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0 any fixed number. Let E := {z
Eel
h n Ip(z)1 :::; (-) }. e
Then there are disks Bl. ... , B n , B;:={zllz-z;l:::;r;} such that
and
SECOND CHAPTER
34
Bound along a circle With help of Boutroux-Cartan lemma we consider bounding log+ 1!(z)1 pointwise in terms of T(r, I). Theorem 2.10 Let! be meromorphic in Izi < R and choose r such that ()r < R. Then there exists a radius p such that
1
r
- 1 and ()r < R we have
1 n(r, I) ~ log ()N(()r, I).
(2.40)
If f has a pole at the origin, then the inequality holds in the form
logr 1 n(r,l)+n(O,l)log() ~ log ()N(()r,l).
(2.41)
Proof We have N(()r, I) =
(or n(t, I) - n(O, I) dt + n(O, I) log(()r)
10
t
~[n(r, I) - n(O, fl
l r
Or
dt
t
+ n(O, I) log(()r) o
which gives (2.41).
Corollary 2.2 Let f be meromorphic in Izl < R such that f(O) #- 00. Choose () > 1, 0 < r < R such that ()r < R. Then there exist a constant C(()), only depending on (), and a radius p depending on f and satisfying that for all cp
./0
~
p
~
r, such
(2.42)
Proof We estimate ()+1 ()+1 ()+1 2 () _ 1 m(()r, I) ~ () _ 1T(()r, I) ~ () _ 1 T(() r, I) and, by Lemma 2.2 n(()r, I)
Replacing () by
()1/2
~ lo~()T(()2r, I).
Theorem 2.10 implies (2.42) for some p,
C(()) := JO + 1 JO - 1
./0 ~ p ~ r, with
+ log 4JO(JO + l)e_l_. JO - 1
log JO
o
SECOND CHAPTER
36
Representation theorems
We forinulate two theorems concerning the possibility of representing a meromorphic function f as a quotient of analytic functions fI! h such that the growth of Ii's are controlled. Definition 2.3 A function f, meromorphic in chamcteristic in Izl < R, if sup T(r, f) < 00.
Izl < R is said to be of bounded
r 1 there is a constant B(O) with the following property. If f is meromorphic for Izl < 00, then there are entire functions h, h such that f = hi h and for i = 1,2 we have for all r > 0
T(r, Ii) ::; B(O) T(Or, f). Proof This theorem is in [Mi], see [Ru] for an exposition.
o
Comment 2.1 Theorem 2.10 is taken from [Ya] (Lemma 4.2) where it is used in a discussion on an inequality of Chuang Chi-tai bounding T(r, f) in terms of T(r, /'). Comment 2.2 In addition to the Nevanlinna characteristic function T(r, f) there are other related characteristic functions in the literature. In particular when the values are considered as points in the Riemann sphere and the distances are measured accordingly, the theory gets a different, more geometric flavor.
THIRD CHAPTER Keywords: Subharmonic functions, vector valued analytic and meromorphic functions, matrix and operator valued meromorphic functions, finitely meromorphic. Analytic vector valued functions We shall next generalize the characteristic function T for operator valued meromorphic functions. The first concept is going to be denoted by Too and it is defined as such for Banach space valued functions; for operator valued functions we just use the operator norms. The discussion shall touch the properties of subharmonic functions, some of which we present below. Before that, however, let us recall what we mean by vector valued analytic and meromorphic functions. If J is defined in a domain n c C taking values in a Banach space X, then it is analytic if
(3.1) lim _l_[J(z) - J(zo)] z - Zo exists for all Zo E n. The limit of the difference quotient is in the norm topology. Furthermore, J is called meromorphic if apart from poles it is analytic and around any pole b there is a smallest positive integer m = m(b) such that z-zo
Z 1--+
(z - b)m J(z)
(3.2)
is analytic at b. It is a well known and important result that if the limits are assumed in the weak topology only, they actually exist in the norm topology as well, and so "weakly analytic are analytic".
Subharmonic functions It is an important starting point for our discussion that if J is analytic taking values in a Banach space, then the mapping
u :z
1--+
u(z)
:= log+
IIJ(z)1I
(3.3)
is subharmonic.
Definition 3.1 Let n be a domain of C. A function u from n to R U { -oo} is said to be subharmonic on n if it is upper semicontinuous and satisfies the mean inequality 1 111" u(zo + rei'P)dv; (3.4) u(.zo) :5 -2 7r
_11"
whenever the closed disc B(zo, r) is contained in n. Furthermore, it is harmonic if both u and -u are subharmonic. We recall that u is upper semicontinuous if for 37
THIRD CHAPTER
38
allzoEO
limsupu(z) $ u(zo). %-+Zo
We can now state the following result.
Theorem 3.1 Suppose I is analytic from a domain 0 to a Banach space X. Then the functions IIIII and log 11/11 are subharmonic in O. Proof Clearly IIIII is continuous when formula we have I I(zo) =-. 271"~
1
I
I d. Let a be a unit vector in this intersection. But then k+l
a E ker B and writing a =
E
j=l
vjavj we obtain k+l
IIA - BI12 ~ II(A - B)a11 2 = IIAal1 2 ~ LO"]lv;aI 2 ~ O"~+l j=l
o
completing the proof.
Remark If A is not a square matrix, then it can be augmented to be a square matrix by adding a suitable number of columns or rows consisting of zeros. Basic inequalities for singular values and eigenvalues
In the following we shall denote by Mm,n the space of complex matrices, consisting of n columns of length m. The following basic but as such a simple lemma can be proved using unitary invariance and the so called interlacing property for the singular values of submatrices. Lemma 4.1 Let C E Mm,n, Vk E Mm,k, W E Mn,k be given, where k min{m,n}, and Vk, Wk have orthonormal columns. Then
(a) O"j(V':CWk) ~ O"j(C) j = 1,2, ... ,k, (b) Idet Vk*CWkl ~ O"l(C) ... O"k(C).
and
~
52
FOURTH CHAPTER
o
Proof See [Ho-J2), Lemma 3.3.1.
Let Aj = Aj{A) denote the eigenvalues of A, O'j = O'j{A) singular values and recall that we number them in the order of decreasing absolute values. Theorem 4.5 (H. Weyl, 1949) If A E Md, then k
k
j=1
j=1
III Ajl :5 II O'j
(4.7)
for k = 1,2, ... ,d,
with equality for k = d.
Proof Let A =diag (AI, ... ,Ad)' By the Schur Decomposition Theorem there exists a unitary U and a strictly upper triangular N such that
A = U{A + N)U*. Let Uk E Md,k denote the k first columns of U. Then we have
A+N=U*AU= (Uk:Uk
~)
with some matrices E, F, G. Since A + N is upper triangular, F = 0 and Uk AUk is upper triangular. Now we apply Lemma 4.1 with C := A, Vk = Wk := Uk and conclude
k
k
k
j=1
j=1
j=1
III Ajl = IdetUkAUkl :5 II O'j(Uk AUk ):5 II O'j. When k
= d the SVD gives us
Idet AI = Idet U det E det V* I = det E
o
and the equality in (4. 7) follows.
If the singular values of A and B are known, what can be said about the singular values of AB? We formulate the answer in the square matrix case. For the general case, see [Ho-J2), Theorem 3.3.4.
Theorem 4.6 (A. Horn, 1950) If A, B E Md, then for k = 1,2, ... ,d k
k
j=1
j=1
II O'j(AB) :5 II O'j {A)O'j (B)
(4.8)
with equality for k = d.
Proof Let AB = YEW· be the SVD of AB and put Vk for the k first columns of V, and W k for those of W. Then
V: ABWk = diag{O'I{AB), ... ,00k{AB)).
(4.9)
Consider BWk E Md,k' It can be written, using polar decomposition, as
BWk=Uk R where Uk E Md,k has orthonormal columns and R E Mk is positive semidefinite satisfying
53
FOURTH CHAPTER
Then Lemma 4.1 gives k
det R2 = det(W; B* BWk) ~
II Uj (B* B). j=l
But uj(B* B) = Uj(B)2 and thus det R2 ~
k
Il uj(B)
2
. From (4.9) we obtain
j=l k
II uj(AB) =ldet(Vk' ABWk) I j=l
=ldet(Vk* AUkR) I =ldet(Vk' AUk)lldet RI· But by Lemma 4.1 Idet Vk' AUk I ~
k
Il uj(A)
and since detR ~
j=l
k
Il uj(B), j=l
(4.8) follows. For k = d there must be equality as Idet(AB) I = Idet Alldet BI and d
d
j=l
j=l
Il Uj = I Il Ajl.
D
The singular values of a sum of two square matrices can be easily estimated with help of Theorem 4.4.
Lemma 4.2 If A, B E M d , then for 1 ~ j, k
Proof Let A j -
17 Bk-l
~
d, j
+k
~
d + 1 we have
be as in Theorem 4.4. Then, since
rank(A + B) ~ rank(A)
+ rank(B) = j
- 1+k - 1 = j
+k -
2,
D
Definition 4.1 For A E Md put for k = 1,2, ... ,d k
IIIAlllk := L:uj(A). j=l
These are sometimes called Ky Fan-norms. For k = 1 we have the induced operator norm IIAII (spectral norm), and with k = d we have the trace norm, also denoted IIAI11' IIAlltr.
Theorem 4.7 For k = 1, ... ,d, III . III k is a submultiplicative norm in Md, i.e., IllABlllk ~ IllAlilk IIIBlllk.
FOURTH CHAPTER
54
o
Proof See e.g. [Ho-J2], section 3.4.
The total logarithmic size of a matrix
In the value distribution theory log+lfl separates the large values of If I from those of small ones. When looking at meromorphic functions F: z 1-+ F(z) E Md we need to be able to do the same thing. Definition 4.2 For A E Md, put d
s(A)
:=
L log+ uj(A). j=1
We may call it the total logarithmic size of A.
In order to study simple properties of s(A) we need the following simple technical tool. Lemma 4.3 Let a1 ~ a2 ~ ... such that for k = 1,2, ... ,d we have
~
ad
~
k
k
j=1
j=1
0, {31
~
{32
~
...
~
{3d
~
0 be given
II aj ::; II (3j. Then d
d
Llog+(aj)::; Llog+({3j).
j=1
(4.10)
j=1
Proof If a1 ::; 1, then (4.10) holds. Otherwise, if we put ad+1 := 0, then let 1::; m ::; d be such that am ~ 1 but a m+1 < 1. Then, with k := m, d
m
m
Llog+aj = Llogaj ::; Llog{3j. j=l
j=1
j=1
But m
d
m
Llog{3j ::; Llog+{3j ::; Llog+{3j,
j=1
j=1
j=1
o
and (4.10) follows.
