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such that 1-1= HI.!) , where Hq> ! def P_ q> f , f e. H~,
HA-PLITZ OPERATORS
and P-
H~
•
89
stands for the orthogonal projection onto Moreover, I ~I=(H~ HIjI) (I", In §3 relations of the spectral theory of Toeplitz operators to some geometric characteristics are exposed (angles between model subspaces, free interpolation, geometric properties of systems of exponentials and of reproducing kernels of model spaces). In §4 some very recent results on the problem of spectral multiplicities for Toeplitz operator are discussed. I am indebted to Prof. S.Khrllshchev for valuable language consultations •
••• We conclude this Introduction with a list of formulae linking Hankel and Toepli tz operators with each other as well as with some other operators. Despite their simplicity most of the formulae will be very useful in what follows. The first four are immediate consequences of the definitions of H~ and
'rljl • 11411" ,
(jl E
L00.
for 'I171E: Hoo,
(j>eLoo •
for ~
feH. (4) ieL00,
91
HA-PLITZ OPERATORS
(5)
H~ H~ =
l'KfI"I, y* +
VH~ H141 V*
for u,e Loo ,
\ u\ = 1 a.e.; here T\.I. = Y I 'T\.(. I is the standard polar decomposition of ~1V (V is a partial isometry) and PE stands for the orthogonal projection onto E , Ec. H~. (Hruacev, Peller; see [24J). Proof. H~~ u = I -Tu, Tu =1- YI rtvl~V*
= 1- Vy*+ Y(I - ITu, I~) y* = = I - yy* + YH~ Hu V*"= = PK~~V* -t- YH~ ~ u, y*. • (6) Ql-Isf=f(M Q ) Pe for ;t€H co
,
=
G being an
inner function; here Ps= PKe , where Ke =def ~ ~ e l'IHt\:1 is a model space and Me is a model operator (the compression of the shift onto Ke ), MQ ~ = PG~~, q,e 1(9 . Proof. Pa=9P- G on H~ and ;f(Me)~=P9;f9on Ka ( f ( Me) being defined for polynomials as usual and extended by continuity to all e H00 ) ,from which the desired formula follows. •
s
(7) Q1 JTe g
1 g,
J G"'"
Ii:
on Qt, = H~ (j) Q1' G~ ; here J;f= Proof. For f
6
Hq.
=
KQ~ 1,r,
IH~ ffi (p" -
I KlI
,,~I7!k
)
for every inner functions ;fe L~(,1I').
E9 K9~ ,
e1Jrp~f~ JQ~f= e1JP+QfQ~J9~.f= GfJ~Jelg~9,.5 = 91 ~Q1 f .•
=
92
N. K. NIKOL'SKII
for 19, \jI e
Leo
and 5-, ~ e H~
(A.L.Volberg).
Proof. Multiply by 1."", 146 Z ,and integrate over 'lP the left and the right hand part of the identity. •
1.
SINGULAR NUMBERS
The singular numbers ~)t. , a ~ 0 space operator A are defined by ~n. (A)
=
in! { II A- F' I : f
bounded, rank
Ii' ~
11,
is linear,
j . ~
The following properties of and easy to prove. (a)
of a Hilbert
~1'l1 ( A) , , ~o ( A) =
"
-numbers are classical
A1/
.
