear J-dissipative operators A of the class K(H) with 0 I< ap(A ). Let us suppose the contrary. As in the proof of Theorem 2.6 we conclude from ap(A) = 0 that the operator A has a maximal uniformly definite invariant subspace 2. Suppose for definiteness that 2 E -it' (the case 2' E ,fl- is verified
similarly). We consider the operator A I Y. This is a nuclear dissipative operator relative to the form [ , ] I Y. Since sp(A I 2') = EaXa(A I 2'), we have sp(A 12)r= Ea Im X.(A I 2). By Theorem 2.8 ap(A I .') ;4 0-and we have a contradiction. 3 Before we present results about the basicity of systems of root vectors of J-selfadjoint operators, we introduce.
Definition 2.10: A basis (fa) of a J-space 0 is said to be almost J-orthonormalized if it can be presented as the union of a finite subset of vectors and a J-orthonormalized subset, these subsets being J-orthogonal to one another. Definition 2.11: A basis (fa) is called a p-basis if there is an orthonormalized basis (ea) and an operator T E .f', such that fa = (I + T)ea (a E A ). Theorem 2.12:
Let A be a continuous J-selfadjoint operator of the class
K(H), and let a(A) have no more than a countable set of points of
§2 Completeness and basicity of root vectors of J-dissipative operators 227 condensation. Then:
a) dim J '/ (A) 5 dim , lfo(A) < oo; b) fo(A) =. if and only if s(A) = 0 and Y),(A) = Ker(A - XI) when X
X;
c) f(A)= Jt if and only if Lin(Yx(A)I X E s(A)) is a non-degenerate subspace;
d) if fo(A) =,-W (respectively, 6(A) = M), then there is in .0 an almost J-orthonormalized Riesz basis composed of eigenvectors (respectively, root vectors) of the operator A; e) if --o (A) = W, then there is in' a J-orthonormalized basis composed of eigenvectors of the operator A if and only if a(A) C FR; f) the bases mentioned above can be chosen as p-bases if and only if the
operator A has an invariant subspace Y+ E J(
with an angular
operator Ky,, E 21p.
a) Since fo(A) C f(A) it follows that dim . lf(A) < dim ,lfo(A). We use 3.Corollary 5.21 and an analogue of 3.Proposition 5.14 for J-selfadjoint operators of the class K(H) and we obtain that dim f(A)/fo(A) < oo. Consequently to prove the inequality MI-6(A) < oo. dim Yelfo(A) < oo it suffices to show that dim
Let 21+ E u+ fl h+ be an invariant subspace, existing by virtue of 3.Theorem 5.16, of the operator A, let _T- = 21_ 1I, and 21o = 2+ n 21-. Since ot'l = Lin (21+, 2-) and dim , 21,' = dim 20 < oo, to prove the inequality under discussion it is sufficient to establish that 201, = .9(A 121 ), or, equivalently, that the f space ie = XolI-To coincides with f (A ), where A is the J-selfadjoint operator generated by the operator A. But the latter follows from the fact that ie = C+ [ + ] 21 21 ± I21o are uniformly definite subspaces
invariant relative to A, and the operators A/40± satisfy the conditions of Lemma 2.7. b) Let fo(A) = M. It follows immediately from 3.Formula (5.4), from the
definition of the set s(A) and the fact that 14(A) is J-orthogonal to 2',(A) when X # µ, that s(A) = 0 and 2),(A) = Ker(A - XI) when X ;d X. Conversely, let s(A) = 0 and 2),(A) = Ker(A - XI) when X ;4 X. We again
use 3.Formula (5.4) and without loss of generality we shall suppose that a(A) C R. Since s(A) = 0, all the kernels Ker(A - XI) are non-degenerate, and therefore 2a (A) = Ker(A - XI). Hence, by virtue of Lemma 2.1, ','(A) (= fo(A)) is also non-degenerate. But since dim fo(A)WWW = dim ,/fo(A) < oo and Afo(A)11J C fo(A)111, we have fo(A)[L] = (B}, i.e..W = fo(A). c) If f (A) = ', then 6,(A) is non-degenerate, and by Lemma 2.1 Lin (2a(A) I X E s(A)) is a non-degenerate subspace. Conversely, let Lin(2),(A) I X E s(A)) be a non-degenerate subspace. Again
by virtue of Lemma 2.1 6(A) is non-degenerate and ' = 6(A) [+] f (A)1, moreover Af(A)WWW C f(A)1-, AI f(A)t1 E K(H), and ap(AI f(A)WW) =0.
228
4 Spectral Topics and Some Applications
Hence it follows that f (A )f11 = L + [+] _ , where .+ are maximal (in ('(A)1'1) uniformly definite subspaces invariant relative to A. Since a(A 12"±) C a(A), the sets a(A I Y±) have no more than a countable set of points of condensation. By Lemma 2.7 aa(A I Y+ ) = 0-we obtain a contraction with the fact that ((av(A) 12+) U ap(A I Y=) = (av(A)E(A)111) = 0.
d), e). First of all we note that if an operator A E K (H) has non-real eigenvalues, then, by virtue of the neutrality of the eigenvectors corresponding
to them and 3.Formula (5.4), there is in .W no J-orthonormalized basis composed of eigenvectors of the operator A. Now let f (a) = .1 ', and , (A) _ "a [+] ll,
when X E s(A), where A4' ,, C ,",,, dim 4'X < co, ,(4 C Ker(A - XI) and ill,, is projectionally complete (see Exercise 5 on 3.§5). Since s(A) consists of a finite number of points, and by virtue of Lemma 2.1 all the 2,,(A) are non-degenerate when X E s(A ), so, taking 3.(5.4) into account, M = '1 [+] W2, where , and W2 are invariant relative to A, and Ye, = Lin (Y,, (A ), ail "a 114 ;d µ, X E s(A)J. Consequently dim Wi < oo, and a(A 1.02) C IR1 and moreover fo(A I. 2) = f(A I Y2) =.3f2. By virtue of Exercise 15 on 3.§5 the operator A I W2 E K(H). Therefore assertions d) and e) of the theorem will be proved if we show that the conditions a(A) C 91 and fo(A) =.e imply the existence in
.w' of a J-orthonormalized Riesz basis composed of eigenvectors of the operator A. We verify this.
Let (Y+, Y-) be a maximal dual pair invariant relative to A, and and 2a = Ker(A - XI) n Y±. It can be verified, just ,
let Y± E h ±
in the proof of assertion a), that fo(A I Y+) = 2'±. Since Ker(A - XI) [ 1 ] Ker(A - µl) when X # µ, and Y + E h ± , it follows that among the subspaces Y,; for different X there are only a finite number of degenerate ones. Let these be 2',;,, 2, ... , Y +K and let 9a = Yo,> + Y;,,,,, as
where'o,a, is the isotropic part of 9', , i.e., Yo,,,, = 2X fl .a;, and the subspace a, is definite and completes 2'o,x, into Ya . Since Lin (Yo,,,, } i is a finitedimensional subspace, it follows that dim .-W/f, (A) < co, where 61 (A) = C Lin (Y, , T,-, Y 1 ,,,, 21;>,, I µ ;4 Xi, i = 1, 2, ... , n ) is a non-degenerate subspace relative to A, and moreover the maximal (in f, (A)) uniformly definite subspaces 6'1± ( A ) = C Lin (2µ , Y;,,, I µ ;4 X1, i = 1, 2, ... , n) are invariant relative to A. We now note that the operators A I f i (A) satisfy the conditions of Lemma 2.7 relative to the scalar products ± [ , ] I f i (A) respectively,
and dim fl(A)I1] < oo, and moreover
fo(A I f,(A)I') and the
kernels Ker(A I 6'1(A)1'1 - XI) of the operators (A I e', (A) 1] - XI) are non-
degenerate. Consequently in each of the subspaces f; (A ), f,- (A ), and 91,, (A I f I (A ) [1) there are J-orthonormalized Riesz bases composed of eigen-
vectors of the operator A. The union of these bases will then be the required basis.
f) Before proving this assertion we point out that, if two bases differ on a
§2 Completeness and basicity of root vectors of J-dissipative operators 229 finite number of elements and if one of the bases is a p-basis, then the second one will also be a p-basis. In proving assertions d) and e) it was essentially established that if F (A) (respectively go(A )) coincides with 'W, then there is in .1Y an almost J-orthonormalized basis composed of root vectors (respectively,
eigenvectors) of the operator A, and all these vectors, with the exception,
perhaps, of a finite number, are characteristic vectors for A and lie in pre-assigned invariant subspaces Y+ E ff + n h ± of the operator A. Let this operator have the invariant subspace 2'+ E af+ fl h+ with the angular operator KY-, E .gyp, let (fc ) be the J-orthonormalized part of a Riesz basis
composed of root vectors (or eigenvectors) of the operator A, and let ( f, } C 2 + , where T_ = 2+ ('I. Since the system (f4 } U (fq } differs from a
basis in J on a finite number of elements, it can be constructed into a J-orthonormalized basis in ' which will differ from the original one on a finite number of elements. Therefore without loss of generality we shall suppose that ( and we shall prove that it is a U If. +,) is a J-orthonormalized basis in p-basis. Since (fa } U (ff } is a basis, we have 2 + = C Lin (ff }, and therefore 11 Ksr+ 11 < 1. From 2.Formula (5.3) we construct the operator U(Kv-,), which is positive, J-unitary, and such that U(Kv-+) - I E 19p. Since (f4 } U (ff } is a J-orthonormalized basis, (e.' } U ( e,-), where e« = U-1(Ky-.) fa , also is a J-orthonormalized basis. But since e, by construction lie in .jy + , it follows that (e,+) U (e.- } is an orthonormalized basis. It remains to notice that the inclusion U(Kv-,) - I E Yp implies the inclusion U-1(Kr.) - I E .9p, and therefore U { ff } is a p-basis.
Conversely, suppose there is in . an almost J-orthonormalized basis composed of root vectors (or eigenvectors) of A and that it is a p-basis. We shall prove that there is an invariant subspace Y+ E af+ fl h+ of the operator A with an angular operator K. E gyp. To do this we separate from the basis the J-orthonormalized system (ff } U if-,,) composed of the definite eigenvectors of the operator A, with (f+ } C j P± + . We form the subspaces C Lin (ff ). They are uniformly definite, invariant relative to A, and such that def Y') < co. By 3.Theorem 4.14 the dual pair (Y+", 9 1)) can be extended into a maximal dual pair (Y+, Y_) invariant relative to A. It is clear from the construction that _T+ E lf+ fl h+. We verify that E J'p. Since dim +/ 1) < co, we can suppose without loss of generality that .?'" = Y+, i.e., (f« } U is a J-othonormalized p-basis. Let f, = (I+ T)g±, where gq } is an orthonormalized basis in , T E Jp, and eq = U-' (Kv--) f, is an + . From 2.Formula orthonormalized basis in M composed of vectors ea E . (5.3) it can be seen that Ky-. E SPp if and only if U(K1-+) - T E 9'p. We
establish this lost result. Since e,± = U-'(Ky-.)(I+ T)ga , it follows that U-' (K,,.)(1 + T) = V is a unitary operator, and therefore I+ T= V(V*U(K,,-) V) is the polar decomposition of the operator I+ T. Hence, (I+ T*) (I+ T) = V*U2 (K,-_) V, and therefore U2 (K,--) - E .f p, which implies the inclusion (U(K,,.) + I)-' (U2 (K,-_) - 1) = U(K, -) - I E SP,-
0
230
4 Spectral Topics and Some Applications
Remark 2.13: Since completely continuous J-selfadjoint operators of the class K(H) and all a-selfadjoint operators with a spectrum having no more
than a countable set of points of condensation satisfy the conditions of Theorem 2.12, the assertions in it also hold for them. Moreover, for completely continuous operators it is possible, by Corollary 2.3, to write 1o(A) instead of Lin[Y),(A) I X E s(A)] in the formulation of Theorem 2.12. 4 In conclusion we investigate the question of the completeness of the system of root vectors of definitizable J-selfadjoint operators.
Lemma 2.14: Let A be a bounded J-non-negative operator having a spectrum with no more than a countable set of points of condensation. Then the equality '(A) =.Ye is equivalent to non-degeneracy of ?o(A).