Theorem 4.8 For A, B E Md we have
s(AB) ::; s(A)
+ s(B).
s(A + B) ::; 2(s(2A)
+ s(2B)).
(4.11)
(4.12)
FOURTH CHAPTER
55
Proof Put OJ := O"j(AB) and (3j := O"j (A)O"j (B). Then (4.8) allows us to apply Lemma 4.3 to conclude d
s(AB) ~ I)og+(O"j(A)O"j(B)). j=l But 10g+(O"j(A)O"j(B)) ~ log+ O"j(A) + log+ O"j(B) and (4.11) follows. To obtain (4.12) notice that for any nonnegative a, b we have 1 log+ '2(a + b) ~ log+ a + log+ b.
Since O"j(A) = !O"j(2A) we have by Lemma 4.2 1 0"2j-1(A+B) ~ '2(O"j(2A)+O"j(2B)).
Therefore log+ 0"2j-1(A + B) ~ log+ O"j(2A) + log+ O"j(2B) holds for and since 0"2j(A + B) ~ 0"2j-1(A + B) we obtain (4.12).
o
When we deal with large dimensional matrices or with operators in trace class we often want to write them as [ + A. Corollary 4.4 s(I + A + B) ~ 2(s([ + 2A) Proof Write
[+ A + B
= (![
+ s(I + 2B)).
+ A) + (![ + B) and use
(4.12).
o
Theorem 4.9 Let A be invertible. Then
s(A) = S(A-1) + log Idet AI. Proof We have by Theorem 4.5 d
II O"j = Idet AI
j=l and thus log
d
d
j=l
j=l
II O"j = I)og O"j =
d
d
1
j=l
j=l
3
L log+ O"j - L log+;-: = log Idet AI·
But ;. 's are the singular values of A-1 and so substituting J
d
1
s(A-1) = '"'log+~ 0". j=l 3 gives the result. It is convenient to put s(A -1) = 00 if A is not invertible.
o
FOURTH CHAPTER
56
Theorem 4.10 A is unitary if and only if s(A)
+ S(A-l) = o.
Proof If A is unitary, then the SVD is A = U with E = I, and s(A) = s(A -1) = O. Reversely, if s(A) + S(A-l) = 0 holds then uj(A) = 1 for all j and the SVD is A = UEV· = UV· as E = I. But UV· is unitary and we are done. 0
Some basic properties of the total logarithmic size We start by studying how the total logarithmic size behaves in similarity transformations. Let A = SBS- 1. Then by Theorem 4.6 k
k
j=1
j=1
IT uj(A) ::; IT Uj(S)Uj(B)Uj(S-I).
(4.13)
But Uj(S-I) = I/Ud-j+l(S) and if we define Kj(S) := Uj(S)/Ud-j+l(S),
then we obtain from (4.13) k
k
j=1
j=1
IT uj(A) ::; IT Uj(B)Kj(S).
Notice that Kl (S) = IISIlIIS- 1 11 is the condition number of S. Since Kj(S) ~ Kj+1 (S) we obtain using Lemma 4.3 d
d
~)og+uj(A) ::; I)og+(uj(B)Kj(S)) j=1
j=1 d
d
::; Llog+uj(B) j=1
+ Llog+Kj(S), j=1
so that s(A) ::; s(B)
+ c(S),
(4.14)
where c(S) is defined in (4.15). Definition 4.3 For invertible S put d
c(S):= Llog+Kj(S), j=1
the total (logarithmic) conditioning of S. Theorem 4.11 Let S, Rand T be invertible matrices. Then (b)
c(SR- 1) =0 if and only if SR- 1 is unitary c(SR- 1) =c(RS- 1),
(c)
c(ST- 1) ::;C(SR-l) + c(RT- 1).
(a)
(4.15)
FOURTH CHAPTER
57
Proof (a) c(SR- 1) ~ 0 always, but if c(SR- 1) = 0 then in particular = 1 and SR- 1 is unitary. Reversely, for a unitary SR- 1, O'j (SR- 1) = 1 for all j and c(SR- 1 ) = o. (b) is trivial from the definition. (c) follows from writing ST- 1 = SR- 1RT- 1 = (SR- 1)(RT- 1) and using /'i,1 (SR- 1)
c(AB) ~ c(A)
+ c(B),
(4.16)
which holds for any invertible matrices A, B. To obtain (4.16) notice that k
k
II /'i,j(S) = II
O'j(S) . j=IO'd-j+1(S)
j=1
We have
k
k
k
j=1 d
j=1 d
j=1 d
II O'j(AB) ~ II O'j(A) II O'j(B). .n
.n
.n
U.(~B) ~ u.tA) u}B)' Thus with m = d-k+l J=m ' J=m' J=m' we obtain by multiplying both sides But in a symmetric way,
k
k
j=1
j=1
II /'i,j(AB) ~ II /'i,j(A)/'i,j(B) which implies (4.16) with help of Lemma 4.3: d
d
I)og+/'i,j(AB) ~ L)og+(/'i,j(A)/'i,j(B)) j=1 j=1 d
d
~ L)og+/'i,j(A) j=1
+ I)og+/'i,j(B). j=1
o Corollary 4.5 If A, B are invertible, then (4.16) holds. When we think of SR- 1 as the similarity transformation which takes RAR- 1 into SAS-l = (SR- 1)(RAR- 1)(RS- 1) then we may think of c(SR- 1) as the "distance" between the similarity transformations Sand R. In particular, the following shows the continuity of the total logarithmic size s in similarity transformations. Theorem 4.12 If Sand R are invertible, then for all A Is(SAS- 1) - s(RAR- 1)1 ~ c(SR- 1).
Proof From (4.14) we have s(SAS- 1) ~ s(RAR-l)
+ c(SR- 1)
and likewise
s(RAR-l) ~ s(SAS- 1) + c(RS- 1). Since c(RS-l) = c(SR- 1), the claim follows.
o
FOURTH CHAPTER
58
We conclude that s(A) behaves in a natural and controlled way under similarity transformations. Next we ask, in what ways we can possibly estimate the norm IIAII and s(A) in terms of each others. Since O'l(A) = IIAII, we have trivially IIAII ~ exp (s(A)) and
s(A) ~ dlog+ IIAII. However, if we know the function s(zA), then the norm can be obtained accurately. In fact, 1 IIAII = sup{lzl I s(zA) = o}.
(4.17)
Let us now look at the power An and the exponential e zA . The behavior of s(An) is related to the spectral radius formula: lim IIAnIl 1/ n = p(A) = max{IAI I A E O'(A)}.
n-+oo
(4.18)
Let A = diag(A1(A), A1(A), ... ).
Theorem 4.13 We have lim .!.s(An) = s(A). n
n-+oo
(4.19)
Proof The claim follows from the following generalization of (4.18): O'j(An)l/n ~ IAj(A)I, which is due to Yamamoto (1967), and generalizes to compact operators, see [Ro], Proposition 2.d.6. 0
Theorem 4.14 For z E C, Izl = r, A E Md, s(e zA ) ~ r11A111'
(4.20)
d
where IIAI11 = ~ O'j(A). j=l
Proof Since ezA = lim(1 + ~A)n, we have s(e zA ) ~ liminf n s(1 + ~A). But O'j(1 + ~A) ~ 1 + ~O'j(A) by Theorem 4.4 and thus log+ O'j(1 + ~A) ~ ~O'j(A), and the claim follows. 0 We have observed that s behaves nicely when two matrices are multiplied together, but estimates for the sum are necessarily somewhat more complicated. Consider the sum of two matrices. If A = B = I, then s(A) = s(B) = 0, but s(21) = dlog 2. By Lemma 4.2 we have
59
FOURTH CHAPTER
and since 0"2k ::; 0"2k-1 we have d
Ld/2J
j=l
j=l
L log+O"j(A+B)::;2 L
10g+(O"j(A) +O"j(B))
::;2(s(A)
+ s(B)) + rank(B) log 2.
(4.21)
Another grouping of the indices in Lemma 4.2 is also useful. Theorem 4.15 For A, BE Md we have
s(A+B)::; 2(s(A) +s(B))
+ rank(B) log 2
(4.22)
and
s(A + B) ::; s(A)
+ s(B) + rank (B) (log+ IIAII + log 2).
(4.23)
Proof Inequality (4.22) is in (4.21) while (4.23) follows from
O"j(A + B) ::; IIAII
+ O"j(B),
j::; rank(B)
and from
O"j+k-1(A + B) ::; O"j(A) + O"k(B) = O"j(A), where k > rank(B). Thus Llog+ O"j(A + B) ::; rank(B)(log+ IIAII
+ log 2) + Llog+ O"j(B) + Llog+ O"j(A). D
Finally, if we add something small into A we do have an estimate without additional terms, so that we see the natural continuity. Continuity Lemma 4.4 If A, B E Md, then
Is(A) - s(B)1 ::; IIA - Bill.
(4.24)
Proof If a and b ~ 0, then 110g+(a) -log+(b)1 ::; la - bl· So, we have
Is(A) - s(B)1 ::;
L
Ilog+ O"j(A) -log+ O"j(B)1
::; L 100j(A) - O"j(B)I· The trace norm 11.111 has the following property. Form from the singular values two diagonal matrices E(A) and E(B) respectively, arranging the diagonals in the usual decreasing order. Then
IIE(A) - E(B)111 ::; IIA - Bill see [Ho-J1J, p. 448. This completes the proof.
D
FOURTH CHAPTER
60
Direct sum, Kronecker product and Hadamard product Given two matrices A E Md1 , B E M~, operating in C d1 , C d2 respectively, their direct sum A E9 B is the linear mapping in C d1 E9 C d2 which maps as (x, y) E C d1 E9 C d2 to (Ax, By) and can be represented with a block diagonal matrix
The singular values of AE9B E Mdl+d2 are clearly O'j(A), O'k(B), j = 1,2, ... ,d1, k = 1,2, ... , d 2 • Therefore we have the following result:
s(A E9 B) = s(A) + s(B).
(4.25)
The Kronecker product of two matrices A E Mm,n and B E Mp,q is denoted by A ® B and is given by
anB A®B=
a
(
1n
B) E Mmp,nq
:
:
am 1 B
amnB
where =
(a~1
a~n) ..
A.
.
.
am1
amn
Remark In this notation the rank-l matrix xy*, with x, y E Cd becomes xy* = x®y*.
In multilinear algebra it is customary to write x ® y for the bilinear mapping. Our notation here follows the matrix analysis tradition where x is thought of as a column vector and x* as a row vector. Lemma 4.5 (A ® B)(C ® D) = AC ® BD. Proof This is of course under the assumption that the dimensions match so that the ordinary products make sense. To prove this, split into blocks and multi~
0
Corollary 4.6 If A E Md 1 and B E (A ® B)-1 = A- 1 ® B-1. Remark In general A ® B prove it:
=f.