(b) t1.m~"\1, (A) = 0 iff A is a compact opera11< tor (i n s y m b 0 1 s : A€ f'00 ). (c) ~"" (A> is the 14-th point from the right of the spectrum of the modulus I AI = (A * A) 1/" (eigenvalues are counted with multiplicities, the first point of the continuous spectrum t>eo ( A) is counted as an eigenvalue of infinite multiplicity). (d) If Ce'r'eo t then C=E ~l'\,(C) (.,X""')1/n. ( { 11.~ on the circle 'll' (1 n s y m b 0 1 s q>eLoo<E.,-E:I,» such that 1-1 = H
£-, 5e. Ht( E1 ) Moreover,
II H~II
=
cli~t (ql,
•
H E1 - - E?v») . CO (
A matrix analogue of AAK theorem (dim E.«oo, dim Eg, < 00 ) is contained in a very' interesting paper by J.Ball and W.Helton [4J. The most general case has recently been proved by S.R.Treil (Leningrad), [)5]. To begin with, we state a generalization of the Kroneoker theorem. Theorem (Treil. 1984). Let ~eL co( E,,-E t ) , Ek (K = 1, t ) be Hilbert spaces. Then H
Bote, that in the scalar case (dim E~= d~mE~ (Kl
we have Cr., kH- ..r. = C~, k", and the general formula gives rank Hill = ~ k1\. = dfi P_ 4> , as it is prescribed by the Kronecker theorem. Now, put for the convenience d,e.g.. 'V = "f..tltl1!k H'" for every --+- E~) -valued rational function '" and denote by 1?w the set of all \II I S with deg l/f ~ 11, • 4A.
(s
Ek
Theorem (Treil. 1984). Let 4 e Lco (E,,- E~ ) , (k = 1,2) be Hilbert spaces. Then
~11r1\-( = (J,111. ....1 i ",-t-1 + ... with 0.."'+1*0 t m, ~ 0 • The above identities yield
for 14?; 0 wi th I e,h\+1
t
K~
I=
,and hence HlI' 1 = ~tK+1 l. tIII+ •.. I Q.m .....1 I • Putting / ( = m we obtain 0
and the result follows. Now, let us turn to some recent results concerning Questions 1 and 2 (see D6] for details). Theorem (S. Treil). Let {:JIi tll } \1. ~ 1 be a bounded sequence of distinct positive numbers. Then there exists a Hankel operator HIf such that
H
0 is an arbitrary non-increasing sequence of non-negative numbers then there ensts a Hankel operator \-I ~ wi th ~lII j({TJ}~ at'S- J-,1 tI.+1 wi th
Ae
=
"PK
er) +
:It +1.+ 1 (1;-
t "+1)
~-l AI'" l. - A ).
/\ \ \)'1"'"
Al1. 1 ,and let I-{ = ~ ~"
tl+1
= HflQ,n.(~K ~
k-1
I
.6
0"
is minimal under the
Blaschke condition ~ (1-1 A\) . .: :.
00 ).
•
AE6'"
Corollary 6. codim span
=d,,:w{, Ken,treB ' = Ke iff reB
(k e (.(eJ H~-------" L~(O,oo) ea.H~--------"'''~ ~1.Q~ H~---------l~-L ?(a,oo)
K~ H~ 9Q, Ht_____ • ei~x, GQ,/~ _ _ _ _ _ _--'lI~- Lteo, (1,)
pQ. -----------------~,.,- (.f-t(o,a) f)
(up to unessential constants;
kQQ (A,
.)
,.,
Thus, ru transforms corollaries 3-6 following propositions.
1.16 B
1-(0,0)
e, iJloc,
into the
is left invertible iff there exists a constan% c ~ 0 such that Jool :f/?"~ c for 15-/1"
te
~p'lUt. L~.Il(,U5,,/lt. Ae6'
A... f'
.hH'
118
N. K. NIKOL'SKII
for some (i, C.,. 0 and for any finite numerical family to.,>. J • It is easy to see that the family fi' , being a basis, is minimal, and even uniformly minimal; the latter means
It is not hard to check that for a family of repro. -1 ducing kernels f,). = ( 1- A~) ,A € 6" , we have
Corollary 7. If t(1_X~,-f : basis in its span Kg ,then
Ae6" I is a Riesz
( (C) => Riesz basis property) is also true, but this is a much deeper result which is essentially due to L.Carleson. There are some useful and transparent reformulations of the Carleson condition (C) in terms of the distribution of ~ in the disc; see [17] for a discussion. Thus, the problem of exponential bases { e iJA:X- : ,ftA(,ii,'t"} :Ln L~(O, (0) is solved. However, the situation is not so simple for L~ spaces on finite In fact, the converse conclusion
119
HA-PLITZ OPERATORS
intervals of the real axis. W10g we can deal with t LeO, Q,) , (;t > 0 • So, the problems are: to describe all the Riesz bases of exponentials t tt~~ ~ JA e ~ J , 'r c t in the whole L~ ( 0, at) as well as in span Lt(o,Q.) (~ip~ ~; e:. '!;')
•
The method of
Toep1itz operators described below works only in the case of the frequencies 'r contained in a ha1fplane Jm ~ 7 C ,or ~m % 0 . Theorem 8. Let 9 be an inner function and 6"c /D • If the family { ke (ft., ) ..:. Ae ()' } is a Riesz basis in its span, then 1(1- A1, )-f : ,.\ e 6' J is a Riesz basis too (or, equivalently, the condition (0) is satisfied). Referring for the proof to [171 we give here an easy and short reasoning which is sufficient for our main particular case 9 = 9Q,' £t > o. Namely, under the additional hypothesis ~~
Ae
~
I E) ( }.,) I
-AI)~ P~(A)=dimKtJL('r4'-AI). This pair of functions can playa role of·the local multiplicity (as a similarity invariant of an operator) on some classes of ~tf 's. Indeed, one of the theorems of [7] says that if'
'\It.