Since 2'0(A) [1] 2,,(A) when µ ;e 0, it is clear that non-degeneracy of (A) (= i) implies non-degeneracy of Yo(A ). Conversely, let .?o (A) be non-degenerate. Then 6(A) is also nondegenerate. For, if ','(A) is degenerate and if xo E e (A) fl t (A)I1I, then [Axo, xo] = 0. Consequently xo E Ker A, which implies the degeneracy of
Yo(A).
Let us assume that 4"(A) ;d °, i.e.,
(0). We consider the
operator A' = A I e(A)WWW. This operator has the following properties: it is
positive relative to the G-metric [ , J I 6'(A)111; aa(A' I = 0; and a(A') consists of not more than a countable set of points. The first two of these assertions are trivial, and so we verify the third. To do this we note first that if X E p(A), then X E p(A'). From 2.Corollary 3.25 and Exercise 7 on §1 it follows that if X0(0) is an isolated point of the spectrum of the operator A, then the
range of values of the operator A - XOI is closed, Ker(A - XoI) is projectionally complete, and Ye = Ker(A - XoI) [+] Ker(A - XOI) W1l, and moreover Xo E p(A I Ker(A - XoI)W1). Hence we conclude that if Xo(#0) is an isolated
point in a(A), then XoEp(A'). By hypothesis v(A) has no more than a countable set of points of condensation and therefore a(A') consists of not more than a countable set of points. Let µo be an isolated point of the spectrum of the operator A', and let -y,, be
a circle of sufficiently small radius with centre at the point µo. Then dX is a G-orthogonal projector on to the 'G,,-space' Ae,,, = P,04" (A) [11 invariant relative to A', and a(A' I ,,0) = [µo) . The operator Aµ0 = A' W . is G,,,, positive, and µo 0 ap(A,,0). By 3.Lemma 3.16 we extend Aµ, into a f-non-negative operator Aµ, which has 14o as the only point of its spectrum. In accordance with Exercise 7 on § 1 Aµ0 = t ol,,0 when µo ;e 0 and A20 = 0 when µo = 0, i.e., µo E op(A')-but µp(A') = 0, so we have obtained a contradiction.
Let A be a bounded J-selfadjoint operator, defmitizable by the polynomial p(X) with the roots [X;) i and let o(A) have no more than a Theorem 2.15:
§2 Completeness and basicity of root vectors of J-dissipative operators 231
countable set of points of condensation. Then the equality ','(A) =,w is equivalent to non-degeneracy of Lin
Since f(A) = f (p(A)) and 9?o(p(A)) = Lin(2),,(A));, it is sufficient to use Lemma 2.14.
Remark 2.16: Since for a J-non-negative operator A the equality 9'),(A) = Ker(A - XI) holds for all X ;4 0, so, on replacing in the formulations of Lemma 2.14 and Theorem 2.15 the root subspaces by the corresponding
kernels Ker(A - XI) of the operator A, we obtain a criterion for the coincidence of fo(A) with W. In contrast to the case of operators of the class k(H) completeness of the system of root vectors or even of the eigenvectors of a J-non-negative operator
does not guarantee the existence of a basis composed of such vectors. However, the following theorem holds. Theorem 2.17:
Let a bounded J-non-negative operator A satisfy the condi-
tions of Lemma 2.14. Then Anx = EcX « [x, fa]fa sign X., where n > 2, (Xa) C ap(A), X. 3e- 0, and (fa) is a J-orthonormalized system composed of the eigenvectors of operator A: Afa = Xa fa; if moreover A is a J-positive operator, then Ax = Z.X. [x, fa] fa sign X. (here the series converge with respect to the norm of the space W). In accordance with Theorem 1.5 Ax = Sx + 17-P dE,x. From Lemma 2.7 we conclude that
E-x_-aa where (fa) is the J-orthonormalized system of eigenvectors of the operator A corresponding to the eigenvalues (Xa) : Af« = Xafa, Xa # 0. Consequently I - Ea=
17.P dE,x = EcXc [xifc] fa sign X,,, where the series converges with respect to the norm of ,' (we notice at once that if A is a J-positive operator, then S = 0 and therefore Ax = > c Xc [x, fa] f« sign X.\). We use Theorem 1.5:
Anx= An-`IS+ _
Xn. [x, fa ] f c Y
P
dE)x= An-1
sign X.
P dEx
when n = 2, 3, ....
Exercises and problems I
Let A be a completely continuous J-dissipative operator of the class K(H), let AjE %'j and limn-,, nsn (A) = 0. Prove that f(A) = .h" if and only if 'o(A) is non-degenerate (Azizov [4], [8], Azizov and Usvyatsova [2]).
4 Spectral Topics and Some Applications
232
Hint: Use Lemma 2.4 and the definite analogue of this assertion ([XI], Theorem 4.4.2). 2
when n - Co. Prove Let A E v'. fl K(H), OA = a1p, p > 1, and o(n that ('(A) = .h' if and only if'o(A) is non-degenerate [Azizov [4], [8], Azizov and Usvyatsova [2] ).
Hint: Use Exercise 6 on 2.§2, Lemma 2.4 and the definite analogue of this assertion ([XI], Theorem 5.6.1). 3
Let A E y'- fl K(H), OA = jr1p, p > 1, and for some a for the operator B = [e'°`A]r let s. (B) = o(n-'/ P) when n - oo. Prove that the equality f (A) =.1 is equivalent to the non-degeneracy of S'o(A) (Azizov [4], [8], Azizov and Usvyatsova [2]). Hint: The same as for Exercise 2 except that [XI] 5.6.1 is replaced by [XI] 5.6.2.
4
Let
5
Prove that if in a Pontryagin G`)-space Y there is at least one almost
be a Pontryagin G(')-space (i.e., 0 E p(G('))). Prove that there is in .W' at least one `almost G(''-orthonormalized p-basis' if and only if I G(') IE yP (Azizov and Kuznetsova [1]). G(')-orthonormalized p-basis, then any other almost G(')-orthonormalized basis is also a p-basis. In particular, if G(')2 = I, then any G(')-orthonormalized basis is a p-basis for any p > 0 (Azizov and Kuznetsova [1]).
6
Give an example of a J-non-negative bounded operator B ¢ 1. such that B2 E J'P (0 < p < oo). (I. Iokhvidov [8], Azizov and Shlyakman [1]). Hint: In 2 Example 3.36 put B, E .PP and take as B2 the linear homeomorphism mapping Y, on the .Y 2, B, > 0, B2 > 0-
7
Prove that if A is a bounded J-selfadjoint operator definitizable by the polynomial p(X) with the roots (k;);, then the following assertions are equivalent: a) C\(X1) 1 CP(A); b) there is an integer q > 0 such that AQp(A) E .f'm; c) Ap(A) E ,1 ,. (V. Shtraus [3]; Azizov and Shlyakman [1]).
8
Let A E .1 be a GI')-selfadjoint operator in W (0 5 x < oo), and let 0 E a,(A). Prove that then 411(A *) = .fit') and that there is in .W(') a Riesz basis relative to the norm I x II, =III GI 112X II composed of root vectors of the operator A ([31).
9
Let A E .y'm be a G-selfadjoint operator in W, let 0 E a, (A), and let at least one of the sets (- oo, 0) fl a(GA) or a(GA) fl (0, oo) consist of a finite number of normal
eigenvalues. Prove that then rr (A ") = W and that there is in # a Riesz basis relative to the norm II x II, =III GA I "2x II composed of root vectors of the operator A Q III] ).
Hint: Relative to the indefinite form [x, y], _ [Ax, y] the space .# is a G'-space with G(') = GA. Use the result of Exercise 8. 10
11
Let ,Y be a K")-space, with 0 E p (Wt' ), let J ' I Ker W(') be a Pontryagin space, and let A be a completely continuous W(')-selfadjoint operator. Prove that there is in .# a Riesz basis composed of root vectors of the operator A if and only if /'o(A) n C Lin(21a(A)I X ;d 0) = (B) (Azizov [7]).
Prove that if A E .y',0 is a r-selfadjoint operator and f (A) * IL, then it is impossible to choose in the subspace f (A) a Riesz basis composed of root vectors of the operator A (Azizov [7] ). Hint: Prove that the inequality OF (A) ;d n, implies the inequality V'o(A) n C Lin(.V',,(A) I X ;d 0) ;d [0), and use the result of Exercise 10.
§3 Examples and applications §3
233
Examples and applications
In this section we give some examples showing the impossibility of weakening the conditions in some of the theorems given in §2, and also 1
examples showing some applications of the results in the preceding section. For our first purpose we need the following. Theorem 3.1: Let A = (AR + iAj) be a continuous operator acting in a Pontryagin space II,,, and let AR and A, be a-non-negative operators. Then non-degeneracy of ?o(A) is equivalent to the inclusion Ker A n:3?A C wA.
Let Yo(A) be non-degenerate. In accordance with 2.Theorem 2.26 Yo (A) = A 'o [+] , ifo, where /{ "o is finite-dimensional and invariant relative to A, and -Ito C Ker A is non-degenerate. Consequently A"o is a non-degenerate subspace. By construction (see the proof of 2.Theorem 2.26) the kernel of the operator A, = A I .4'0 is neutral and it is the isotropic part of the kernel of the
operator A. Since A and A, are a-dissipative operators, it follows from 2.Corollary 2.17 that Ker A = Ker A`, and Ker A, = Ker A'1. This implies the equality Ker A fl 3A = Ker A fl (Ker A`) [11 = Ker A fl (Ker A) [11 = Ker A, = Ker A, n RA,.
Since RA, C RA, it follows that Ker A n 4A C A. Conversely, let Ker A fl.A C w A. If xo E Yo(A) n 2o(A )[1), then [Axo, xo] = 0. Using the fact that AR and AT are ir-non-negative we obtain that xo E Ker A. Consequently, since xo [1] £o(A ), we have xo [1] Ker A(= Ker A`), and therefore xo EJIA, i.e., xo E Ker A n 4A. By hypothesis Ker A n 4A C RA, and therefore there is a vector yo such that xo = Ayo. The vector yo E '0(A) and therefore 0 = [xo, yo] = [Ayo, yo]
Again using the fact that AR and AI are 7r-non-negative, we obtain that (Ayo = )xo = 0, i.e., 91o(A) is non-degenerate.
The theorem just proved enables us easily to construct examples demonstrating that in the theorems about completeness (see §2) the condition of non-degeneracy of Lin [Yx(A) I X E s(A )] does not follow from any of the other conditions. We give one such example, and leave the reader to construct others. Example 3.2: (cf. Corollary 2.9 and Exercise 3 on §2 when p > 2). Let B be a nuclear operator in an infinite-dimensional space W, with Ker B ;4 (0), and let
the operators i (B+ B*) and (I/2i)(B- B*) be non-negative. In Ker B and B I RB we fix on vectors xo (11 xo
= 1) and yo (11 Yo I I = 1) respectively. Since
B is a dissipative operator, xo 1 yo. This in turn implies that the operator
J: J(«xo+ayo)=axo+ayo,
JI(Lin(xo,YoI)` = -I
234
4 Spectral Topics and Some Applications
is selfadjoint and unitary. By means of this operator we introduce the form [x, y] = (Jx, y), turning the space ,Y into a Pontryagin space with one positive
square. The operator A = JB is nuclear, and AR = J[(B+ B*)/2] and Aj = J[(B - B*)/2i], and so AR and A, are ir-non-negative operators. Since Ker A = Ker B and yo 1 Ker B, so xo [ 1 ] Ker A, i.e., xo is an isotropic vector in Ker A, and therefore xo E Ker A n A. But xo 0 ,3 A, for otherwise yo would be in ?B. Consequently, by Theorem 3.1, Yo(A) is degenerate. 2 The following example shows that in Theorem 2.8 the condition that the operator A be 7r-dissipative cannot be replaced by the more general condition that A is a J-dissipative operator of the class H.