Md2
are invertible, then so is A ® Band
B ® A so observe the order: A-I ® B- 1 • To
(A ® B)(A- l ® B- 1 ) =AA- l ® BB- l = Idl ® Id2 =(A- 1 ® B- 1)(A ® B).
= Id1d2
FOURTH CHAPTER
61
We can now compute the spectrum of A ® B for A E Mdl' B E Md2. Let Ax = AX and By = f,Ly, where x, yare eigenvectors, and A, f,L eigenvalues. We have by Lemma 4.5 (A ® B)(x ® y) = (Ax ® By) = Af,LX ® Y so that Af,L E O'(A®B), and x®y E C d1d2 is a corresponding eigenvector. If we take all products Ajf,Lk with multiplicities when needed, we have obtained all eigenvalues of O'(A ® B). This is easiest to check using the Schur decomposition. In fact let L, M be upper triangular and U, V unitary so that
A = ULU*,
B = VMV*.
Then U ® V is unitary (by Corollary 4.5) and L ® M is upper triangular, with Ajf,Lk'S on the diagonal. The conclusion follows as by Lemma 4.5
(U ® V)* (A ® B)(U ® V) = L ® M. In order to obtain the singular values of A®B observe that since (A®B)* = A*®B*, we have (A ® B)*(A ® B) = (A* A) ® (B* B) and the previous result on eigenvalues implies that the singular values of A ® B are obtained as products of singular values of A and B:
where j
= 1, ... ,dl , k = 1, ...
,d2 • We obtain the following result.
Theorem 4.16 If A E Mdl' BE
then
Md2'
s(A ® B) ~ d2 s(A)
+ dl s(B).
(4.26)
Proof Consider the product of all singular values of A®B and replace O'i(A® B) with ui(A ® B) where, Q := max{a, I}. Then s(A ® B) is the logarithm of the new product. But ui(A ® B) = (O'j(:A);;;(B)) ~ Uj(A)Uk(B) and doing this for every o'i(A ® B) yields the following product: dl
d2
IT Uj(A)d IT uk(B)d 2
j=1 The logarithm of this is d2 s(A)
1•
k=1
o
+ dl s(B).
Given two matrices A,B E Mm,n with elements (aij), (bij ) respectively, their Hadamard product A 0 BE Mm,k is defined by
(A 0 B)ij = aijbij . This is sometimes called the Schur product or the entrywise product. An important and simple result related with this product is that the Hadamard product of positive semidefinite matrices is always positive semidefinite. If A, B E Md are positive definite and the eigenvalues are ordered decreasingly then k
k
j=1
j=1
IT Aj(A)Aj(B) ~ IT Aj(A
0
B),
for k = 1,2, ... , d
FOURTH CHAPTER
62
(see [Ho-Jl], p. 316). For our purposes the reverse inequalities are of interest. To that end, for any A, BE Mm,n, already Schur showed that
0'1 (A 0 B)
~
0'1 (A)O'l (B).
Let A E Md' Put r1(A) ~ r2(A) ~ ... ~ rd(A) for row sums as follows: (rk(A))2 d
is the kth largest number among L: laij 12, i = 1, ... ,d. Let Ck (A) be defined j=l similarly for column sums. Then the following holds: for A, B E Md: k
IT O'j(A
j=l
k
0
B) ~
IT cj(A)rj(B),
k = 1,2, ... ,d
j=l
[Ho-JI], p. 355). This allows an easy upper estimate for s(A Lemma 4.3
0
B). In fact, by
d
s(A 0 B) ~ I)og+[cj(A)rj(B)]. j=l
(4.27)
Comment 4.1 In the discussion above we have used [Ho-J2] as a basic reference. Comment 4.2 Notice that s(A) is not of the form log+ IIAII in any operator norm. In that sense it is really a different "tool". Many of its properties appear here first time. Comment 4.3 The total logarithmic size generalizes for bounded operators A in Hilbert spaces. In fact, let O'j(A) :=
inf
rank(B)<j
and then set
IIA -
BII,
00
s(A)
L)og+ O'j(A). j=l For example, with compact operators K we always have s(K) < 00, and if K is in the trace class, that is, IIKll1 := L:~1 O'j(K) < 00, then also s(1 - K) < 00. Many properties of the total logarithmic size can be proved simply by approximating techniques. We refer to [N08]. See also the subsection "Extension to trace class" in the next chapter. :=
FIFTH CHAPTER Keywords: Inversion identity, trace class, finitely trace class meromorphic, Schatten class. The total logarithmic size is subharmonic We shall consider here the subharmonicity of the total logarithmic size of an analytic Md-valued function. To that end we shall first consider a problem related to the eigenvalues instead of the singular values. We follow closely [A2]. Let {>.j(AHt denote the eigenvalues of A E Md, indexed so that
If F is now an analytic function in a domain n we know from Theorem 3.2 that log+ IA1(F(z)1 is subharmonic as IA1(A)1 gives the spectral radius of A. However, the corresponding function with the other eigenvalues need not be subharmonic.
Example 5.1 Let
F(Z)=(~ ~) so that the eigenvalues are {I + z, 1 - z}. Thus IA1(F(z))1 = max{11 + zl, 11- zl} while IA2(F(z))1 = min{11+zl, 11-zl} and we see that IA2(F(z))1 is not subharmonic as it violates the mean value property at the origin with a small enough radius. Lemma 5.1 Let F be analytic from a domain
n into Md.
Then the functions
k
Uk(Z)
:=
L log IAj(F(z))1
(5.1)
1
are subharmonic for k = 1,2, ... , d. Proof Fix Zo E n and choose an eigenvalue /-to E a(F(zo)) and take a small enough radius s > 0 such that the closed disc B(/-to,s) contains no other eigenvalue of F(zo). Then we can fix a small 8 > 0 such that for Iz - zol < 8 no eigenvalue of F(z) touches the circle {}B(/-to, s), which is possible as the eigenvalues are continuous and there is only a finite number of them. When counted with multiplicities, let rno denote the multiplicity of /-to, so that d-rno eigenvalues of F(z) stay outside of B(/-to, s). If rno = 1 then /-to(z) is analytic in z, but for rno > 1 it may happen that the eigenvalue is not analytic. In such a case the eigenvalue splits into several eigenvalues, say, into /-t1 (z), ... , /-tmo (z), each of which is analytic in a small punctured neighborhood of zoo Notice that some of these eigenvalues can be multiple copies 63
64
FIFTH CHAPTER
of each others, but then they stay as copies and each one is separately analytic. In any case, if one defines a function h around Zo by setting
and for 0
0 and an integer mj such that
bj(z) = cj(l + o(1))r2m; as Izl = r ---+ O. Consider now bl decreasingly we have
= L~ Aj. As the eigenvalues are numbered
which further implies Cl
-d
~
l'
1m
. f Al(Z)
III
-2- ~
z-+o r
ml
l'
1m sup
Al(Z)
- 2 - ~ Cl'
z-+o r
ml
FIFTH CHAPTER
67
For the coefficient b2 we have in the same way
),1),2::; b2::; (~),1),2 This implies
< l'lIDm . f
C2 -C1 (~) -
< lim sup
),2(Z)
z->O
r 2 (m2- m l) -
z->O
< -c2 d
),2(Z)
r 2 (m2- m d -
C1
Continuing this way we see that if ),j is not identically 0, then there exists constants aj > 0 and an integer kj such that
a~ < lim inf ),j(z) < lim sup ),j(z) < ~. z->O r 2k j
J -
-
z->o r2kj
-
a~ J
Taking the logarithm and dividing by 2 gives log aj ::; lim inf (log Uj (F(z)) z->O
1
+ k j log -r )
::; lim sup (loguj(F(z)) + k j z-+O
log~) r
::; log ~ aj
Since the eigenvalues were ordered decreasingly there is a largest J such that k j < 0 for j ::; J. Summing over j then gives J
0: ::;
lim inf ( " log+ uj(F(z)) + " k j log ~) z->O
~
~
r
j=1 J
::; lim sup (Llog+Uj(F(z)) 0:
- 'E.;=1
:=
r
j=1
Z-+O.
where
+ Lkjlog~)::;(3
'E.;=11ogaj and (3 := 'E.;=11og
;j'
Thus, in particular, J.L(O) .-
0
k j is an integer.
The proof actually gave somewhat more. Namely that limsup can be replaced by lim and that the limit process is controlled with bounds. Lemma 5.5 If F is as above, then .
1 ) p. (zo=lm
Z-+Zo
and there are constants 0: ::;
0:
s(F(z)) 1 log p-=r IZ-ZOI
(5.8)
and (3 such that
lim inf (s(I - F(z)) - J.L(zo) log I 1 I) z-+zo z - Zo
::; lim sup (s(I - F(z)) - J.L(zo) log I 1 Z-+Zo
z - Zo
I)::; (3.
Proof The inequalities are explicitly available in the previous proof and the limit in (5.8) is obtained by dividing the estimates by log(l/Iz - zol). 0 We shall need an auxiliary function.
FIFTH CHAPTER
68
Izl
0 we have
Too(r, (I - ZA)-1) $ crP + O(logr). Since A E Sp there exists an m, large enough so that 1
00
- L aj(A)P < c. p j=m+1 Then, however, we can proceed as follows:
Too(r, (1 - ZA)-1) $T1(r,I - zA) $ logM1 (r, I - zA) m
$ Llog(l + raj (A)) j=1 $O(logr) + crP •
1
00
p
j=m+1
+ -rP L
aj(A)P
Here we used the inequality 10g(1 + x) $ ~xP, valid for x > 0 and 0 < p $ 1. In the general case, let k be a positive integer such that k < p $ k + 1. Then in particular Ak+1 E S1, and in fact 00
00
Laj(Ak+1)m $ Laj(A)P, j=1 j=1 see e.g. Corollary II.4.2 in [Go-K]. We have, compare with Theorem 5.7,
Too(r, (I - ZA)-1) $ T 1(r, 1- Zk+1 Ak+1) + k 10g(1 + riIAII). Here we proceed as above and in particular use 10g(1 + r k+1aj(A k+1)) $ k + 1 rPaj(Ak+1)m p to split the sum at a proper place in order to have the growth again bounded by crP + O(logr). 0 Recall that in Example 1.5 we had a self adjoint operator A such that its eigenvalues were
Aj = -
(:jr
Thus w = 1/2, but A E Sp only for p > 1/2. Since V 2 is a rank-1 perturbation of A, the same applies to V 2 .