Example 3.3: A Volterra J-dissipative operator A of the class H with a nuclear J-imaginary component and sp AI = 0. Let W = W' O+ .,Y- be a J-space, where W ± L2(0, 1). We put A;i = 0 when i P6 j,
A = II AjiMI ?i= 1,
All = -A22=2i I
r
ds
As is well-known (see, e.g., [XI], 4.7.4) A11 is a Volterra disipative operator
and (112i)(A11 - Ail) is a one-dimensional operator. So A is a Volterra J-dissipative operator with a two-dimensional J-imaginary component AI and sp AI = sp(A 11)i + sp(A22 )J = 0. It remains to verify that A E H. To do this it
± are its only maximal semi-definite invariant suffices to establish that subspaces. In the present case the latter is equivalent to the fact that for any it follows from BA 11 = A22B that B=O (see 3. Lemma 2.2). Let 2iBJ0 ' f(s) ds= -2iSo (Bf)(s) ds. Differentiating both sides of this
bounded operator B
equality with respect to t we obtain Bf = - Bf, i.e., Bf = 0 and therefore B = 0.
3 We recall that a function K(s, t) of two variables defined on a square [a, b] x [a, b] is called a Hermitian non-negative kernel if for any finite set of points (t1) i of [a, b] and for any complex numbers (;) i the sum E"J_,K(t;, t;);; is non negative, and it is called a Hermitian positive kernel if Ej",j= jK(tj, t;)E; ; = 0 if and only if , = 0, i = 1, 2, ... , n (for a more general definition see 4.§3.11 below). The function K(s, t) is called a dissipative kernel if (1/2i)(K(s, t) - K(t, s)) is a Hermitian non-negative kernel, and a strictly dissipative kernel if (1/2i)(K(s, t) - K(t, s)) is a Hermitian positive kernel. Let 9 = L2(a, b) (cf. Exercise 5 on 1.§2), i.e., the space of all w-measurable
functions sp such that lab yp(t)I2 dw(t)I < oo, where w is a function either non-decreasing or non-increasing on [a, b], and the scalar product (,p, >G) is given, up to sign, by the relation (,p, >G) = Jo p(t)>G(t) dw(t). Moreover, we I
§3 Examples and applications regard p =
235
if
Jb I P(t)-0(t)12 dw(t) = 0 a
Let a(t) be a function of bounded variation on [a, b]. In Exercises 5 and 6 on 1.§2 it was proved that the space L2, (a, b), where w(t) = WI (t) + W2 (t), and
a(t) = wi (t) - u)2 (t) is the canonical representation of a(t) in the form of the difference of two non-decreasing functions, and the space is provided with the do(t) is a J-space. In particular, if wi(t) is a piecewiseconstant function with x points of growth, then b) is a Pontryagin space with x positive squares (see Exercise 7 on 3.§9). In this and the following paragraphs we apply the results obtained in §2 to the investigation of integral operators A = Jb, K(s, t) do(t). form [gyp, '] = JQ
3.4: Let K(s, t) be a dissipative kernel continuous on [a, b] x [a, b], let A and A, be integral operators defined by the relations
Theorem
A = L K(s, t) da(t),
A, =
K(s, t) dt, J aa
a
let o(t) = WI M - w2(t), and let w, (t) be a piecewise-constant function with x
points of growth. Then if the system of root vectors of the operator A, corresponding to its non-zero eigenvalues is complete in C[a, b], then the system of root vectors of the operator A corresponding to its non-zero eigenvalues is complete in b). If in addition a(t) has no intervals of constancy, then the system mentioned of root vectors of the operator A is complete in C [a, b].
It follows from the definition of the operator A that it is a nuclear 7r-dissipative operator. Therefore by virtue of Corollary 2.9 the root vectors of the operator A corresponding to its non-zero eigenvalues will be complete in L2,(a, b) if and only if 0 ¢ ap(A). We prove that 0 0 ap(A). Let cpo E Ker A. Then po E Ker A`, i.e., JaK(s, t)(po(s) do(s) = 0. Let 1(t), ("2(t), ... be the root vectors of the operator A, corresponding to the non-zero eigenvalues and forming a system complete in C[a, b]. It follows from the definition of root
vectors that ;(t) E MA,, i.e., there are functions ti(t) E C[a, b] such that (t) = (AIE;)(t) (i = 1, 2, ...). Then b
[0o, ]'i] =
b
po(s)(AjEj)(s) du(s)
soo(s)(";(s) du(s) = J a
J a
b
a
b
'PO (s) J
K(s, t)E;(t) dt da(s) a
b
b
;(t) a
K(s, t),po(s) du(s) dt = 0. a
Since C[a, b] is denose in
b), we have po = B, i.e., 0 ¢ ap(A).
4 Spectral Topics and Some Applications
236
Now let a(t) have no intervals of constancy, and let -q,,'92, ... be a system, complete in y (a, b), of root vectors of the operator A. Since q j E A, the ?7i are continuous functions, and in b) there are functions > i such that n; = Ai,&; (i = 1, 2, ...). Let 4' be a continuous linear functional on C[a, b] such
that 4'(,q;) = 0 (i = 1, 2, ...). By virtue of Riez's theorem on the integral representation of a linear functional on C[a, b] there is a complex function of bounded variation CD such that (AOi)(s) dw(s)
4)(AO;) = J b a
= r b rb
K(s, t)O;(t) da(t) d(5(s) a
a
= J ba Oi(t)
a a
(i = 1, 2, ...)
K(s, t) &Z(s) du(t)
b), and the function w = wl + wz like a has no intervals JoK(s, t) d(,)(s) = 0 for all t E [a, b]. But then Ja ;(t) Ja K(s, t) d(:w(s) dt = 0 (i = 1, 2, ...). Since [ j';} is complete
Since if (A) = of constancy,
in C[a, b] we conclude that 4i = 0. This is equivalent to the completeness of (,t;} in C[a, b]. 4
Now let K(s, t) be a Hermitian positive kernel, and let a(t) be an
arbitrary function of bounded variation on [a, b]. We bring into consideration the iterated kernels K(")(s, t) = K(s, t), K(n) (s, t) = fab K(n-,) (s, l)K(l, t) da(l) (n = 2, 3, ... J
Moreover, we assume that the kernel K(s, t) generates a bounded operator A = JQK(s, t) da(t) having no more than a countable set of points of condensation of the spectrum. For example, if I K(s, t) 15 c < oo and if the function K(s, t) is continuous in each variable when the other is fixed, then (see, e.g., I. Iokhvidov and Ektov [1], [2]) A is a completely continuous operator and therefore has a single point of condensation of the spectrum. Let 71, denote the eigenvectors of the operator A corresponding to X. Re 0. In accordance with Theorem 2.17 for every function xE L2. (a, b) we have
A"x=
Xn [x,,7.],a(t)sign X. LT
b
_ E va' ?a (t) a
X(T)17,(T) da(T)sign X.. a
The function K(")(s, t) belongs to
b) with respect to each of the
variables. Consequently b
X_
AK(s, t) a
a
X.,)a(t),ta(s)sign X.. a
§3 Examples and applications Similarly AKA") (s, t)
X.
237
(n = 2, 3, ...
a
Here the series converge in the norm of L2,(a, b). We note also that, by definition, K"") = AK("-'). So we have proved
Theorem 3.5: If a Hermitian positive kernel K(s, t) and its iterations K(") (s, t) belong to b), and if the operator A = J K(s, t) da(t) is continuous with not more than a countable set of points of condensation of the spectrum, then when n > 2 Kt"t (s, t) _
X?7a(t),la(s)sign Xa,
(3.1)
a
where (na) is a J-orthonormalized system of eigenvectors of the operator A corresponding to Xa # 0, and the series (3.1) converges in the norm of the
space L2a, b). Theorem 3.5 has been obtained as a simple consequence of Theorem 2.17. By applying additional and different methods, going beyond the scope of our
book, for an integral operator and iterations of the kernels generated by it more precise results can be obtained (see, for example, Exercises 3-5 below).
5
In S. Krein's article [1] it is shown that the problem of the oscillations of a
heavy viscous fluid in an open fixed container reduces to the study of an operator-valued function
L(X)=XG+C-'H-I,
(3.2)
where G and H are completely continuous selfadjoint operators of finite order, i.e. G E 91P, HE 9q, p, q < oo, and G > 0, and H > 0. A whole series of general non-selfadjoint boundary problems with a parameter X in the equation and in the boundary conditions reduce to the spectral analysis of a similar operator-valued function. Here we show one of the ways of analyzing an equation (3.2) based on applying Theorem 2.13. For other results in this direction see Exercises 6 and 7 below.
Definition 3.6: In equation (3.2) let G = G ` and H = H* be bounded operates acting in a Hilbert space .'. A point Xo E C is said to be a regular
point for the function L(X) if 0Ep(L(Xo)); otherwise it is a point of the spectrum of the function L(a). A vector xo is called on eigenvector of the function L(X) if there is a Xo E C (Xo is an eigenvalue) such that L(Xo)xo = 0. The vectors x,, x2.... , x," are said to be associated with the eigenvectorxo and
the set (x;)o' is called a Jordan chain if CYJL(X0)
Z j!ax j xk-l = B ,=o
(k = 0, 1, 2, ... , m).
4 Spectral Topics and Some Applications
238
Following M. Krein and Langer [2] we introduce a scheme of argumentation which reduces the spectral analysis of the function L(X) to the spectral
analysis of a certain J-selfadjoint operator. We shall suppose that G > 0. After replacement of the variables X _ - µ ' - a the function L (X) becomes the function
L,(µ)
µ(1 +µa)
[µ2(a2G+ H+ aI)+µ(2aG+ I) + G].
We put
a>inf(b>01 Fb= b2G+H+b1>0). Then
Li(µ)= - µ(1 +µa) Fa (µ2l+µBa+Ca)FQ/2, 1
where
Ba=Fa "2(2aG+I)FQ v2)> 0,
=Fa'2GFa''2
Ca
It can be verified immediately that Xo is an eigenvalue of the function L(X) if and only if µo = - (Xo + a) - ' is an eigenvalue of the function L2(µ) =,U21+ µBa + Ca, called a quadratic bundle. Moreover, (xo, xi, ... , is a Jordan chain of the function L(X) if and only if is a Jordan chain of the bundle L2(µ). (FV2xo, FcV2x,, ... ,
_ .i+ O+ e-,
We bring into consideration the J-space the J-selfadjoint operator 0 Caln
Aa =
Ve
_' and
CQi2l (3.3)
-Ba
acting in it. It is easy to see that the regular points, the spectrum, and in
particular the eigenvalues of the bundle L2(µ) and of the operator Aa coincide. Moreover, if (yo, yl, . . . , y.,.) is the Jordan chain of the bundle L2(µ) corresponding to the eigenvalue µo, then the vectors Co /2 Yo
C.1/2 Y1
ILOyo
µ0y I + Yo
Cav2
µoyn, + yin- I
form a Jordan chain of the operator Aa, and conversely if zi') zf2)
is a Jordan chain of the operator A., then yo=
I AO
z62),
(z(?)yi= 1 (zf2)- yo),..., y,,,= 1 µo µo
is a Jordan chain of the bundle L2(µ).
y,,,-I)
§3 Examples and applications
239
Thus, a one-to-one correspondence has been established between the Jordan
chains of the function L(X) and the operator A. Here we denote by symbols .4->, and -lix
KerL(X) fl ((2XG- I)Ker L(X))1
KerL(X)fl (Lin((2XG-I)xk+Gxk_,)o")1 respectively, where (xo, x,, ... , x,,,) are all possible Jordan chains of the and
bundle L (X) corresponding to the eigenvalue X, and x_ 1 = 0.
Let the function (3.2) be given, where G > 0, G E 99.,, Theorem 3.7: H = H* is a bounded operator, and the set a(H) is no more than countable. Then:
1) each of the operators (3.3) generated by the function L (X) belongs to the class H;
2) dim .'/f(Ao) < dim Y16-o(A,,) < oo; 3) fo(AQ) =.W'
if and
only
(211G11)-'IxIs211H11;
4) f (A0) = 1Y if and (211 GII)-'
only I XI s211H11;
if
.
"\ = (0)
for all
X
such
that
if
/1"a = (0)
for all
X
such
that
5) If fo(A.) = W (respectively, f (Aa) = M), then there is in W an almost J-orthonormalized Riesz basis composed of eigenvectors (respectively, root vectors) of the operator Aa. If fo(A,,) = X, then there is in W a J-orthonormalized Riesz basis composed of eigenvectors of the operator
A. if and only if the function L(X) has no non-real eigenvalues. Moreover, the bases mentioned above can be chosen as p-bases if and only if G E 9'p12-
1) Since it follows from G E Y. that Co/2 E 9'- so (cf. 3.Theorem 1.13, El 2.Remark 6.14, and 3.Theorem 2.8) the operator A has at least one maximal
non-negative invariant subspace 9+ with an angular operator K. In accordance with 3.Lemma 2.2 Kr--CQ"2K, + C"2 + B,,K'-- = 0, and since B,, )> 0, so Kv~ = - BQ'(CQ"2 + K,-CQ,2K,--) E q.. Consequently (cf. Exercise 3 on 3.§5), A E H, and moreover, Kv,. E .%'p if and only if CQ12 E 9p, which
in turn is equivalent to G belonging to the class .9'p,2 (see [XI], 3.7.3). Assertions 2) to 5) will now follow from Theorem 2.13 if we show that
the set a(Ao) is no more than countable, that the non-real eigenvalues of
the
operator
Aa
and
the
set
(t=-(X+a)-' 1(211G1-'slxl, ;e (0). Then, in particular, some xo E Ker ((2Xo G - I)xo, xo) = 0 for L (Xo), and therefore Xo = (xo, xo)/2(Gxo, xo), which in turn implies the equality (xo, xo)z =
4(Gxo, xo)(Hxo, xo), and we have a contradiction with the fact that the function (3.2) is strongly damped.
Exercises and problems 1
2
Investigate the possibility of generalizing Theorem 3.1 to the case of a Krein space.
let K(s, t) be a Hermitian positive kernel continuous on the square 0 5 s < t < b, let R(s, t) be a function bounded on this same square and continuous with respect to each variable when the other is fixed, let R(s, t) = R(s, t), and let a(t) generate a
Pontryagin space according to the form [,o, ,&J = la do(t), and let A = Jo (R(s, t) + iK(s, t)) do(t). Prove that the equalities F (A) = L2 (a, b) and JQ K(t, t) do(t) = E; Im k;, where (a;) is the set of eigenvalues of the operator A, are equivalent. (Azizov [8] ). 3
Consider on [a, b] x [a, b] a Hermitian bounded kernel K(s, t) continuous with respect to each variable when the other is fixed, and a function of bounded variation
a(t) 0 const. We shall say that the kernel K(s, 1) is a-non-negative if K(s, t)g(s)g(t) da(s) da(t) >, 0 for all g E a
Y,
a
where )/" is the lineal of all those functions from L2 (a, b) (o = Var a) which are generated by continuous functions from C[a, b]. Prove that a-non-negativity is equivalent to either of the conditions: a) J b K(s, t) dr(s) dr(t) > 0 for any (complex) function of bounded variation r on [a, b] with E, C Ea (E, and E. are the sets of points of variation of the functions r and a respectively); b) Y- k=, K(t;, tu)Eksk ,>O for all n E N, ti, t2, .. ,,, E Ea, and
(I. Iokhvidov and Ektov [1], [2]). 4
Let K(s, t) be a a-non-negative kernel satisfying the conditions of Exercise 3. Prove that the assertions `K(2 (s, t) = 0 on E. > Ea' and 'K(2 (s, t) = 0 on [a, b] x [a, b]' are equivalent (I. Iokhvidov and Ektov [1], [2]).
5
Prove that the integral equation p(s) = XJo K(s, t),p(t) da(t) with a a function of bounded variation, and a a-positive kernel, has x < oo positive characteristic points if and only if the positive variation of the function a has exactly x points of growth (M. Krein [1], I. Iokvidov and Ektov [1]).
6
Let S=S*E.y'., T=T`E.y'm, -1Ep(s),0>iao(T)andA=(I+S)T.Prove that 6 (A) = .0 and that in i' there is a Riesz basis composed of root vectors of the operator A ([XI] ); if S E .vo, then there is in .W' a p-basis composed of root vectors
of the operator A (Kopachevskiy [1]-[3]). Hint: The operator A is 7r-selfadjoint relative to the form [ It only remains to use Remark 2.13 and Exercise 4 on §2. 7
,
] _ ((I+ S)
'
,
).
Prove that if L (k) is the function (3.2), G = G * E .v'm, H = H* E .'/'. and Ker G =
Ker H= (0), then:
Remarks and bibliographical indications on Chapter 4
243
1) the system Xk(k)
{
(-ly k
j=0
j+1 xk-j(t)
1 of `special vectors', where (xo(X), ..., xk(k) ( is the Jordan chain of the function L(X) corresponding to the eigenvalues k, is complete in .JY (Askerov, S. Krein and Laptev [ 1 ] );
2) in ,Y there is a Riesz basis composed of vectors of the special form (Larionov [6], [7]); 3) If G E yo, HE 'Q, and r = max(p, q), then there is in -Y an r-basis composed vectors of the special form (Kopachevskiy [1]-[3]).
Remarks and bibliographical indications on chapter IV §1.1. The J-spectral function was introduced by M. Krein and Langer [1] for the case of IL, and later Langer [4], [5] transferred the investigation to the case of Krein spaces. The definitions in the text, given by Azizov, modify Krein and Langer's definitions in order to accommodate them to operators of the K(H) class (see Exercises 1 and 2). §1.2-1.3. Definitizable operators in H. were introduced by I. lokhvidov and
M. Krein [XV], and in a J-space by Langer [4], [5], [9]. All the results of these sections are due to him. We borrowed from Bognar [5] the elegant proof of Theorem 1.5. In Bognar [5] these are also given bibliographical references
to other proofs and, in particular, references to M. Krein and Shmul'yan's proof [3] based on considerations from the problem of moments. In connection with criteria for the regularity of the J-spectral function see Langer [4], [5), Jonas [1]-[7], Jonas and Langer [1], [2], Akopyan [1]-[3], and Spitovskiy [ 1].
§ 1.4. Definition 1.11 and Theorem 1.12 for the case of operators acting from one space into another are modifications of corresponding results of Potapov [ 1], Ginzburg [2], M. Krein and Shmul'yan [3]. On square roots of J-selfadjoint operators see Bognar [ 1] and, more fully, Bayasgalan [ 1]. §2.1. The results in this paragraph are due to Azizov. Corollary 2.3 was published in the paper by Azizov and Usvyatsova [2]. §2.2. Lemmas 2.4 and 2.5, Theorem 2.6 and Corollary 2.9 in the case A E Y. were published in the paper by Azizov and Usvyatsova [2]. The presentation in the text is due to Azizov. To him also is due Theorem 2.8, which is a transfer to IIX of a corresponding result from [XI]. Lemma 2.7 is, apparently, well-known, though we have not found its formulation in print.
§2.3. Theorem 2.12 is due to Azizov [13]. For the formulation of the problems about p-basicity he is obliged to Kopachevskiy, who first began the
study of this question for the case of indefinite spaces in connection with problems of hydrodynamics (see, e.g., Kopachevskry [ 1]- [3]). The concept of p-basicity itself was introduced by Prigorskiy [ 1]. We mention also that the
244
4 Spectral Topics and Some Applications
questions of completeness of a system of root vectors of 7r-selfadjoint operators A E .`/'- with Of aa(A) were investigated for the first time by 1. lokhvidov [2]; the existence of a Riesz basis of root vectors of such operators was proved essentially in [XI]; the criterion for completeness and basicity of these vectors without the condition 0 0 ap(A) was given by Azizov and I. Iokhvidov [1]. On other conditions for completeness and basicity and for a historical survey see [IV] and also the Exercises. §2.4. The results of this paragraph in so general a formulation are due to Azizov. In the case when the set a(A) has no more than a finite number of points of condensation, Theorem 2.15 and Remark 2.16 (even in the case of Banach spaces with an indefinite metric) were obtained earlier by Azizov and Shtraus [ 1 ] , and Theorem 2.17 when A 2 E 9 " . is due to Kuhne [ 1 ] (see 1. lokhvidov [ 8], Ektov [ 1 ] ). As above, for a historical survey and for other results we refer the reader to the survey [IV] and to the Exercises on this section.
§3.1-3.3. All the results in these paragraphs are due to Azizov. Some of them were published in Azizov's paper [8]. Theorem 3.4 for the case of a Hermitian kernel was obtained earlier by I. Iokhvidov [2]. §3.4. Theorem 3.5 in the text was proved by Azizov; for the case A E 3'm see M. Krein [1], I. Iokhvidov and Ektov [1]. §3.5. Theorem 3.7 and Corollary 3.10 were proved by Azizov [13] (cf.
Askerov, S. Krein and Laptev [1], M. Krein and Langer [2], Larionov [6], [7], Kopachevskry [1]-[3], Azizov and Usvyatsova [2]). As regards other investigations of operator bundles and application, of an indefinite metric see Langer [4], [6], [10], [12], Kostyuchenko and Orazov [ 1 ] , and others.
5 THEORY OF EXTENSIONS OF ISOMETRIC AND SYMMETRIC OPERATORS IN SPACES WITH AN INDEFINITE METRIC
In § 1 the apparatus of Potapov-Ginzburg transformations is developed, and its application to the theory of extensions is demonstrated. §2 is devoted to another approach to the theory of extensions of isometric operators in Krein spaces. In §3 generalized resolvents of J-symmetric operators are described.
§1
Potapov-Ginzburg linear-fractional transformations and extensions of operators
One of the methods allowing extensions of (Jr, J2)-isometric operators to be constructed is the application of Potapov-Ginzburg transformations or, 1
briefly, PG-transformations. Definition 1.1: Let .,Y, =X1' Q+ .YP,- and .YP2 = .02' Q+
. z be the canonical decompositions of a J,-space ., and a J2-space .02 respectively, and let 1(3 = .Wi O+ )Yz and ,Y4 = ,Yz OO i"i be a J3-space and a J4-space constructed from them with J3 = It Q+ - Iz and J4 = Iz (j - IF. A transformation w+ :.r°1 O+ .#2 -- H3 .W'4 carrying a vector (x,, x2) into a vector
245
5 Theory of Extensions of Isometric and Symmetric Operators
246
(X3, X4), where
XI=Xj +Xl
X2=X2 + X 2
,
X3=X1 +X2
,
,
X4=X2 +X1
,
,+ (i = 1, 2), is called a PG-transformation. E As well as the transformation w+ having the form
x,
w+:
1 QIY2--,IY3 Q+'4,
W
other PG-transformations can be considered: U)
:.Y1 G. 02-x,03 Q W4,
W±:
w (XI,X2)= dim 9l `l] , then we put Vo = (V-' )`, where V-' is an arbitrary extension of the operator V subspace in ILl
on to W2 which maps 9 I'l injectively on to a
Thus, for the description of all (JI, J2)-bi-extensions of an operator VE St(.1,.2) the results of Exercises 13-17 on §1 can be used.
2 We pass on now to study the problem of the possibility of (JI, J2) unitary extensions of operators VE St(. 1, JY2). Example 2.8 given below of an operator V E St(.W1, ."2) with 9v = Rv (and therefore 9 fr1 = 3 *) shows that, in contrast to the Hilbert space case (see, e.g., [I] ), in the case of a Pontryagin space (see Exercise 3 below) information only about the properties of 9 Vll and f'l is not always sufficient to make conclusions about the possibility of extending an operator V into a (J1, J2)-unitary operator. But first we introduce some preliminary materials. Definition 2.5: Let (91+, 2-) be a maximal uniformly definite dual pair in a J-space . , and suppose the subspaces A"+ E -ff+ and ./I-- =.4'[+-,I (E 11-)
contain no infinite-dimensional uniformly definite subspaces. Then the number v (.,l "+) = dim(./l "+ n Y+) - dim(,il n Y-) is called the index of the subspace . 4'+.
Lemma 2.6: In Definition 2.5 v (. 'l '+) does not depend on the choice of the (Y+, Y-). maximal uniformly definite dual pair Let
P1( l"+)=dim(.,l"+ fl.,Y+)-dim(./l"_ n.w- )
§2 Extensions of standard isometric and symmetric operators
257
and
v2(. '+) =
dim(.'{"+ n q+)
- dim(.'1i. fl g'-).
We verify that v, (,/1'+ ) = P2(,4'+ ).