Powers and their resolvents The proof of Theorem 6.5 was based on
T 1(r, 1- Zk+1 Ak+1) = T 1(r, (I - Zk+1 Ak+1)-1), valid for k + 1 ::::: p. We shall next study the asymptotic behavior of kT1 (r, (I - zk Ak)-1) as k grows. Given a compact A, we denote by {Aj(A)} the sequence of its eigenvalues, indexed so that i>'1(A)1 ::::: IA2(A)1 ::::: ... and each eigenvalue repeated according
SIXTH CHAPTER
80
to the dimension of the corresponding eigenspace. If the operator has only a finite number of eigenvalues, then the sequence is continued by setting Aj{A) = 0 for the larger indeces.
Lemma 6.2 If A E SI, then 00
N l (r, (I - ZA)-I) = I)og+ IAj(A)rl.
(6.12)
j=1
Proof Choose r and take the Riesz spectral projection of A including all eigenvalues which are larger than, say, in modulus. This gives a finite rank operator A r . Then A - Ar can be approximated arbitrary well with another finite rank operator and this shows that N l (r, (I - zA)-I) only depends on A r . (Compare with the Continuity Lemma 4.4.) But since this is of finite rank, it is unitarily similar to a finite dimensional upper triangular (that is, a sum of diagonal and nilpotent) operator. But then the nilpotent part can be made arbitrarily small by another suitable similarity transformation and we conclude that Nl only depends on the eigenvalues. In fact, if S denotes a similarity transformation, and if B = SCS- l with d =dimB, then
r!1
s(B) :$ d log (lISIIIIS- l ll) + s(C) and therefore the multiplicity J.L{ Aj ~A)) is not affected by the similarity transforma&a 0 Observe that the right hand side of (6.12) makes sense for all compact operators as it is always a finite sum for any fixed r. We introduce the following notation. Given a sequence {Aj} converging to zero we set N(r,{Aj}):= I)og+ IAjrl· j
Now the following holds.
Theorem 6.6 Assume A E Sp with some p. Then lim -k1T1(r, (I - zk Ak)-I) = N(r, {Aj(A)}). k--+oo
(6.13)
Proof Recall that if A E Sp then A k E SI for k ~ p. Then for such k T l (r, (I - zk Ak)-I) and T l (r,! - zk Ak) are both well defined and equal. The proof is given by several simple lemmas, some of which have some independent interest. Lemma 6.3 If A is compact, then (6.14)
Proof of Lemma 6.3 We formulated this as Theorem 4.13 for Md. This version can be found as Proposition 2.d.6 in [8]. 0 0 The aim is to show that 1 k k kml(r,I-z A )-+N{r,{Aj(A)}).
(6.15)
SIXTH CHAPTER
81
This would imply (6.13) as Tl(r, (I - zk Ak)-I) = T l (r, I - zkA k )
and, trivially, Nl(r, 1- zk Ak) = o. We shall first reduce the claim to a finite dimensional problem. Since our basic claim is about a limit with a fixed r, we can without lack of generality set r = 1 in the following. Choose a small 0 < 0 < 1. Then take a spectral decomposition of A = Al EEl A2 as follows:
1. ( A2 := -2 'In
)"(>./ - A)-ld)".
11>\1=1-6
By the spectral radius formula we have for large enough n
~ 1- ~.
IIA2'II!.
Lemma 6.4 Assume that A E Sp and p(A) < 1. Then we have lim ml(1,I - zk Ak) = 0
k-+co
as k -
00.
Proof of Lemma 6.4 If p(A) < p < 1 then for large enough n we have IIAnl1 ~ pn. If also n ~ k where k such that Ak E Sl, then we can estimate as Izl = 1, which shows that
o
The claim follows.
Lemma 6.5 If A E SI and B is of finite mnk and they opemte in invariant subspaces H A, H B respectively with HAn H B = {O}, then s(I + (A EEl B)) ~ 8(1 + A) + s(I + B) + rank(B) (log(1 + IIAII) + log 2).
Proof of Lemma 6.5 This is clear by (4.23) and (4.25).
(6.16)
o
If A = Al EEl A2 as above and rankA I = d then Lemma 6.5 gives ml(1,I _znAn) ~ml(1, 1-
+d(log(1
zn Ai) + ml(1, 1- zn A2')
(6.17)
+ IIA2'11) + log 2).
This follows because An = Af EEl A2 allows US to apply Lemma 6.5 with _zn An in place of A. By Lemma 6.4 we have limn-+ co ml(1, 1- zn A 2 ) = 0 and since IIA211- 0, then inequality (6.17) implies lim sup .!.ml(1,I - znAn ) n-+co
n
~ limsup .!.ml(1,I n-+co
n
znAi).
82
SIXTH CHAPTER
What we need still to prove is the reverse inequality liminf !ml(l,I - zn Ar) n-+oo n
~ liminf !ml(l,I n-+oo n
zn An).
(6.18)
and that the limit exists and satisfies (6.19) Consider first (6.18). Let P denote the spectral projection: Al = PA. Then for ~ d we have aj(I + A 1 ) ~ 1lPllaj(I + A)
j
while for j > d we have aj(1 + A 1 ) = 1. Thus s(I + A 1 ) ~ s(1 + A) + dlog IIPII.
Applying this to _zn An in place of A gives (6.18). In order to prove (6.19) observe first that by construction Nl (1, (I - zAd- 1 = N(I, {Aj(A)}). And recall that we have set r = 1. For Izl = 1 we have -1
+ aj(An)
~
aj(1 - zn Ar)
~
1 + aj(An)
which implies, as Al is of rank d,
By Lemma 6.3 we know that
which proves (6.19). The proof of Theorem 6.6 is now completed.
D
We shall close this topic with similar results for Too. Here it is natural to look at general bounded operators in a Banach space X. Definition 6.1 Suppose A E B(X). We denote by Poo(A) the smallest radius such that (I - zA)-l is meromorphic for Izl < 1/ Poo(A). Theorem 6.1 If A E B(X), then (1 - ZA)-l and (I - zk Ak)-l are meromorphic in the same discs: Poo(A) = Poo(Ak)-k and
(6.20) while
SIXTH CHAPTER
83
Proof Write, with,pj := 2rrj/k,
(I - zk Ak) = (I - zA)(I - ei 1, for all 0 :s: c < "(. Clearly, we have C = 0 only when A = O. We close this topic by a consequence of resolvent being bounded in the unit disc, as in (6.28).
Theorem 6.11 If (6.29) then 00
B
:s: L IIAnl1 :s: 4B(1 + B).
(6.30)
i=1 Proof The idea ofthe proof is simple. Knowing the value of Moo(r, (I -zA)-1) allows us to use the estimate (6.27) with r > 1 such that B(r - 1) < 1. In fact, we obtain from 1- zA = z(I - A) - (z - 1)1 that -1 1 +B
Moo(r, (I - zA)
If we choose r := 1 +
2k we obtain
00
B =
):s: 1- (r-1 )B.
II L
00
Ai ll-1:S: L IIAili i=O i=1
00
:s: LMoo(r, (I -
zA)-1)r- i
:s: 4B(1 + B).
i=1
o What if small perturbation means small in norm One application of knowing the growth function Too (r, (I - zA) -1) is given in the following chapter: we show that there exists a sequence of monic polynomials {Pi} such that the decay of Ilpi (A) II is related to the growth of the resolvent. Above we saw that the speed of growth of Too(r, (I - ZA)-1) is robust in low rank perturbations of A. In practical computations we would however not use A itself, sayan integral operator representing the inverse of some differential operator but rather a discretization of it, say Ah. In such a case it is of interest to know what happens to the growth function, under the assumption that E := A - Ah is small in norm. The first observation is that knowing IIA - Ahll and Too(r, (I - zA)-1) alone does not imply much. In fact, Too carries no information on the dimensions of invariant subspaces related to the poles, and thus an arbitrarily small perturbation of A can split the pole into arbitrarily many poles. Thus in these terms, all we can say is that, if Too(r, (I - zA)-1) stays small for Izl :s: R o, so that we can conclude that (I - zA)-1 is actually analytic in that disc, then a simple perturbation result is possible, as a corollary of Theorem 6.9.
Corollary 6.2 If for r
:s: Ro,
then (I - ZA)-1 is analytic for r
:s:
(6.31) Ro and the following estimate holds (6.32)
SIXTH CHAPTER
86
for r ::; Ro where ~ is given in Theorem 6.10. If now
11(1 -
z(A + E))-lll
RoIIEII < ~,
then
::; ~ _ £IIEII'
(6.33)
In order to be able to estimate Too(r, (I - z(A + E))-l for larger r we must pose further restrictions on either A or on E. We shall assume that A is in the trace class.
Theorem 6.12 Assume that A E 8 1 and E E B(H). Then for
1
Too(r, (1 - z(A + E))- ) ::;
zA.
r11E11 < 1
rllEl1 rllAlh + (1 + rllAll1) 1 _ rlIEIl'
(6.34)
Proof This follows from Theorem 6.2 by choosing F(z) = 1- zE and G(z) = 0
Comment 6.1 Much of the material of this chapter is from [N08] and [N09]. About Theorem 6.5 there are early related results in the Russian literature, see e.g. [Ma] and the references given there. Comment 6.2 Theorem 6.11 is from [N09]. It would be interesting to know the exact constant(s) in (6.30), say in the form 00
L IIAil1 ::; aB + bB2. i=l
We know that this requires a
~
2, b ~ 4/9.
SEVENTH CHAPTER Keywords: Infinite products, quotient representations, spectral polynomials, Krylov solver, robust error bounds. Combining a scalar function with an operator In the following we consider functions fA which are obtained by combining a bounded operator A E B(X) with a scalar meromorphic function f as follows
fA: z For simplicity, we shall assume that at most with finite order p, that is,
1-+
f(zA) E B(X).
(7.1)
f is meromorphic in the whole plane and grows (7.2)
for all f > O. Likewise, we assume that A is almost algebraic (equivalent with assuming that the resolvent (I - zA)-l is meromorphic in the whole plane) and such that the resolvent grows at most with finite order w, i.e.
(7.3) for all f > O. For example, operators in Schatten class Sp grow at most with order p by Theorem 6.5. Now, assuming additionally that f is analytic at the origin, fA is analytic for small z. If either f is entire or A is quasinilpotent, then fA is actually entire. Otherwise singularities can occur but these are all poles. In general we can define fA as follows. We assume for simplicity that f is analytic at the origin. Thus it has an expansion 00
f(z) = I:ajz j j=O
for
Izl < Ro with some Ra :::; 00.
If p(A) denotes the spectral radius of A, then 00
fA(Z) = I:ajAjzj
(7.4)
j=O
converges for Izl < Rajp(A). Outside of this disc fA is then extended by meromorphic continuation. 87
88
SEVENTH CHAPTER
Theorem 7.1 Let 1 be a meromorphic function in the whole plane such that 1(0) i= 00, and such that it grows at most with finite order p. Let A E B(X) be an almost algebmic opemtor such that its resolvent grows at most with finite order w. Then IA in (7.4) is a well defined B(X)-valued meromorphic function in the whole plane such that it grows at most with order max{p, w}, that is,
(7.5) for all {3 > max{p, w}. Proof Let us consider first the cases in which with Theorem 2.1, that if
1A
is entire. Recall, compare
00
G(z) = L:Bjzj j=O
then G is entire of order at most w if and only if for all .lim l/(W+E) IIBj1l1/j = 3-+ 00
€
> 0
o.