Let K+ be the angular operators of the subspaces A-+ and Q± be the angular operators of the subspaces 9?± . From Exercise 17 on 1.§8 and Exercise 13 on 2.§2 it follows that K+ is a 4)-operator and I K+ I = I+ + S, where S E Y.. From Exercise 14 on 2. §2 it therefore follows that K+ - Q+ is also a 4)-operator and ind K+ = ind(K+ - Q+). It remains to observe that ,4'+ n ± = Ker K+, A"± fl 2± = Ker(K+ -Q±), and K+=K*, K+-Q+=K*-Q*, i.e., -ind K+ and v2 (-4"+) = - ind(K+ - Q+ ).
Lemma 2.7: If VE St(.Y(1,.Y2), d'vE M+(JY1), .9lvE,11+(."2), and 1'v, Rv, gW and 91VI contain no infinite-dimensional uniformly definite subspaces, then V admits a (J1, J2)-unitary extension if and only if v(Vv) = v (91 v ). Moreover every maximal extension V' E St(.Y1, '2) of the operator V will be (J1, J2)-unitary. Let 17 be a (J1, J2)-unitary extension of the operator V. We introduce in
.Y'2 a scalar product (x, y)2' = (V-'x, V-'y)i (x, yE.)2) equivalent to the original one. Since [x, y]2 = (Jzx, y)2, where Ji = 17J1V-' = Jr', the components of the new canonical decomposition 2 = 'Y2" O+ .VZ 'will be the subspaces ,Yf F.01'. Consequently dim(vv n i ) = dim(V(cv n .)Yi ) = dim(3 v n ,y2,,) and dim(91*1 fl e-) = dim(V(Qo ] fl Yi 1)) _ dim( V, fl W '), and therefore v(V v) = v(,R v). Now let P(Vv) = v(Rv) Without loss of generality we shall suppose (see the proof of Theorem 2.2) that the decompositions (2.1) and (2.2) of the subspaces cv and Rv are connected by the relations VCA; = Ri, i = 0, 1, 2.
Since c1 = Iv n .iYi and 1 = 91v fl .w2+, we have dim(iv n i) = dim(3ly n . w2+). Consequently dim(cv, n W,-) = dim(f1w fl. l-) = -v(/v)+dim(Vvn -wi v(JRv)+dim(.91vn,Yzdim(.RVI n.YZ ) = dim(MV n . i ).
Following the scheme of the proof of Theorem 2.2 we deduce that the operator V admits an extension V' E St( 1, . 2) with 9W n .01- and wv n Wi . Hence V VJ and [1I are negative subspaces of equal finite dimension and therefore it is now easy to construct a (J1, J2)-unitary extension of the operator V' and it will also be an extension of V. Let V' E St(.,*P1, -02) be an arbitrary maximal (J1, J2)-isometric extension
of the operator V, and suppose that V' is not a (J1, J2)-unitary operator. Since V;,1l and ,j-] have the same sign, we conclude from Corollary 2.3 that we can, without loss of generality, suppose the operator V' to be (J1, J2)-semiunitary and dim R Vi] ;4 0.
We consider the J2-space ,Y2= V'. 1. From what has been proved, P(Vv) = v'('), where P'(') is the index of Mv in .W''. But by hypothesis
258
5 Theory of Extensions of Isometric and Symmetric Operators
and it is easy to see that v(3'v) = P' (3 ) - dim .91(1j-so we v(V v) = have obtained a contradiction. We pass on now to the construction of an operator VE St Y' with (v= ,3v, but which does not have J-unitary extensions.
Example 2.8: Let V1, LA2, (!Yl)2 be infinite-dimensional separable Hilbert spaces. We form the space ,Y = 91 Q+ CA2 Q+ (GO ')2 and introduce in it a J-metric (see 1.Example 3.9) by means of the operator
J=
J22
0
0
J33
0
V36 (I3 - J33)
0
(J3 - J33 1/2
)1/2
V36
,
- V*J03 V36
constructed relative to this decomposition of the space ye, where J22 = 12, J33 E Y., 0 < J33 < 13, and V36 is an isometry mapping ('W1)2 on to 92. Let be the eigenvalues of the operator J33, and let X1 >, X2 >, limk-., Xk+1/Xk = a > 0; let [ fk) be an orthonormalized basis in 92 composed of the eigenvectors of the operator J33: J33fk = Xkfk (k = 1, 2, . . .); and let [ ek 11 be an orthonormalized basis in 91. We define on 9 v = 91 O+ 92 an operator V E St W by putting V e k = ek+1 (k = 1 , 2, .
.
.),
Vf1= Xl el,
V f k = Xk+1/Xkfk
(k = 1, 2, ...).
By construction My = cv. The operator V admits no J-unitary extensions V, for if it did the operator VI V2 (E St(,'1, ,1W2)) acting from the J1-space .)r1 = C42 O (4 i )2 into the J2-space W2 = Lin [ el) + CA2 O+ W)2 with Ji = J I Yei, i = 1, 2, would have (J1, J2)-unitary extensions, and this is impossible by virtue of Lemma 2.7 since P(92)= 0 and v(Vi2) = 1.
The above example shows that in answering the question whether an operator VE St(.'1,,W2) has (J1, J2)-unitary extensions one has to take into account not only the characteristics of the spaces CA I1l and RI'l but also the `action' of the operator V. Let be a maximal uniformly definite dual pair in Ye1, let V V = C/o [+] 1/ 1 [+] V2 be the ('l', 21 )-decomposition of the domain of definition of an operator V E St(,*'1 i 46), and 91 v = 3o [+] V!Y 1 [+] Vc12 be the decomposition of its range of values. We choose in .-W2 a maximal uniformly
definite dual pair (Y2, Yz ) such that Vw v fl 99i ) C Y27+, and we introduce the following constants v+ (V) and v _ (V ): if there is in V[,1l or in an infinite-dimensional uniformly negative (respectively, uniformly positive) subspace; (2.4) dim(V I-L] fl Y ) + dim( V/2 fl £Z) - dim(. 1[,1] fl Pz ) otherwise. 0
v±(V) =
§2 Extensions of standard isometric and symmetric operators
259
Remark 2.9: If there are no infinite-dimensional uniformly definite subspaces in Vv, v, 9111 and R(,1], and if 9v E,.tl+(.W'1), and i?vE tl+(,°z), then
v_(V)=0 and v+ (V) = v(Av)- v(C'v). Our immediate purpose will be to prove that v±( V) are independent of the choice of the dual pairs (-Ti', 2 ), i = 1,2. We preface this with the following proposition.
Lemma 2.10 Let . be a J-space, f a subspace of it, and let U be a J-semi-unitary J-bi-non-contractive (respectively, J-bi-non-expansive) extension ;e (0) implies that there is in c(1) of the operator I I V. Then the condition an infinite-dimensional uniformly negative (respectively, uniformly positive) subspace.
Let !J = go [+] c1 [+] 92 be the )-decomposition of the subspace J, and let U be a J-semi-unitary J-bi-non-contractive operator. Since 90 and Col] are invariant relative to the operators U and U`, it follows in accordance
with Exercise 20 on 2.§4 that the operator U induces in the i-space _ Col]
!I o
a
J-bi-non-contractive
Jsemi-unitary operator
U1 9 = J1 9, where I = 9/ go. Moreover, ML'I ;;d
(0)
U,
and
if and only if
U ;e (0); and 9[1] contains an infinite-dimensional uniformly negative subspace if and only if (1] _ 9 [1]' go has this property. Therefore we shall suppose without loss of generality that go = (0), i.e., ', and with it also iI [1] and g', are non-degenerate subspaces. In accordance with 2.Proposition 4.14 we have for the J-bi-non-contractive operator U that M I l (= Ker U`) is a uniformly negative subspace, and so without loss of generality we shall suppose that 91Il C -W-. We shall assume
that W # (0) and that, contrary to the proposition, V[l] does not contain infinite-dimensional uniformly negative subspaces, i.e., (see Exercise 7 on 1.§6) the Gram operator of every non-positive subspace from c l1] is completely continuous. Let U = II U jII ? j=1, J = II J;i ?i= 1 be the matrix representations
of the operators U and J relative to the decomposition W = SI (1 91 . Since mil[ 1 C Y and ci CRC, we have J& C V`(= Jc [1] ), and U I = 9lu2, n ci 1 = i i I n c 1. By direct calculations taking Exercise 7
on 1.§3 into account we verify that the fact of the operator U being J-semi-unitary is equivalent to the relations a) .V1, - u22 C . IrZZ, U12 = - J21 Jzz' (Iz - U22); b) Uzz Jzz' Uzz Jzz = Iz.
Since Ker Uzz =
u,z n `I' 1, and condition b) is the condition for the operator be to Jzz-semi-unitary, so
( Uzz) ` = Jzz' Uzz Jzz
(Ker U2*2)111). Let CA1 = c+ [+] V_' be the canonical decomposition of 'J 1, and let P+' be the projectors on to V +, P+ + P_ = 12. It follows from the relation a) and the complete continuity of J2'21 Y_ that the operator
T= P_Jzz'(I2- U22 )J221 ci_' =I_ - P_Jzz'UzzJzzI f'
260
5 Theory of Extensions of Isometric and Symmetric Operators
is completely continuous, and therefore P_ Jzz' U22 J22 9' = I_ - T. Since Jzz' U22 J22 is a J22-semi-unitary operator, we have Ker(P_ Jzz' U22 J221 CA') _ (0) and therefore i?j - r= LA-. Hence we conclude that A2' U22 J22 C4' is the
maximal non-negative subspace in 9 1. But the subspace Ker Uzz (6 {6) ), which is J-orthogonal to ,?(uiz)', is also negative-and we have obtained a contradiction. The case when U is a bi-non-expansive operator is proved similarly. Lemma 2.11:
In Definition (2.4) v±(V) do not depend on the choice of the
dual pairs (?; , 2' ), i = 1, 2. We shall verify that v+( V) does not depend on the choice of the dual pairs the argument for v- (V) is similar. If there is an infinite-
dimensional uniformly negative subspace in mfr], then v+(V)=0 by definition does not depend on the choice of the dual pairs. Suppose all the uniformly negative subspaces in Vl] are finite-dimensional. Without loss of generality we shall assume that the decompositions (2.1) and
(2.2) are connected by the relations Vii; = 3;, i = 0, 1, 2, and therefore the
constant p+(V)v, calculated with respect to the dual pairs i = 1, 2, coincides with the difference dim(9f1' fl;)-dim(3[1I fl .z). We bring into consideration a J,-space ;, and a J2-space 46, putting , y , [ O+ ].
1,011
0]0 1,0-
,W2[O] I
where dim
-_I
P, I,
j, IYe
G)
_
±K± if v, dim 91wl. Con-
sequently W-1 is a regular extension of the operator V.
In conclusion we introduce some definitions concerning J-Hermitian operators, and we formulate in Exercises 6-13 the corresponding results 4
obtained by applying as above the Cayley-Neyman transformation. Definition 2.19: We call an operator B a J-bi-extension of a J-Hermitian operator A if A C B and the graph I'A is isotropic in I'B relative to the form
[(x1, x2), (Y1,Y2)]r= i([x1,Y21 - [x2,Y1] (2.Formula (1.4)).
The set of closed J-Hermitian operators A with points of regular type (Xo, Xo) (Xo 5;d ko) will be called Xo-standard and will be denoted by the symbol St(.h"; Xo) (= St(Jf; ko)).
Definition 2.20: We shall say that an operator A E St(,'; 4) admits a x-regular (0 < x < cc) Xo-standard extension if there is a Pontryagin ax-space II with x negative squares such that in the J-space = W C+ II
(J = J @ H.) a J-Hermitian operator A has a J-self-adjoint extension A E St(,*; Xo). When x = 0 such an extension will be called a regular extension.
Exercises and problems 1
Let V E St(.r#1, k2). Prove that v, (V-1) = - v, (V) (Azizov). Hint: Use Formula (2.4) and Lemma 2.11.