(7.6)
If 1 is entire, then 1A is entire, too, and it follows from the inequality IlajAjIl1/j::; IIAlllajl1/j that the Taylor coefficients of IA cannot decay slower than those of I. Consequently the order cannot increase. Likewise, if A is quasinilpotent then limj-+oo IIAj Il1/ j = 0 and the fact that 1 is analytic near the origin guarantees that for some G we have lajl1/j ::; G. Thus now the coefficients can be estimated by
lI aj Aj ll1/ j ::; GIIAj I11/j and the decay is now dominated by the decay of the coefficients in the resolvent. Again we have an entire function with order at most that of the resolvent. Fix now {3 >max{p, w}. If 1 has only a finite number of poles, then form a polynomial n
p(z) :=
II (1- z/bj )
j=1 so that pi is entire and hence of order less than (3. Estimating the inverse of p( zA) is easy: n
Too(r,p(zA)-1) ::;L:Too(r,(I - :.A)-1) j=1 J n
=
L: O(r/lbjl).B) = O(r.B).
(7.7)
j=1
By construction pf is entire and of order less than (3. Thus
Too(r,IA) ::;Too(r,p(zA)-1) + Too(r,p(zA)/(zA)) ::;O(r.B) + O(r.B). What remains is the general case of 1 having infinitely many poles. Let us denote these by {bj }, ordered so that Ibjl ::; Ibj+11 and each pole is repeated as many times as the multiplicity requires. Without loss of generality we can assume that {3 is not
SEVENTH CHAPTER
89
an integer and that it is close enough to max{p, w} so that they have a common integer part: (7.8) m ~ max{p,w} < f3 < m+ 1. We shall form an entire function
"1 = 211AII with II(>..! - A)-III ~ 1/IIAII.
0
Lemma 7.2 For m < f3 < m + 1 there exists 0(3 such that log IIE(zA, m)1I ~ 0(3 IIAII(3r(3
holds lor all A
(7.10)
and all r > O.
E B(X)
Proof of Lemma 7.2 The proof is divided into two parts, depending whether 211 A II r is smaller or larger than 1. Assume that 211Allr ~ 1. Then we can denote by F the function
F(z) = 10g(E(zA, m)) = -
~
L...J
j=m+l
1
..
-:-A3 Z3
J
which is analytic in this disc. Clearly for these values IIF(z)1I
~
E ~IIAlljrj ~ j=m+l
2l1All m +l r m+1
~ 2I1 AII(3r(3.
J
But E = eF so that IIEII ::; e llFll and thus
10gIlE(zA,m)lI::; IIF(z)1I ::;2I1AII(3r(3 which is of the form required. Assume then that 211Allr ~ 1. Here we base the estimation on Lemma 7.1 and on the fact that the claim holds in the scalar case. In fact, we have log IE(z, m)1 ::; c(3r(3
(7.11)
where c(3 = ~ for m = 0 and c(3 ::; e (log(f3 + 1) + 1) otherwise, see (5.6.13) and (5.6.16) in [NOl].
SEVENTH CHAPTER
90
We now apply Lemma 7.1 to the function E{z, m) and obtain
Since
211Allr 2: 1 we have log IIE{zA, m)11 SCi32i311Alli3ri3 + log 2
S{ci3 + log 2) 2i3l1Alli3ri3.
o
The proof of Lemma 7.2 is thus completed.
:;t
We shall now start estimating
1 we set
C2 := 2(
va;:- + 1)/( va;:- - 1)2.
If A E 8 1 and E E B(H) are given, then there are monic polynomials {pj} depending on A, E and C 1 , satisfying for j ~ C211AlldllEII
IIpj(A + E) II
~ eC2 (C1C2~AIl1er
(7.34)
and for j ~ C211AlldllEII
Ilpj(A + E) II ~ eC2 (HIIAlh/Il E IJ)(CIIIEll)j .
(7.35)
Proof Here the polynomials Pj are not obtained from just one function X but rather from a sequence of such functions, X7)' with help of Lemma 7.5 as in the proof of Corollary 7.1. By Theorem 6.12 we have for rllEIl < 1 1 rllEIl Too(r, (I - z(A + E))- ) ~ rllAll1 + (1 + r1lA1l1) 1 _ rllEIi'
(6.34)
Let C1 > 1 be given and choose () > 1 such that ()2 = C1 • Then for "I ~ 1/(()IIEII) we may assume X7) given so that (7.22) holds. But then for r ~ "I we have
Too(r'X7)(z)(I - z(A + E))-l) ~ 2Too (TJ, (I - z(A + E))-l)
(7.36)
and further ()+1 log+ Moo (TJ/(), X7)(z)(I - z(A + E))-l) ~ () _ 1 2 Too ("I, (I - z(A + E))-l)
2(() + 1) ~ (() _ 1)2 (1 + ()TJIIAlld Choosing "I = 1/()IIEII we obtain
IIpj(A + E) II
~ exp (~~()-+1~~ (1 + IIAlldIlEII)) (()2I1EII)j
which holds for all j. For short, put c := 2(() + 1)/(() - 1)2. Then with j < ciiAlldllEIl we have TJj := j/(c()IIAI11) ~ 1/()IIEII and we obtain
IIpj(A + E) II
~e
C
(c()2 11:11 Ie
r.
o Robust bounds for Krylov solvers Krylov subspace methods is a class of iterative methods for solving linear systemsof equations. Among them conjugate gradient method is widely used for positive definite problems, while GMRES and QMR are examples of methods suitable for general nonsingular problems. A typical step of such an iterative method involves applying a matrix to a vector and doing linear algebra operations in the low
99
SEVENTH CHAPTER
dimensional subspace created. The methods are often used with preconditioning. For example, suppose we have a nonsingular problem in the form
Bx=c. If we additionally have an approximate inverse for B, Le. we have an M such that M B = I - A with I - A invertible and A "small" , then we can write the equation equivalently in the form (7.37) x=Ax+b where b = M c. Often the preconditioner is not given explicitly but requires running a short subroutine. One special property of good Krylov methods is the following. If A is small except possibly in a low dimensional subspace, then the methods converge rapidly. Traditionally the convergence analysis in the case of conjugate gradient method has been based on approximation theory on the spectrum - a technique which cannot be used for highly nonnormal problems. Our analysis covers both cases simultaneously. In fact, a low rank perturbation of A may change the operator from self adjoint to highly nonnormal and in such a case one would have to change the method e.g. from conjugate gradient method to GMRES, but the error bounds remain essentially unchanged. Practical computations in the low dimensional subspaces created assume inner product structure. Our bounds, however, are based on spectral polynomials: they are upper bounds for the best polynomials and they can be formulated in general Banach spaces. We outline now our setting. Given a bounded operator A and a vector b we may create the sequence {Ajb}~o. If 1 ¢ a(A), we can ask for approximations to the solution of (7.37) from the subspaces
Kk(A,b):= span{Ajb}J:J. There are a lot of different methods for different kind of problems which associate an approximation Xk E Kk(A, b) for (7.37). These typically aim to satisfy
Ilxk -
AXk -
bll :::; Ily -
Ay -
bll
(7.38)
for all y E Kk(A, b). We shall give a bound assuming that (7.38) holds exactly.
Lemma 7.6 If x satisfies (7.37), then
IIx - yll :::; 11(1 - A)-llilly -
Ay -
bll.
(7.39)
Proof The claim follows from
(x - Ax - b) - (y - Ay - b) = (I - A)(x - y) = -(y - Ay - b).
o Note that any vector y in Kk(A,b) can be written in the form y = qk-I(A)b with some polynomial qk-l and that all vectors of this form are in Kk(A, b). It then follows from (7.38) that if Pk-l is any polynomial of degree k - 1 and we set Yk := Pk-I(A)b then necessarily
IIx -
xkll
:::;11(1 - A)-IIlIIYk - AYk - bll :::;11(1 - A)-IIiIII - AIIII(I - A)-lb - Pk-I(A)bll :::;11(1 - A)-III III - AIIII(1 - A)-l - Pk-I(A)lllIbll
SEVENTH CHAPTER
100
We conclude that if we can give an estimate for Ek := inf 11(1 - A)-l - p(A) II
where the infimum (actually minimum) is over all polynomials p of degree less than k, then (7.40) IIx - xkll ~ Ek ll(1 - A)-III 111 - Alillbil where Xk satisfies (7.38).
Theorem 7.4 Assume A is almost algebmic and {ai} is a sequence such that for all j = 1,2, ... IIpi(A)1I ~ Co
+
( c ew)i/W
(7.41)
holds where Pi (A) = Ai + alAi - l + ... + ai' Then X(z) = 1 + alZ + a2z2 entire. Assume also that 1 - A is nonsingular and that X(l) -:f O. Then
+ ...
is
(7.42)
Proof We have 1
00
.
(1 - zA)-l = - () Lpi(A)zJ X z i=O and so Ek
~11(1 1
k-l A)-l -
xt1)
~Pi(A)1I
00
~ IX(l)IL IIPi(A) II J=k
o
which implies the estimate (7.42).
We call this error bound "robust" as it has the following property: we have shown above that there are spectral sequences whose decay is bounded by the growth of Too(r, (1 - zA)-l). Then we have shown that this growth is insensitive in low rank updatings. Thus the only part in the error bound which is obtained by combining (7.40) and (7.42) which is not robust is in the term 11(1 - A)-lll/lx(l)l. Of course it may happen that some low rank updating brings a problem nearly singular, and then this would be large.
A bound for spectral projectors In the first chapter we discussed shortly Riesz projections: 1 . [ (AI - A)-IdA p = -2 1n
lr
(7.43)
where r surrounds an eigenvalue of A. Here we consider the following situation. We ask whether it is possible to give a bound for such a projection in terms of the growth function of the resolvent. To that end, let A be a bounded operator in
101
SEVENTH CHAPTER
a Banach space X and assume that the resolvent (1 - zA)-1 is meromorphic for Izl < R ~ 00. Choose any radius r < Rand () > 1 such that ()r < R. Then we take
(7.44) where p =
1/s satisfies
r
v'o ~ s ~ r
(7.45)
and is such that I (>.1 - A) -111 can be controlled along r in terms of its characteristic function Too (()r, (1 - zA)-I).
Theorem 7.5 Given () > 1 there is
C(()) < v'o + 1 + 10 4ev'o( v'o + 1) - v'o-1 g v'o-1
(7.46)
such that the following holds. Let A be a bounded linear operator in a Banach space such that the resolvent (1 - ZA)-1 is meromorphic for Izl < R ~ 00. Then for any r such that ()r < R there exists an s satisfying (7.45) so that for cp E (-7r, 7r]
(7.47) Proof Observe that the claim is essentially the same as in Corollary 2.2. So is the proof, too. However, we have here an operator valued function and therefore u:
z ~ log (
Ip(z)III(1 -
zA)-III)
is only subharmonic; here again we denote by P the monic polynomial vanishing at Zj = l/bj for poles bj with Ibjl 2: 1/()r. Thus we cannot use Poisson-Jensen formula as in the proof of Theorem 2.10 but we get the exactly same inequality by arguing as follows. Since u is subharmonic it stays below the harmonic function h S h(se'°t ) := 21 111" P( -() , t - cp)u(()re'°t )dt
_11"
7r
r
and so we obtain ° 1 111" P( -() S log 11(1 - se"P A)- 1 II ~, t - cp) log 11(1 - ()re'°t A)- 1 Iidt
27r
_11"
~l
(()r)2 - bk seicp og ()r(seicp - bk)
+~
r
k=1
The rest is then identical to that of the scalar case.
o
This is related to projections onto invariant subspaces as follows. Assume p is such that (7.48) a(A) nrp = 0 and denote 1 (7.49) Pp = ~ (>.1 - A) -1 d)". 7rZ
1 rp
Then Pp projects onto the invariant subspace corresponding to the part of spectrum which is smaller than p in modulus.
SEVENTH CHAPTER
102
Corollary 7.5 Given 0 > I there is C(O) satisfying (7.46) such that the following holds. Let A be a bounded linear operator in a Banach space such that the resolvent (I - zA)-l is meromorphic for Izl < R ~ 00. Then for any r > 0 such that Or < R there exists p such that I r
VB
~p~-,
r
(7.48) holds and
log IIPpl1 ~ C(O) Trx;;(Or, (I - ZA)-l). Furthermore, the number of eigenvalues outside r p is bounded by
(7.50)
I
noo(r, (1 - ZA)-l) ~ logO Too (Or, (I - ZA)-l). Proof This is clear by Theorem 7.5.
D
Comment 7.1 Almost algebraic operators were discussed in [NOI]. Bounds for Krylov solvers, based directly on Too(r, (I - ZA)-l), were discussed in [Hy-N], without help of theorem of Miles.
EIGHTH CHAPTER Keywords: Approximate polynomial degree, approximate rational degree. Approximate polynomial degree of an analytic function If p is a polynomial of degree = d, then
log+ M(r,p)
= (1 + o(l))d log r
as r
--+ 00
and reversely, if log+ M(r, 1)/ logr is bounded as r --+ 00 then I is a polynomial. Suppose we look at a given analytic I in a tiny neighborhood Izl :$ r. Then obviously, just one evaluation of I, e.g. at origin is sufficient to approximately represent I. In a larger disc one needs more evaluations. Likewise, we may want to know, how the work increases with increasing accuracy. This is achieved simply by looking at 1/c in place of I. We shall make this precise by introducing the following notation and terminology.
Definition 8.1 Let
I
be analytic for
Izl < Ro :$ 00.
Put for r < Ro
do(r, 1) := min{deg pip is a polynomial and such that M(r, 1- p) :$ I}.
We shall call do the approximate polynomial degree. We can relate do to M as follows.
Theorem 8.1 Suppose ()r
I
is analytic lor
Izl < Ro
:$
00.
Then lor () > 1 and
< Ro we have 1 + (M(()r, 1)) do(r,1) < log () log () _ 1
+ 1,
(8.1)
and
log+ M(()r, 1) :$log+ M(r, 1) + do (()r, 1) log() + log 2.
(8.2)
Proof Let d be an integer such that 1 + M 1 + M log()log () -1 :$ d < log()log () -1
+ 1,
(8.3)
where for short M = M(()r, 1). Then ()d =
M
M
exp(dlog()) ;::: exp(1og+ () _ 1) = max{() _ l' I}. d
Let us now put p(z) := the coefficients satisfy
E
akzk where ak's are the Taylor coefficients of
k=O
103
I.
Since
104
EIGHTH CHAPTER
see (2.10), we obtain with the help of (8.3) 00
2:
M(r,f-p)~
JakJr k
k=d+l 00 O-d ~ M ' " O-k = M L...J 0-1
k=d+l
1 0-1 ~ M 0 -1 min{ 1\{' 1}
~ min{l, 0 ~ 1 } which implies the first claim. In order to prove (8.2) observe that if M(Or, f -p) with deg p = d, then
M(Or, f)
~
1
M(Or,p) + 1 ~ Od M(r,p) + 1 ~
~ Od(M(r, f)
+ 1) + 1
which implies the second claim. Here we used the inequality
M(Or,p) ~ OdM(r,p) which is a special case of Bernstein's lemma and can in this form be concluded as follows. The function g(z) := z-dp(z) is analytic and bounded for JzJ ~ r. By the maximum principle we have
(Or)-dM(Or,p)
=
sup Jg(z)J ~ sup Jg(z)J Izl~(lr
Izl~r
= r-dM(r,p).
o Our main interest in formulating Theorem 8.1 is the fact that we shall later be able to formulate an analogue of it for meromorphic functions, approximated by rational functions. However, in that case we cannot in general code the growth in terms of the Taylor coefficients. The following examples illustrate the inequalities (8.1) and (8.2). Example 8.1 If p is a polynomial of degree d, then clearly do(r,p) ~ d for all r with equality for all r large enough. Consider first (8.1) with a fixed r:
do(
)
_1_1
+ (M(Or,f)) 0_ 1
r, p < log 0 og
Letting here 0 -+
00
+
1 = _1_1
log 0 og
+ ((Or)d(1+0(1))) 0- 1
gives
do(r,p) < d + 1 or, as both are integers,
do(r,p)
~ d.
On the other hand, from (8.2) we obtain, for r > 1
do(r,p) logr ~ log+ M(r,p) - C with C = log+ M(l,p)
+ log 2.
0
+
1
105
EIGHTH CHAPTER
While log+ M(r, f) is bounded for an entire f only when f is constant, do(r, f) is bounded for polynomials and thus for very slowly growing functions do(r, f) is essentially slower than log+ M(r, f). It is then not without interest that for entire functions of positive order, do(r, f) grows with the same speed as log+ M(r, f) (without any logr term) and that also the type can be correctly recovered from do(r,f).
Example 8.2 In order to see that do(r, f) codes both the order and type faithfully, it essentially suffices to consider the function
We show the following:
do(r, F) = (1 + o(l))'Tewrw,
as r
-+ 00. d
Let do be fixed and put for short d := do(r, F). If P(z) =
E Cjzj
is the corre-
j=O
sponding approximating polynomial, then the Parseval's identity gives us d
1 ~ M(r,F - p)2 ~
00
L laj - cjl2r2j + L j=O
lakl2r2k
k=d+1
~ lad+11 2 r 2 (d+1).
This gives us immediately
do(r, F)
~
'Tewrw - 1,
for all r > O.
To bound do(r, F) from above, we use inequality (8.1). By Theorem 2.2 we have for 1/2 ~ c > 0 and r > 0 13 log+ M(r, F) :5 (1 + e)'Trw + log+( -w). c Inequality (8.1) now implies with ():= exp(l/w) that there exists Ce such that
o
holds for all r > O.
Theorem 8.2 If f is entire of order w, then log do(r, f) . w = 11m sup 1 . r-+oo
ogr
(8.4)
If f is of finite positive order w, and of type 'T, then 1 l' do(r, f) 'T = 1m sup . ew r-+oo rW
(8.5)
EIGHTH CHAPTER
106
8.2.
Proof We leave this as an exercise: try to modify the discussion in Example D Some properties of the approximate polynomial degree
If we want to approximate f within tolerance e, i.e. that M(r, f - p) ~ e, then the minimum degree possible is given by do(r, fie). From (8.1) we obtain with () > 1, r fixed such that ()r < Ro,
1 1 1 do(r, - I) ~ - 1 ()log+ - + C e og e
(8.6)
where C = C(r, (), I) is independent of c. Observe that, apart from C, the right hand side of (8.6) depends on f only through (). We can relate these notions to the standard setting in approximation theory. To that end, let Ed(r, 1):= inf M(r, f - p). deg(p)~d
It is well known that if f is analytic in a slightly larger disc, then Ed(r, I) decays fast with increasing d, see e.g. [Wa], p. 75. Here is a simple version with explicit constants. Theorem 8.3 Assume f is analytic in Izl < such ()r < Ro. Then for d = 0,1,2, ... we have E ( f) d
r,
< M(()r, I) -
() -1
Ro ()-d
~ 00.
Choose r
1, (8.7)
.
Proof Let () := M~9r~j)' so that by Theorem 8.1 for e E (0,1] we have () 1 1 do(r, - I) < - 1() log - + 1. e og e Denote
cd
(8.8)
:= ()-d. Then by (8.8)
and D
In the following we formulate some simple inequalities for do(r, I). Expressions of the form do(r, c(r)1) are to be understood as follows: we look at functions z 1-+ c(r)f(z) with fixed r for Izl ~ r and consider c(r) as a constant in the approximation process. Theorem 8.4 Let f and g be analytic for Then
do(r, f + g) do(r, fg) where cf
:=
Izl < Ro
~ 00
~
max{do(r, 21), do(r,2g)}, ~ do(r, cgl) + do(r, cfg),
max{3M(r, I), va}, cg := max{3M(r, g), va}.
and ()r
1. (8.9) (8.10)
EIGHTH CHAPTER
107
Proof If M(r,21 -p) ~ 1, M(r,2g-q) ~ 1, then M(r, I +g- !(P+q» ~ 1, while 1 deg 2(P+q) ~ max{deg(p), deg(q)}. To prove (8.10), suppose M(r, 1- p)
~
1/cg and M(r, 9 - q) ~ 1/cf. Then
M(r,/g - pq) M(r,/g - Iq) + M(r, Iq - pq) ~ M(r, f)M(r,g - q) + M(r,g)M(r, 1- p) + M(r, 1- p)M(r,g - q) 111 ~"3 +"3 +"3 = 1, ~
while deg(pq)
~
deg(p) + deg(q).
0
Theorem 8.5 Let I be analytic lor Izl < Then do(r,J')
Ro
~ 00
and () > 1 such that (}r < Ro.
~ max{do((}r, (() ~ l)rf) -
1, O}
(8.11)
and do(r,f) ~ do(r,rl')
+ 1.