2
Suppose that VE St(./P1, IY2), that V has at least one (J1, J2)-unitary extension, and that ciH and RI-LI contain no infinite-dimensional uniformly definite subspaces. Prove that then every maximal (J1, J2)-isometric extension V' (E St(. 1, .,Y2 )) of the operator V is (J1, J2 )-unitary (Azizov). Hint: Prove that x ± (V') = 0 for any extension VE St(.10'1,./,0Uo(UoUo)-` - I)12i Im Xo,
where lx traverses the set of all J-bi-extensions of the operator I I WA - 41; if, in addition, Ao E St(.W; ko), then R,,0(A) _
I)12i Im Xo,
and when AO = Ao the operator A will be J-symmetric (respectively, maximal J-dissipative, J-self-adjoint) if and only if ix, is a J-semi-unitary (respectively, J-bi-non-expansive, J-unitary) operator (Azizov [14]). Hint: Use the results of Exercise 16 on §1, Exercise 6 and 2.Formula (6.29). 13
Let A E St(./P; ko) and Im ko < 0. Prove that A admits extensions A =Ac E St(.M; Xo) with exit into some J-space .W D .W'. Moreover the following conditions are equivlent: a) A admits x-regular ko-standard extensions A = A `;
b) A admits regular ko-standard extensions 4= c) A admits maximal J-dissipative extensions A E St(. Y; ko) with (Azizov [14]).
Hint: Use Theorem 2.18 and the result of Exercise 6.
Xo E p (A )
266 14
5 Theory of Extensions of Isometric and Symmetric Operators Let .w be a J-space, N' +O 1P a J-space, where J = J Q+ 1, and let A be a J-symmetric operator. Prove that A admits Jself-adjoint extensions (cf. Exercise 13).
Hint: Use the scheme A -* JA -. JA O+ - JA -* JA O+ - JA = J JA 15
- JA.
VV= let the (.YYt , .W1 )-decomposition and Let V E St(.YPI, IY2 ), Jo [+] J, [+] V2 and the (.Yr z , .Wz )-decomposition W p = .moo [+] .-?, [+],R2 have the
property V11; = , ,, i = 1, 2. Then the following conditions are equivalent:
a) the operator A has at least one (J,, J2)-unitary extension; b) there is a (J,, J2)-unitary operator W such that WJv= Rv and WJ, = -R,;
c) there is a (J,, J2)-unitary operator U such that UJv=.wand UJ2=.-R2; d) there is a (J,, J2)-isometric operator V, mapping 91F1i on to .1,1.11 and in the Krein space VJ2 [ + ] V, (J i1i )2 there is a dual pair (Y" Y'-) maximal in that space and such that SP± fl VJz= (B) and Y'± fl V,(Ji1))2= (0) (Azizov [14]).
§3
Generalized resolvents of symmetric operators
Before describing generalized resolvents of J-symmetric operators we introduce and prove a number of auxiliary propositions which are, incidentally, of some independent interest. One of the concepts generalizing the 1
concept of an `extension' of operators is that of a `dilatation' of operators. An extension is a process which can take place both within the limits of the given spaces or with emergence from them; but a dilatation takes place necessarily with emergence from the original spaces. Definition 3.1: Let YW; (i = 1, 2, 3) be Hilbert spaces, and let T: ,-'de2 be a bounded operator with JT= .YW1. We call a bounded operator T:.Yy1 O+
O+
Y3 a dilatation of the operator T if T= 11
T;;jj i;= 1,
where T,1 = T and T21, T12, T22 are operators such that T12 T 2 t T21 = 0 when
n = 0, 1, 2, .... If also the Wi are J;-spaces, and if T is a (J1, J2)-unitary operator with it = J, O J3, JJ = J2 O J3, then we call T a (J,, J2)-unitary dilatation of the operator T We say that such a dilatation is x-regular if .W3 = II,Y with x negative squares; we shall call 0-regular dilatations regular.
Remark 3.2:
If under the conditions of Definition 3.1 Tis an extension of the
operator T (this is equivalent to the equality T21 = 0), then, clearly, T is a dilatation of the operator T. However, not every dilatation is an extension. We shall convince ourselves of this later, for example, in Theorem 3.4. Lemma 3.3:
Let T:.-WI -' X2 be a bounded operator with JT =,W,, and
be a dilatation of . let T: -W, 2 1Y3 3 T: .Ye,_O .W'3 O+ J 4 , Y'2 ( W3 O+ tea be a dilatation of
it,
and
let
the operator T.
Then T is a dilation of the operator T By hypothesis the operators T and T admit the follwing matrix represen-
§3 Generalized resolvents of symmetric operators
267
tations relative to the corresponding decompositions: T= II T%JII?J=I,
T=IITJII'J
with
T21=0,
and I
I T3 I I ?= I T33II T 3 j 1
1
=o = 0 (n = 0, 1, 2, ...).
Hence II Tu IIJ=2(II T,J 11 J=2)"II TJ1 IIJ=2 n-I
T12 T22 T21 + T13 T33 131 + T13 k=1
T33T32Tzz ` kT21
= 0 when n =0, 1, 2, ..., and it only remains to use Definition 3.1.
Theorem 3.4: Let T:,W, .W'2 be a bounded operator acting from a J,-space WI into a J2-space i 2 with CA T ='I. Then there is a J3-space W3 and a (J1 (@ J3, J2 O+ J3)-unitary operator T:.O, O+ W3 - -W2 O+ .03 which
is a dilatation of the operator T Let J, - T*J2 T = Jo I J, - T*J2 T I be the polar decomposition of the selfadjoint operator J, - T*JzT, let moo =3j, - T'J2T and let Jo = Jo I .Wo
Then Jo = Jo = Jo 1. We bring into consideration the Jo-space WO' = Q+
0
-.W o', where ,moo
= Yeo and Jo= (
k=0
JAkt, Jak)= Jo (k=0, 1,2,...),
and we define an operator T, :, O+ 06' -'2 O+ 0o by the formula
T1(x;xo,xl,...)=(Tx;IJI-T*J2TI1i2x,xo,X1,...).
(3.1)
By construction T, is a continuous operator with IT, = 'Y, 0,06' and it is (J1 O+ Jo, J2 O+ Jo)-isometric, i.e., T, is a (J, Q+ Jo, J2 O+ Jo)-semi-unitary
operator. The subspace moo is invariant relative to Ti, and therefore T, is a dilatation of the operator T. By virtue of Theorem 2.18 there is a J-space .W
such that the operator T, admits a (J, O J3, J2 O J3 )-unitary extension T: .01 O+ .3 -'2 O 03, where .3 = Wo O+ ' and J3 = JO' O+ J. It only remains to use Lemma 3.3. Corollary 3.5: For an operator T under the conditions of Theorem 3.4 to admit a regular dilatation it is necessary and sufficient that it be a (J,, J2)-binon-expansive operator. El The necessity follows from Exercise 34 on 2.§4. The proof of sufficiency can be carried out using the same scheme as in the proof of Theorem 3.4 taking into account that Jo = I I iJ, - T'J.T and that the operator T, is
268
5 Theory of Extensions of Isometric and Symmetric Operators
(J, O+ J6, J2 O+ J6)-bi-non-expansive. Consequently, by virtue of Theorem 2.18 we can put J = I. 2 Here we shall generalize the concept of a dilatation (see Exercise 4 below) and investigate special classes of holomorphic operator-functions.
Let T(µ) be a function holomorphic in the neighbourhood of 0 with values in a set of continuous operators acting from a space .I', into a Definition 3.6:
(B space .W'2 with VT(,) _ . 1, and let T= II T,, II ?i=1 :. 3 -,02 (9 '3 be a bounded operator with 1'r = 1Y1 O+ J °3. We shall say that the function T(u) 1
is generated by the operator T (or that the operator T generates the function T(µ)) if in some neighbourhood of zero T(µ) = T + µT,2(13 - µT22)- `T21.
Definition 3.7:
In Definition 3.6 suppose the .Yt'i are J,-spaces (i = 1, 2, 3) and
that T is a (J, O+ J3, J2 O+ J3 )-unitary operator. Then we shall say that the
function T(µ) belongs to the class H(MI, .2). If, in addition, .W'3 is a Pontryagin space with x negative squares, then we shall say that T(µ) belongs
to the class fl W',,.Y'2). (In cases where it will not cause confusion, the symbols 01, W2 will be omitted from the designations of classes). Lemma 3.8:
Let T= I I Tii
11 i=
i :.Yf, 0 JP3 - W2 O+ .k3
and
J4
W1
T=IITUII1i=i:.Y,
be bounded operators with #T = .N, O+ .YY3, VT= .-W, O+ .Yr'3 O+ .Yea, and let
T be a dilatation of the operator T Then T and T generate the same operator function T(µ) : Ye,
Ye2.
In accordance with Definition 3.6 we have to verify that T11 + µTlz(I3 - µT22) - `Tzl
= T + It II Tii II i=z(I3 G4 - µ II Tii
II'i=2)
`II
Ti,
II'=2.
(3.2)
The condition 'T is a dilatation of T' is equivalent in the present case to the system of equalities T3 T 33 T3j = 0 (i, j = 2, 3), and therefore II T,iHI]=2(II TiiII'i=2)'II Ti li=z= T12Tz2T21
(k = 0, 1,2).
It only remains to note that in a sufficiently small neighbourhood of zero the equality (3.2) is equivalent to the coincidence of the series k= =0
µkT12T22T2,
and
k=0
µk11 TUIIJ=2(11
TiiII'i.i=2)kII Ti, II' =2.
§3 Generalized resolvents of symmetric operators Lemma 3.9:
269
Let T(µ) :.WP, --+ k2 be an operator function holomorphic in
the neighbourhood of zero. Then there is a space IY3 and an operator T= II T, II ?i=1 : h"1 T
O+
3 such that T(µ) generates the operator
--W2 O+
3
We consider the function, holomorphic in the neighbourhood of zero,
K(µ) = (T(µ) - T(0))µ with µ 5;6 0 and K(O) = T'(0). By Cauchy's formula we have for it, when r > 0 is sufficiently small, K(µ)
tai
r-µ
t1=
2ir
N)
A
K(rfl d(' = 1
K(reI
d4p
(I it
I < r).
µ
J
Let .03 be the Hilbert space of weakly measurable functions x3 (e "0) on [ - a, n] with values in .Y2 and with the square-summable norm IIx3(e'")I12dtip, in which only a finite number
of the xa i;e 6. For elements f and g = EX E
e),g), we define a form
Y/,
If, glo = Ea.µE 111, (K(X, µ)x>., yµ). By hypothesis this form has x negative squares, and if P is degenerate and Yo is it isotropic part, then the completion of the factor-lineal Y/Yo will be a Pontryagin space with x negative squares
(cf. Exercise 6 on 1.§9)-and we denote it by fl,(F). From now on we shall identify elements from 9 with the corresponding elements from £lYo. On the set
1v= j.f If= aE Yj14 e),xxEY, Y xx=9} XE 14
we define an operator V : Vf= EXE 111, XE)IXX- Since
[Vf, V.f]o= >
x
X.µ E Y/,
X.1+ E Y/,
1-Xµ =
xµ
x
p E Y/
X E Y/, L
F(X)x,\,
F(X) + F`(µ)lxx, 1- Xµ J
xµ
J
([F(k)xa,x,,]+ [xa, F(µ)xµ])
[F(X)x),, X E Y/,
\\1 J
1
µ E Y/,
J
µ E Y/, 1- XtL. xµJ
['Z/,
+
kµ
1-Xµ 1
µ E Y/,
I
xa, F(y)x, ]
XX, F(A)x,
X E W,
=[f,flo, Visa 7r-isometric operator in II, (F). Its range of values fit v consists of vectors g = EX E Y/, e,,ya E 9' such that yo = 0 and Eo , a E Y,, y,\/ X = 0. This set is dense in
n, (F). For, if e; x -' eox, ('e),x
--+ 0 and x E ., then, using Exercise 13 on 1. §9, we obtain 0, and therefore any element E,, E Yn exx7, E ' (xo = 0) is
approximated by the elements EXE Y/, eaxa - (et Eo4 XE Y,, x>/X E R v, and since
Y' = II, (F), it follows that ,;? v = rI, (F) also. It follows from the last relation that is non-degenerate, and this implies that V is invertible and hence
(taking 2.Corollary 4.8 into account) that the operators V and V-1 are
§3 Generalized resolvents of symmetric operators
271
continuous. We keep the same notation for the closures of these operators, the second of which will be 7r-semi-unitary. We bring into consideration an operator IF : Ye -p 2' by putting Fx = (1 /, 2)eox, x E M. For any x, y E Y and X E WF we obtain [Fx, Fx]o = [(F(0))Rx, y], [rx, exy]o = (1/,2) [F(0) + F`(X))x, y], and therefore IF is a continuous operator (see Exercise 6 on 1 §9). Consequently there is the operator r': rl (F) .Y and
FT = (F(O))R,
r `c),x = (1 ,,!2) (F` (0) + F(X))x.