(8.12)
Proof Differentiating the Cauchy integral
r
I(z) = ~
I(() d(
2n i1z- 0 while
_
1
T(r, (z - a) n) = n log Tal for 0 < r :5 1 -Ial. Thus, in general we cannot bound d in terms of T alone. Since trivially d(r, f) 2': n(r, f) we may try to bound d in terms of both T and n. Theorem 8.8 For () > 1 let ()i > 1 be such that ()1 ()2()3 = (). Suppose J is meromorphic Jor Izl < R:5 00. Then Jor ()r < R we have
d(r, f) < C1 «())T«()r, f)
+ 2n«()r, f) + C2 «()).
where
C1 «()) = _1_()2 + 1 [2 + 10g()1 ()2 - 1
and
+ 1]
()3 ()3 -
1
1 ( + 1 ()2 + 1 ) C2 «()) = - 1 () log ( - ( )1) + -()1 10g2 + 1. Ogl
1-
2-
(8.21)
EIGHTH CHAPTER
110
Proof Let () > 1 be given. Choose (}i > 1 such that (}l (}2(}3 = () and assume that r is such that 1] := (}r < R. To start, let q be a finite Blaschke product such that it is analytic in Izl < 1], vanishes at the poles bj of f in that disc so that g:= qf
is analytic there and Iql = 1 along Izl = 1] and of minimal degree. Thus deg(q) n(1], f) and T(1], q) = O. By Theorem 2.4 we have 1
1
T(1], -) = log - I1 = N(1], f)· q Ck
~
(8.22)
In fact, if f is regular at origin, then this is part of Lemma 7.5, since then
q(O) =
IIn .J..b j=l 1]
so that 1 ~ + 1] log Iq(O)1 = {:rlog Ibjl = N(1],f).
f
if, on the other hand, Ie q = ~q. So, writing
has a pole at origin of degree k, then q is of the form
q(O)
q () z =
-kz 1]
k
+ Ck+l Z k+l + ...
and using (2.28) we again get (8.22). Put p := (}1(}2r and consider the NevanlinnaPick interpolation problem: Find a w, analytic in Izl
~ p
such that
w(bj ) = g(bj ) for j = 1, ... , n(p, f) and such that M(p, w) is minimal (with natural modifications if some poles are multiple). It is well known that the solution is unique and that the solution is a rational function of degree at most n(p, f). Furthermore
M(p,w)
~
M(p, g)
(8.23)
since 9 itself is a feasible function. By construction
w
1 = -(g - w) q q
f - is analytic for Izl ~ p =
(}1(}2r
log+ M((}l r ,f _~) q
~
~ ~
and we can estimate it pointwise as follows:
+ 1 T(p,/ _
(}2 (}2-
1
+ 11
(}f)2 2-
(}2 + 1 01 2 -
w) q
(T(p, ~ ) + T(p, 9 - w)) q
( N(1], f)
+ T(p, g) + T(p, w) + log 2) . (8.24)
111
EIGHTH CHAPTER
Since T(T/, q) = 0 we have
T(p,g) S T(T/, g) S T(T/,j), while by (8.23) we obtain
T(p, w) S log+ M(p, w) S log+ M(p, g) S :: ~~ T(T/, j). Substituting these into (8.24) gives log+ M((}1 r ,I - w) S (}(}2 + 1 ((2+ (}(}3 + 1) T(T/,j) + log 2 ). q 2- 1 3- 1
(8.25)
We have now an analytic function I - w / q which we shall still approximate with a polynomial p. By Theorem 8.1 we have wI + wI + 1 do(r, 1- -) < - 1 () log M((}1 r,/ - -) + - 1 () log ( - ( )1) + 1. q og 1 q og 1 1-
Thus we have approximated
I by a rational function
(8.26)
w / q + P and we have
w d(r, j) S degq + degw + degp S 2n(T/, j) + do(r, 1- -). q
o
This completes the proof.
Corollary 8.1 For () > 1 let (}i > 1 be such that (}1(}2(}3(}4 = (). Assume I is meromorphic lor Izl < R S 00 and 1(0) =1= 00 • Then lor (}r < R inequality (8.20) holds with
and C2((})=-1 1() (log+(-() 11)+(}(}2+111og2)+1. og1 12-
Proof In (8.21) we can now estimate
o Example 8.6 Let us apply these bounds for rational functions. First, if T(r, q) satisfies a lower bound T(r,q) ~ d log+ r - C (8.27) then we have from Theorem 8.7
d logr S T(l, q) + d(r, q) logr + 21og2 + C which gives immediately liminf d(r, q) r ...... oo
~
d.
(8.28)
Reversely, suppose that T(r, q) satisfies an upper bound
T(r,q) S d log+ r + C.
(8.29)
EIGHTH CHAPTER
112
Of course, we can conclude from this that q is a rational function of degree at most d, but we look what the bound in Theorem 8.8 gives. First, the bound contains the term 2n(()r,q). For r ~ 1 we have, see Lemma 2.2, 1
n(r,q)::; 10g()N(()r,q) and so we obtain, by using (8.29) and letting ()
-+ 00,
n(r, q) ::; d. We then show that inf {C1 (())T(()r,q)+C2 (())} ::;3d+1.
8>1
(8.30)
Thus, combined we have
d(r, q) ::; 5d + 1. To obtain (8.30) choose t::
(8.31)
> 0 and take ()2 and ()3 large enough so that C1 (())
::;
3 + t:: .
log ()1
Then (8.29) gives limsup C1 (())T(()r,q)::; (3 + t::)d. 8 1 -+00 But limsupC2 (()) 81 -+00
::;
1
and (8.30) follows.
Example 8.7 We saw earlier that do codes both the order and type of entire functions accurately. For meromorphic functions the approximate rational degree codes the order accurately but leaves a gap for the type. In fact, suppose T(r, J) grows with a positive order w and with a positive type a. Then we obtain from Theorem 8.7
. d(r, J) 11m sup - W -
(8.32) ~ aew, r-+oo r analogously to (8.5). To get an upper bound for limsuPr-+oo d(r, J)/r w we use Theorem 8.8. We take ()1 = e, ()2 = ()3 = 2 so that () = 4e. Thus
C1 (())T(()r, J) ::; 15 (4e)WarW Further, since n(()r, J) ::; T(()er, J) for r lim sup r-+oo
d(r~J) r
+ o(rW).
> 1, see Lemma 2.2, we conclude
::; (15 + 2eW)(4e)W a.
Spijker's lemma Polynomials satisfy Bernstein's inequality
rM(r,p') ::; deg(p)M(r,p) which we used to get Theorem 8.6
rM(r,f')::; (do(r,r!')
+ I)(M(r,J) + 1) + 1.
(8.33)
EIGHTH CHAPTER
For meromorphic by M. Spijker.
113
I an analogous result can be obtained from the following lemma
Lemma 8.1 II w is a rational/unction, then (8.34)
o
Proof The original is in [Sp].
Theorem 8.9 II I is meromorphic in Izl < Rand 0> 1 is such that Or < R, then
rl11"· .) -11" 1!,(re''P)ldcp :5 d(Or, f) ( s~p I/(re''P) I + 1 + 0 _1 1·
211"
(8.35)
Proof Suppose deg(w) = d(Or,f) and M(Or,J - w) :5 1. Then using (8.34) we obtain r 111" Iw'(rei'P)ldcp -r 111" 1/'(rei'P)ldcp:5211" -11" 211" -11" + ~ 111" 1!,(rei'P) - w'(rei'P)ldcp 211" -11"
:5d(Or, f) sup Iw(rei'P) I + rM(r,!, - Wi)
'P
:5d(Or, f)(s~p I/(rei'P) I + 0 ~ 1 M(Or, 1- w)). To have the last inequality we used the Cauchy inequality
rM(r, I'
- Wi) :5 0 ~ 1 M(Or, 1- w).
o Remark 8.1 Observe that if I is rational then for € > 0 we can apply ( 8.35) to ~I and recover (8.34) as d(Or, ~f) :5 deg(f). This scaling technique can also be used in the following result.
Theorem 8.10 Let
I be meromorphic in Izl < R such that it is analytic in
Izl < Ro, and has there the expansion 00
I(z) =
I>k Zk . k=O
Then lor r < Ro we have (8.36)
EIGHTH CHAPTER
114
Proof Let w be rational, of degree d(r, f) such that M(r, f - w) :::; 1. Then 1. ( 1. { Ck = -2 Ck-1w(()d( + -2 Ck-1(f(() - w(())d( 7rZ J1(I=r 7rZ J1(I=r where by partial integration and Spijker's Lemma
111
12 . 7rZ
while
Ck-1w(()d(1 :::; -k deg(w)r- kM(r, w)
1(I=r
I~ (
Ck-1(J(() - w(())d(1 :::; r-k. J1(I=r The claim then follows as M(r, w) :::; M(r, f) + 1. 27rZ
o
Theorem 8.12 below contains a different variant of this mechanism.
Power bounded operators and bounds for the Laurent coefficients Let A be a bounded operator in a Banach space. It is a very basic task to give conditions on the resolvent that guarantee power boundedness
IIAnl1 :::; K
for n = 0,1,2, ...
(8.37)
(see Prologue). A necessary condition is obtained easily from (8.37). In fact, for Izl < 1 we obtain
This is often called the Kreiss resolvent condition and we may write it here as follows:
K
M(r, (1 - ZA)-l) :::; - - for r < 1. (8.38) 1-r This does allow a linear growth IIAnll = O(n). We shall assume that the resolvent is additionally meromorphic in some neighborhood of the unit disc. Together with this the Kreiss condition is sufficient. We make this quantitative by assuming for some () > 1 and L = L(()) < 00 (8.39)
Theorem 8.11 For each () > 1 there are constants C i (()), i = 1,2,3 such that if the resolvent is meromorphic for Izl :::; () and the conditions (8.38) and (8.39) hold, then for n = 1,2, ... (8.40)
Proof This is a special case of the following result on Laurent coefficients of meromorphic operator valued functions. 0
EIGHTH CHAPTER
115
Theorem 8.12 For each () > 1 there are constants Ci (()), i = 1, 2, 3 such that the following holds. Assume that F is a B(X)-valued junction, meromorphic for Izl < () and analytic for 0 < Izl < 1, satisfying the quantitative estimates limsup (1 -lzI)IIF(z)1I ~ K 1%1-->1-
(8.41)
sup Tco(r,F) ~ L.
(8.42)
and r 0 for some a, then we call A defective, otherwise it is nondefective. Theorem 10.11 Let 0 ::f. A E Md. Then for a ::f. 1
"Yoo(a) = "Y00(0).
132
TENTH CHAPTER
Proof Write F(z) - aI = (1- a)(I - l':aA), so that
moo(r, (F(z) - aI)-I) = moo ( 11: ai' F- 1 ) + 0(1).
(10.27)
From the proof of Theorem 10.4 we know that with some integer k we have moo(r, F- 1 ) = k log+ r
+ 0(1)
which substituted into (10.27) gives moo(r, (F(z) - aI)-I) = moo(r,F(z)-I) + 0(1). Hence 'Yoo(a) is independent of a.
D
Corollary 10.1 A E Md is defective, in the sense of Definition 10.4, if and only if 0 is a defective eigenvalue. Proof This follows from 'Y00(0) = 600 (0).
D
Example 10.5 The operator V 2 is quasinilpotent and thus (I - zV2)-1 is entire. Clearly all quasinilpotent operators are defective as 'Y00(0) = 1. Recall that V 2 is a rank-l perturbation of a self-adjoint operator A (see Example 1.5). Theorem 10.12 If A is almost algebraic and defective, then the set of values a for which 'Yoo (a) > 0 contains a circle. Proof If a::j:. 1 is such a value, then set p:= 11 - al. Writing as in (10.27) we see that all b's on the circle p = 11 - bl satisfy 'Yoo(b) = 'Yoo(a) > O. D
We end this with a natural statement on diagonalizable operators. Theorem 10.13 Let A be an almost algebraic operator of at most finite order in a Hilbert space. If it is similar to a normal operator, then it is nondefective. Proof Clearly defectiveness is preserved under similarity transformations. Assume thus that A is normal. Now the statement is essentially that of Theorem 6.4, except that we need to check all a ::j:. 1. Comparing with the proof of Theorem 6.4 we see that moo(r, F- 1 ) = o (log Too (er, F- 1 )). Using again (10.27) yields the result. D Comment 10.1 Some recent developments in the value distribution theory, in particular for holomorphic curves and quasiregular maps, are summarized in [Er].
EPILOGUE Keyplaces: Toronto, Karjalohja. Lecturing and typing in Toronto During October 2001 I gave ten lectures at Fields Institute in Toronto. Each lecture formed the basis of a chapter in this book. My former book [N01] was intended to be an easy-to-read text book on waveform relaxation - but it transformed into a difficult-to-read research monograph on convergence theory for iterative methods in an abstract setting. Likewise, this book was intended to be an easy-to-read text book on matrix valued meromorphic functions - but it transformed into an extended version of [N08] instead, where we alternate between matrix- and operator-valued functions. To be exact, one chapter contains material from two lectures, and the tenth chapter is an exceptional chapter, or simply defective one, written later as a partial and simple minded answer to a natural question: the word defective appears both in the value distribution theory and in linear algebra, so, are they related? Fishing and finishing in Karjalohja A year later I am finishing this monograph at our summer home in Karjalohja. My grandfather worked winters in an insurance company in Helsinki but spent his summers here doing mathematics. During the winters he and Rolf Nevanlinna had their offices close by in Helsinki - at times they even shared an office. Rolf's summer home was across the lake - the ride on a motor boat, at six knots, took twenty minutes. OlIi Lehto has written (in Finnish) a biography on Rolf Nevanlinna which appeared in the fall 2001 when I got back from Toronto. There is an interesting section on the birth of Nevanlinna theory with a discussion on the mutual relations between the two brothers. My grandfather kept diary all his adult life. Unfortunately, the diaries from years around 1925 are missing, but Lehto's book includes diary quotations from later years. I decided to include the Prologue in this book for several reasons. One is this: if a Nevanlinna writes about Nevanlinna theory three quarters of a century after its birth, some explanation is wanted, and I had already published a version of the Prologue in Finnish. During the seven or so years on this project many people have been of great help. I want to thank them all, but especially my hosts and the personnel at the Fields Institute and Bob, Carl, Jarmo, Marja, Marko, Nikolai, Olli-Pekka, Saara, Ulla, Timo and Xiaoushu. 133
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It would be only natural to dedicate this book to the brothers Frithiof and Rolf. However, I dedicate this to my father. He lost his elder brother in the war and was wounded himself. I grew up in the independent Finland. When we go fishing we use a rowboat, and the lake remains quiet.
BIBLIOGRAPHY [AI] [A2] [Bo] [Dr] [Du-S] [Er]
[F] [Go-K]
[H] [Ha-K] [Ho-J1] [Ho-J2] [Hu-N] [Hy] [Hy-N] [Ko] [Ma] [Mi] [Mii] [NF] [N01] [N02] [N03] [N04] [N05]
[N06] [N07] [N08]
Aupetit, B. [1991] A Primer on Spectral Theory, Springer-Verlag, New York. Aupetit, B. [1997] On log-subharmonicity of singular values of matrices, Studia Mathematica 122(2), 195-200. Boas Jr., R. P. [1054] Entire Functions, Academic Press. Drasin, D. [1987] ProoLoJ..aS!!!}jecture of F. Nevanlinna concerning junctions which have deficiency sum ~~Mat:h:~-Q4. Dunford, N an«SchwiiJ:tz', 'J:;.'t'z.[19p3j zan"/lr Operators, Part II: Spectral Theory, Interscience. ." ~. . Eremenko,~. [2002) Va(~~.·R¥.~butio_n'~nd ~otential Theory, Proceedings of ICM 2002, Vol II, Hig~ lM!Jcation ,PresSi.:i>p",~681-:69Q; Faddeeva, V. ~959tComp~~ti"na!.~ods of Linear Algebra, Dover. Gohberg, 1. and KrlJioB;.;~LU.~.l1J,..1ntt"oduction to the Theory of Linear Nonselfadjoint Operators, AMS Translations of Mathematical Monographs, Vol. 18. Halmos, P. R. [1971] Capacity in Banach algebras, Indiana Univ. Math. 20, 255-863. Hayman, W. K. and Kennedy, P. B. [1976] Subharmonic Functions, Vol. I., Academic Press. Horn, R. A. and Johnson, C. R. [1985] Matrix Analysis, Cambridge Univ. Press. Horn, R. A. and Johnson, C. R. [1991] Topics in Matrix Analysis, Cambridge Univ. Press. Huhtanen, M. and Nevanlinna, O. [2000] Minimal decompositions and iterative methods, Numer. Math. 86(2), 257-282. Hyvonen, S. [1997/98] Case studies on growth properties of meromorphic resolvents, Insitute Mittag-Leffler Report No. 18. Hyvonen, S. and Nevanlinna, O. [2000] Robust bounds for Krylov methods, BIT 40(2), 267-290. Konig, H. [1986] Eigenvalue Distribution of Compact Operators, Birkhiiuser. Matsaev, V. 1. [1964] Doklady Akademii Nauk SSSR 154/5, 1034-1037. Miles, J. [1972] Quotient representations of meromorphic junctions, J. Analyse Math. 25, 371-388. Miiller, V. [1987] On quasialgebraic operators in Banach spaces, Operator Theory 17, 291-300. Nevanlinna, F. [1930] Uber eine Klasse meromorpher Funktionen, Den syvende skandinaviske matematikerkongress i Oslo 19-22 August 1929, A. W. Broggers, Oslo. Nevanlinna, O. [1993] Convergence of Iterations for Linear Equations, Birkhiiuser. Nevanlinna, O. [1991] Tiede 2000, vol. 3, s. 51. Nevanlinna, O. [1996] Meromorphic resolvents and power bounded operators, BIT 36(3), 531-54l. Nevanlinna, O. [1996] Convergence of Krylov methods for sums of two operators, BIT 36(4), 775-785. Nevanlinna, O. [1996] A characteristic junction for matrix valued meromorphic junctions, XVIth Rolf Nevanlinna Colloquium, Eds. Laine/Martio, Walter de Gruyter & Co, Berlin, pp. 171-179. Nevanlinna, O. [1998] Juhlien jalkeen, Arkhimedes 3, 12-19. Nevanlinna, O. [1997] On the growth of the resolvent operators for power bounded operators, Banach Center Publications 38,247-264. Nevanlinna, O. [2000] Growth of operator valued meromorphic junctions, Ann. Acad. Sci. Fenn. Math. 25, 3-30. 135
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[N09] [NRl] [NR2] [Ri-V]
[Ru] [Se] [Sp]
[Wa] [Ya]
BIBLIOGRAPHY
Nevanlinna, O. [2001] Resolvent conditions and powers of operators, Studia Mathematica 145(2), 113-134. Nevanlinna, R. [1925] Zur Theone der meromorphen Funktionen, Acta Math. 46, 1-99. Nevanlinna, R. [1970] Analytic Functions, Springer-Verlag. Ribaric, M. and Vidav, I. [1969] Analytic properties of the inverse A(z)-l of an analytic linear operator valued function A(z), Arch. Rational Mech. Anal. 32, 298-310. Rubel, L. A. [1996] Entire and Meromorphic Functions, Springer-Verlag, Universitext. Segal, S. [1996] Nine Introductions in Complex Analysis, North-Holland Publ. Co.. Spijker, M. N. [1991] On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31, 551-555. Walsh, J. L. [1935] Interpolation and approximation by rational functions in the complex domain, AMS Colloquium Publications, vol. XX. Yang, L. Value Distribution Theory" Springer-Verlag, Berlin, and Science Press, Beijing.
Titles in This Series 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4
3
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Olavi Nevanlinna, Meromorphic functions and linear algebra, 2003 Vitaly I. Vol08hin, Coloring mixed hypergraphs: theory, algorithms and applications, 2002 Neal Madras, Lectures on Monte Carlo Methods, 2002 Bradd Hart and Matthew Valeriote, Editors, Lectures on algebraic model theory, 2002 Frank den Hollander, Large deviations, 2000 B. V. Rajarama Bhat, George A. Elliott, and Peter A. Fillmore, Editors, Lectures in operator theory, 2000 Salma Kuhlmann, Ordered exponential fields, 2000 Tibor Krisztin, Hans-Otto Walther, and .Jianhong Wu, Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback, 1999 .Jiff Patera, Editor, Quasicrystals and discrete geometry, 1998 Paul Sellck, Introduction to homotopy theory, 1997 Terry A. Loring, Lifting solutions to perturbing problems in C·-algebras, 1997 S. O. Kochman, Bordism, stable homotopy and Adams spectral sequences, 1996 Kenneth R. Davidson, C*-Algebras by example, 1996 A. Weiss, Multiplicative Galois module structure, 1996 Gt§rard Besson, .Joachim Lohkamp, Pierre Pansu, and Peter Petersen Mir08lav Lovric, Maung Min-Oo, and McKenzie Y.-K. Wang, Editors, Riemannian geometry, 1996 Albrecht Bottcher, Aad DlJksma and Heinz Langer, Michael A. Dritschel and .James Rovnyak, and M. A. Kaashoek Peter Lancaster, Editor, Lectures on operator theory and its applications, 1996 Victor P. Snaith, Galois module structure, 1994 Stephen Wiggins, Global dynamics, phase space transport, orbits homoclinic to resonances, and applications, 1993