(3.4)
Since (see 2.Definition 6.6) the spectrum of the ir-semi-unitary operator V ', with the exception of not more than x eigenvalues, is situated in the disc (t t 15 1), the resolvent (V - XI) - ' exists everywhere in the disc (X I I X I < 1) with the exception, possibly, of not more than x points. Since
(V- XI)(e),x- cox) = Xcox, we have r`cxx-r`eox= XT`(V- XI)- Icox, and so we obtain from (3.4)
F(X) = i(F(0))r+ F`(V+ XI)(V- VI)-'F.
(3.5)
We sum up the foregoing argument:
Let V' be a J-space, let F(X) be an operator function holomorphic in the neighbourhood of zero and taking values in the set of continuous operators defined on ', and let the kernel K(X, µ) = J(F(X) + F`(µ)) (1 - X) have x negative squares. Then: Theorem 3.12:
1) F(X) extends, preserving these properties, on to the disc [X I I X I < 1) with the possible exception of x points; 2) there is a space II,,, a bounded operator r : -+ II,,, and a a-semi-
unitary operator V ' : H - H. such that the equality (3.5) holds for all
X a(V),IXI, (A) the set of all operatorfunctions F(X) :.W' .W, holomorphic in a neighbourhood of the point Xo, which can be expressed in the form F(k) = F1 1
where F(X) = I
- xo F( _ kok ko k 12\I J1)-unitary
kok - k0
+kok -
F,i II ?i=1
\
Fzz l
Fz 1,
extension of the operator
is a (J O+
IA.x,, in the (J (@ J1)-space f O+ . 1; .01 is a certain J1-space; if .1P1 is a Pontryagin space with x negative squares, then we denote this set by II
'A _
rIL(A)
It follows immediately from the definition of the set 11x.(A) that for every k from a certain neighbourhood of the point X0 the function F(X) (EIIao(A )) is a J-bi-extension of the operator IA,XO and the operator-function
G(µ) = F(I )'°I z(1
\
ko - Xoµ
µ) I E H
l
is holomorphic in a neighbourhood of µ = 0. Conversely, suppose that for every k from a certain neighbourhood of the point Xo the operator-function F(X), holomorphic in this neighbourhood, is a J-bi-extension of the operator IA,a,,. Then it belongs to the set II1,,,(A ). For, in accordance with Theorem 3.10 G(a) E H and therefore kok - k0
F(k)=F11+kok-
.0
F
1z
- x0 Fzz 1 I I1 _ ko kkok-xoJ
1
Fz1 ,
and moreover by virtue of Theorem 3.18 we can regard the (J O+ J1)-unitary operator F= II F, II Z;=1 :.k' O+ .k1 -* .Y( O+ .,W1 as simple. (We note that we can take H, as.W1 if and only if G(µ) E H".) We verify that Fis an extension of
5 Theory of Extensions of Isometric and Symmetric Operators
280
the operatorlA,a or, what is the same thing, that F21x = 0 when x E WA - i. Suppose this is not so. Then there is a vector xo E MA - j,r such that F21x0 0, and therefore Y' = C LinI FX"J °m n w'1 * (0) and it is a completely invariant subspace relative to F and F22-and we have obtained a contradiction of the fact that the operator F is simple. Thus we have proved
Let F(X) be a function holomorphic in a neighbourhood of the point X0. Then F(X) E IIx, (A) if and only if, for every X in a certain
Theorem 3.21:
neighbourhood of the point Xo, F(X) is a J-bi-extension of the operator IA,X,. Moreover the conditions (F(X) E Hx'o (A )) and j G(µ) = FI' X01 z(1
are equivalent. The operator F be chosen to be simple.
µ))
E
n'}
F;; i;=1 generating the function F(X) can
Let a J-symmetric operator A E St(,W; X0) with Im Xo < 0, let Ao E St(, ,Y; X0) be a J-bi-extension of the operator A such that Xo E p(Ao) (on the existence of such an Ao see Exercise 10 on §2), and let Uo = K1,,,(A). 5
Theorem 3.22:
The relations R>, = (Ta - XI)-' and Ta=(X0-Xo)[F(X)Uo(UoUo)-i-I]-1+X I
describe all the generalized resolvents of the J-symmetric operator A in the neighbourhood of the point Xo when F(X) traverses the set In the neighbourhood of the point Xo we have R"
= El
[(Tµ-µ1)-1l`.
Let A be a J-selfadjoint extension of the operator A acting in the J-space w' O+ -Y,, and let Xo E p(A). It follows from Exercise 12 on §2 that (A - X01)-' =
1
- x0
[4.00(17C0170-1
- 1],
where U0 is a J bi-extension of the operator U = K,,o (A) coinciding on W' with U01 and Uo I, = (X0/4)11i and !>,, is the corresponding J-bi-extension of the operator IA,XO. In accordance with Hilbert's theorem for the resolvent R,,
1
-
P I - X_
Ilo (1 >,,,Uo(UoUo)-1 - I
Jof
[i U0(UoUO)-'I] I .i
Let Ik,,= II Fiijj?i=I be the matrix representation the operator operator U= has the matrix representation U= 11 U1ijj?i=1, where
Un = FnUo(UoUo)-', and Uiz>'o F,2, x0
.
Then the
i= 1,2.
§3 Generalized resolvents of symmetric operators
281
Therefore in a neighbourhood of the point Xo Hilbert's identity given above can he rewritten in the form r /F(X)Uo(UoUo) I F(X)Uo(UoUo) Xo -
where
Xo - xo
O
\
L
F(X) = F1 1
Xo X - X0
-1
Xo X - X0
+Xo X - Xo F12
IXo X - o
Fzz
Fzi
is an operator-function holomorphic in a neighbourhood of the point X0 which is, for every X from this neighbourhood, a J-bi-extension of the operator IA,X,, i.e., F(X) E II,,,,(A) by virtue of Theorem 3.21. In this same neighbourhood of
the point ko the operators F(X)Uo(UoUo)-' are J-bi-extensions of the 1 E o, (U) that where Rs=(T,-XI) ', Hence, I]-1 + X01. Tx _ (Xo - Xo)[F(X)Uo(UoUo)-' Conversely, let F(X)EIh,,,(A). Then F(X)Uo(UoUo)-` is an operatorfunction holomorphic in a neighbourhood of the point Xo which, by virtue of for all X from Theorem 3.21 is a J-bi-extension of the operator U= this neighbourhood. Therefore
operator U= K,,, (A ), and therefore it follows from
IEa,(F(X)Uo(UoUo)-')
/ T(µ) =
z
F( 1 X0 '
Uo(UoUo)-'
(l
X0-1,A))
is a function holomorphic in a neighbourhood of the point-,u = 0, and therefore '1 and a simple J-unitary operator there is a Ji-space
U=
U;;11 ;=, :
0
, --
.
0 .W'i, with J= J 0 Ji, such that U gener-
ates the function T(µ). It follows from the result given later in Exercise 11 that
we can take as U an extension of the operator U. Therefore by virtue of Exercise 16 on §1 U=I,,oUo(UoUo)-1, where Uo is a J-bi-extension of the operator U coinciding on r with Uo, and Uo I W', = Oo/Xo)h, and 1),0 is the corresponding J-bi-extension of the operator 1A.ao. Since U C Uand 1 E ac(U),
it follows, because the operator U is simple, that 1 E oc(U), i.e., there is a J-selfadjoint operator A such that U = Kx,,(A ). It follows from Exercise 15 on II§6 that A C A, and from the proof of the first theorem that
R,,=(T,,- XI)-', where
Ta=(Xo-&) T Q
I
+ XI
= (Xo- Xo)[F(X)Uo(UoUo)-' - I]-' + Xol. It
then follows from the definition of the generalized resolvent that
R, = (R,,)` = [(T;, - µl)-']` in a neighbourhood of the point Xo. Corollary 3.23: Under the conditions of Theorem 3.22 let A0 be a maximal generates J-dissipative J-symmetric operator. Then the function F(X) E a x-regular generalized resolvent, and conversely, every x -regular generalized resolvent is generated by some function from nx',,(A).
5 Theory of Extensions of Isometric and Symmetric Operators
282
Let the function F(X) E HZ,/(A ). Then
FIIX 12(1-µ))U0E H',
T
and the operator U appearing in the second part of the proof of Theorem 3.22
can be chosen to be a x-regular extension of an operator U= Therefore the generalized resolvent constructed there will be x-regular.
Conversely, let R), be a x-regular generalized resolvent generated by a x-regular extension of the operator A. Then U = Ka (A) = where U0 is a J-bi-non-expansive i-semi-unitary extension of the operator U coinciding with Uo on .W, and Uox = (Xo/Xo)x when xE I ,,. Consequently I>,0UoUo is a i-bi-non-expansive extension of the operator IA,ao and therefore, by virtue of Corollary 3.5 and Lemma 3.8, the function generated by this operator belongs to III (A ). Corollary 3.24: Under the conditions of Corollary 3.23 let J = I, x = 0. Then the function (TX - XI)-' admits a holomorphic extension from the neighbour-
hood of X0 on to C-. On C+ we put (Tx- AI) -' = [(TX- XI)-']`. This extension coincides in C+ U C- with the corresponding regular generalized resolvent of the operator A. By virtue of the result of Exercise 12 on §2
Ta=(X0-Xo)(F(X)Uo-I)-'+X01, and by virtue of Corollary 3.23 we can suppose that F(X) E i.e., there is a Hilbert .4" space and a unitary extension (@.;99, of the operator IAA, such that F= II FjiHI +i= I :,-W O+ X F(X)
= Fi +
.o
Xo
_Xo i - Xo
>-1\0 X - 1\o
F 1z I
X0
X - 1\0
\-i
Fzz
Fz i
Since
II Fzz II < and
),o
X - X0
Xo
X - .o
< 1,
it follows that F(X) admits a holomorphic extension from the neighbourhood of the point Xo on to C-. Moreover the extenson of the function z
G(µ) = F( X01 0 _ 01 E IV( 4 - Xoµ
< 1, and therefore II F(X)II < 1),
which, by virtue of 2.Remark 6.13, implies that the operators Ta are maximally dissipative for all X E C-. Therefore C- C p(TX) (see 2. Lemma 2.8),
and therefore (T,, - XI)-' admits holomorphic extension from the vicinity of
the point Xo on to C-. On C+ we put (T,- XI)-' = [(Tx- XI)-']`. On the other hand, the regular generalized resolvent R,, of the operator A
§3 Generalized resolvents of symmetric operators
283
defined on C U C- does coincide in a neighbourhood of the point X0 with the
function (T,\-XI)-', and therefore R a = (T, - XI)-' when X E C- and µEC+.
Exercises and problems 1
Let.YY, and .Yt'2 be Hilbert spaces, and let T:"Y3 Yj C'r= JJt°,. In [XXIII] an operator T:.Y( O+
.JP2 be a bounded operator with .Yt', O+ 'Y3 with C/T=.yY, ED ,k3
is called a dilatation of the operator T if P,T"I .yt', = T" (n = 0, 1, 2,3 , ...), where P, is the orthoprojector from YY, O+ .Y'3 on to .,Yi. Prove that this definition and Definition 3.1 are equivalent when .Y'i =.YW2. 2
Let T and T be the same operators as in Exercise 1. Prove that the operator Twill
be a dilatation of the operator T if and only if P, (T - XI)-" ,W, = (T- XI)-", n = 1, 2, ..., X E p(T) fl p(T) (see [XXIII] ). 3
Suppose that F(X) is a function holomorphic in a neighbourhood of zero with values in a set of continuous operators acting in a J-space Jr', and that there are: a space II, with x negative squares; a continuous operator r :.YP - 11,; a ir-semiunitary operator V -' : II, - II,; and a J-selfadjoint operator S with !Ys = 'Y; such that F(X) = iS+1'`(V+ X1)(V- xI)-'I', I X I < 1, X a(V). Prove that the kernel K(X,µ)= J[F(X)+ F`(µ)] (1 - Xµ) has not more than x negative squares (cf. M. Krein and Langer [4] ).
4
Let T(X) = To: .01 .V2 be an operator-function holomorphic in a neighbourhood of zero. Prove that an operator T :.yt', O+ 'YO - 2 + . Yo will generate the function T(X) if and only if it is a dilatation of the operator To (Azizov).
5
Prove that an operator To admits a x-regular dilatation if and only if To is a (J,, J2)-bi-non-expansive operator (Azizov [14]). Hint: Use the equivalence of assertions a) and b) in Theorem 3.16 and the result of Exercise 4.
6
Give an example of an operator-function T(X) :.yP, - JI2i holomorphic in a neighbourhood of zero, such that the kernel [J, - T*(µ)J2T(X)]/(I - µX) has x negative squares but nevertheless T(X) ri'.
Hint: Put T(X) - To, where To is a (J,, J2)-semi-unitary operator which is not (J,, J2)-bi-non-expansive, and use either the results of Exercises 4 and 5, or the equivalence of assertions a) and b) in Theorem 3.16. 7
Prove that under the conditions of Theorem 3.16 the function T`(X) E II'. Hint: Verify that if the operator T generates the function T(X), then T` generates the function T`(X).
8
Let T(µ):.;o , -..W2 be an operator-function holomorphic in a neighbourhood of the point µ = 0 with values in a set of continuous operators (i T(,) _ .Y',) acting from the J,-space.YY', into the J2-space .W2. Prove that the following assertions are equivalent: a) the kernels
J,-T*(µ)J2T(X) 1 -µX non-negative;
and
J2-T(X)J, T*(µ)
1 -µA
284
5 Theory of Extensions of Isometric and Symmetric Operators b) in a neighbourhood of zero the T(X) are (J,, J2)-bi-non-expansive and the function w' (T(>,)) = T, (X) admits holomorphic extension on to the disc [XI IXI. (A) was introduced into the investigation by Azizov. Theorem 3.21 is also due to him (see Azizov [14]). §3.5
Theorem 3.22 is due to Azizov [14]. The form of writing the
generalized resolvent Rx, = (Tx - XI) - ' was first introduced by A Strauss [1].
Moreover we have used M. Krein's idea (see [I]) of all the generalized resolvents by means of a fixed extension of the operator. For more detail about
other approaches to the description of generalized resolvents and historical information see [IV] and also Etkin [1], [2]. Corollary 3.23 is due to Azizov [14], and Corollary 3.24 to A. Shtraus [1]; we point out that the proof in the text, due to Azizov, is not based on A. Shtraus's result.
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INDEX
Adjoint operator 49 Almost J-orthonormalized basis 76 Almost normalized system of vectors 76 Amenable group of operators 137 Angular operator 49 Anti-linearity 2 Anti-space 4 Aperture of two subspaces 61 Asymptotically isotropic sequence 42 B-plus-operator 132
Biorthogonal systems of vectors 76 Bounded spectral set 97
Canonical decomposition of a space 14 Canonical imbedding 183 Canonical imbedding operators 22 Canonical projectors 18 Canonical symmetry operator 21 Carrier a(E) of homomorphism 210 Cauchy-Bunyakovski inequality 6 Cayley-Neyman transformation 142 Centralized system of subsets 59 Class H operators 193 Class K(H) operators 197 Closed mapping in a Hausdorff space 177
Complete system of vectors 219 Completely invariant subspace relative to a family of operators 161 Condition L 160 Condition A+ 170 Conditional basis 77 Continuous spectrum of a linear relation 145
`Corner' 170 Critical points 220
Decomposable lineal 34 Deficiency subspace 25 Definite lineal 3 Definite subspace 24 `Definitizable' J-selfadjoint operator 211 Degenerate lineal 5 Dilatation of an operator 266 Direct Cayley-Neyman transformation 144
Dissipative kernel 234 Dissipative operators 91
Domain of definition of a linear relation 85
Doubly strict plus-operator 120 Dual pair 72
Eigen spectral function of a J-selfadjoint operator 211 Eigenlineal of a linear relation 143 Eigenvalues of a linear relation 143 Eigenvector of L(X) 237 Eigenvectors of a linear relation T 143 Extension of a dual pair 72 Field of regularity of a linear relation 44 Field of regularity of an operator 92 Fixed point of the transformation F; 176
Focusing plus-operator 126 Function generated by an operator 268 Function T(µ) belonging to the class ,r(. j,IY2) 268
G-adjoint of a linear relation 146 G-metric 40 G-selfadjoint linear relation 146 G-selfadjoint operator 104 301
302
Index
G-space 40 G (')-space 69 G-symmetric linear relation 104 G-symmetry 101 (G1, G2)-adjoint 85 (G,, G2)-semi-unitary operator 35
(G,, G2)-unitary operator 135 Generalized linear-fractional transformation 176 Generalized resolvent of a J-symmetric operator 279 Gram operator 39 Hermitian form 1 Hermitian kernel 269 Hermitian non-negative kernel 234 Hermitian operator 104 Hermitian positive kernel 234 Hermitian symmetric 2 Hilbert-Schmidt operator 222 Hyper-maximal neutral lineal 28
J-non-contractive operator 117 J-non-negative 107 J-orthogonal complement 21 J-orthogonal projection 45 J-orthogonality 21 J-orthonormalized bases 72 J-orthonormalized system 74 J-positive 107 J-real part of an operator 96
J-space 20
J-spectral function with a set s(E) of critical points 210 J-symmetry 20 Ji-module 216
(Ji, Jz)-bi-extension of an operator 248 (J1, Jz)-bi-non-contractive operator 124 (J,, Jz)-isometric operator 140 (Ji, Jz)-non-expansive operator 129 (Ji, Jz)-semi-unitary operator 134 (Jl, Jz)-unitary dilatation of an operator 266
(J1, Jz)-unitary operator 141 Identical linear relation 143 Indecomposable lineal 34 Indefinite form 2 Indefinite lineal 3 Indefinite metric 2 Indefiniteness 85 Indefiniteness of a linear relation 143
Jordain chain of associated vectors 237
Kernel 85
Kernel of a linear relation 85 Krein space 14 Krein-Shmul'yan linear-fractional transformation 176
Index of a subspace T, 256 Index of a (D-operator 94 Inertia index 38 Intrinsic completeness 32 Intrinsic metric 31 Intrinsic norm 31 Invariant subset relative to a generalized linear-fractional transformation 176 Invariant subspace 158 Inverse Cayley-Neyman transformation
(t+, 'P_ )-decomposition of a subspace 73
Law of inertia 38 Lineal 2 Linear 1
Linear dimension 24 Linear relation 25 Linear space 1 Linearly ordered subset 7
144
U (.Y°) 51
Involution 19
Isometrical isomorphism 4 Isotropic lineal 5 Isotropic vector 5
J-accumulative linear operator 147 J-bi-extension of a J-Hermitian operator 264 J-complete J-orthonormalized system 74 J-accumulative linear operator 149 J-form 21 J-imaginary part of an operator 96 J-isometricity 20 J-metric 20
Maximal completely invariant subspace 166
Maximal dual pair 72 Maximal extension of a pair 72 Maximal invariant dual pair 166 Maximal invariant subspace 166 Maximal J-accumulative operator 147 Maximal J-orthonormalized system 74 Maximal negative lineal 6 Maximal neutral lineal 6 Maximal neutral subspace 28 Maximal non-degenerate lineal 6 Maximal positive lineal 6
Index
303
Maximal W-accumulative operator 147 Maximal W-dissipative operator 91 Maximal W-symmetric operator 104 Minus-operators 129
Q-orthogonal sets 5 Q-orthogonal vectors 5 (Q1, Q2)-isometric isomorphism 4 (Q,, Q2)-isometrically isomorphic spaces
Negative lineal 3 Negative vector 3 Neutral lineal 3 Neutral vector 3 Non-closed definite lineal 32 Non-decreasing linear operators 155 Non-degenerate lineal 5 Non-negative Hermitian kernel 242 Non-negative lineal 3 Non-negative vector 3 Non-positive lineal 3 Non-positive vector 3 Non-strict plus-operator 118 Normal eigenvalue 89 Normal point 89 Normalized system of vectors 76 Normally decomposable operator 139 Nuclear operator 222
(Qt, Q2)-skew-symmetric isomorphism 4 Quadratic bundle 238 'Quasi-inverse' operator 107
4
Orthogonal canonical orthoprojectors 21 Orthogonal sum Q+ 15 Orthonormalized basis 219 Ortho-projectors 18 P-basis 226 PG-transformation 246 Partial ordering by inclusion 7 Partially (J,, J2)-isometric operators 140 Plus-deficiency 122 Plus operator 117 Point of definite (positive or negative) type 210 Point of regular type 92 Point of spectrum of L(X) 237 Point spectrum vp 143 Point spectrum of a linear relation 143 Pontryagin space H, 64 Positive lineal 3 Positive vector 3 Potapov-Ginzburg transformation 245 Projectionally complete lineal 44 Property (D 165 Property 165 Property 1'J 165 Q-biothogonality 10 Q-metric 7 Q-orthogonal complement 5 Q-orthogonal direct sum 14
Range of indefiniteness 38 Range of values of a linear relation 85 Rank of indefiniteness of a subspace 38
Regular critical point of a J-spectral function 211 Regular definite lineal 34 Regular definite subspace 30 Regular dilatation of an operator 266 Regular extension 262 Regular G-metric 40 Regular generalized resolvent 279 Regular J-spectral function 211 Regular points of a linear relation 143 Regular point of L(X) 237 Regular subspace 46 Residual spectrum of a linear relation 144
Resolution of the identity 36 Resolvent set for a linear relation 143 Riesz basis 77 Riesz projector 97 Root lineal 88 Root vector 88
S-bounded operator 94 S-completely continuous operator 94 S-continuous operator 94 s-number of an operator 222 sp A 223 Y, 222 Scalarly commutative family 206 Selfadjoint operator 16 Semi-bounded below 107 Semi-definite lineal 24 Semi-definite spaces 24 Semi-definite subspace 24 Semi-linearity 2 Sesquilinear form 1 Set of critical points of an operator 211 Simple J-dissipative operator 224 Simple J-unitary operator 278 Single-valued linear relation 143 Singular critical point 211 Singular definite lineal 34 Singular G-metric 40
Index
304
Skew-symmetric isomorphism 4 Skewly linked lineals 10 Spectral function 36 Spectrum a
143
Stable J-unitary operator 137 Stable plus-operator 131 Standard operator 253 Strict plus-operator 118 Strictly dissipative kernel 234 Strictly J-dissipative operator 102 Strongly continuous on the right 36 Strongly damped bundle 241 Strongly stable J-unitary operator 139 Subspaces: dual pair 72 Subspaces: extension of a dual pair 72 Subspaces: maximal dual pair 72 Subspaces: maximal extension of a dual pair 72 Subspaces of the h` class 33 Sum of two linear relations 143 Symmetric operator 104 Trace of an operator 223 Unconditional basis 77 Uniformly definite lineal 30 Uniformly J-dissipative operator 102 Uniformly J-positive 102 Uniformly (J1, J2)-bi-expansive operator 126
Vectors associated with an eigenvector of L(X) 237 W-accumulative linear operator 147 W-accumulative linear relation 147 W-dissipative linear relation 147 W-dissipative operator 91 W-Hermitian operator 104 W-isometric linear relation 146 W-metric 39 W-non-contractive linear relation 146 W-non-expansive linear relation 146 W-space 39
W-symmetric linear relation 147 W-symmetric operator 104 W(')-space 69 (W,, W2)-isometric operator 134 (W1, W2)-non-contractive operator 117 (W,, W2)-semi-unitary operator 134 (W,, W2)-unitary operator 134
x-regular dilatation of an operator 266 x-regular extension 262 x-regular generalized resolvent 279 x-regular Xo-standard extension 264 Xo-standard set of closed J-Hermitian operators 264 a-orthogonal complement 67 a-non-negative kernel 242
Uniformly (J1, J2)-expansive operator 126
Uniformly negative lineal 31 Uniformly positive lineal 31
(D-operator 94 4 o-operator 94 (Do-point of an operator 94