2t1t
, =
•
Vr.p=
1 ,and hence
YI.p (lK)
(tv ~
~
; really, here
has a constant valency (i.e. card tp-1(A) ===. const, Aetp( /D) ) and
~ t follows from the possibili ty to write 41 as a cemposi tiol! o , lPo being a conformal mapping of ID onto Jl..o • Then, for every
a.n.
a.u.
pe .fA
J I po lPol1"I~I~= J IPI"w~wo" CJ J I pl1. w "-w , 'Il' 8.n.. aJL and hence for every cy~ ct0-6 L'-('lUdw) 9"'A
~G H'" • Putting tJ,= (1 - tori ~o E.ILo we can see that O, where for each t X t is a random variable w + X t (w) definea on a probability space (~,d)p) and with values in ~n. Many important properties of the process {X t } (often denoted simply by X t ) are given by the finite-dimensional distributions: lltl, ... ,tk(G!X ... x Gk) = P[X tl
c Gl,
... , X tk E Gkl
(1.1)
where ti E:. [0,00) and G.are Borel sets in ~n. Conversely, it is a classical resulf due to Kolmogorov that given a family Vt , ..• ,tk of probability measures on ~nk (for all tl, ... ,tk; k=l,2, ... ) satisfying certain consistency conditions there exists a stochastic process {Y t } whose finite-dimensional distributions coincide with v t 1, ... , tk' This may be used to define n-dimensional Brownian motion as follows: p(t,y)
Let If 0 by
0; Bt f;:. U}
first exit time from U for Brownian motion B. Let t continuous bounded function on dU, the boundary of U.
0.6)
Then f is harmonic in U (this is a consequence of the strong Markov property, see for example [22]) and if au is "reasonable" (e.g. C l ) then lim f(x)
Hy)
for all
y
tau.
(1. 7)
x~y
XEU
-
Note that this formula for f gives the harmonic measure Ax for U is given by
in
particular
that
142
B.0KSENDAL
= PX[B,U
Ax(F)
t
(1. 8)
F] for Borel subsets F of dUo
Next we briefly recall the basic properties of Ito integrals (for details see for example [22]) • For the moment let Bt denote I-dimensional Brownian motion and let I t be the a-algebra generated by { Bs; s < t} for o < t 0
Then we can define for each t
v(t,w) is !t-measurable
> O.
the Ito integral
6t V(s,w)dBs(oo) as a limit in L2(px) of sums ~ ei(w)~B.(w), where 1
1
- Bt . • w -+ ei( w) is !t.-measurable, consfruction is based on the Ito isometry:
ti
(1.19)
EX[]- v 2 (s,w)ds]
EX[(] v(x,w)dB s (w»2]
o
o
and the formula t
EX[(f v(s,w)dBs(w)] = 0
o
(1.10)
for all t.
Similarly one can define Ito integrals in higher dimensions: t fv(s,w)dB s
o
v
t m.nxm
when Bt is m-dimensional Brownian motion. A stochastic integral is a stochastic process form
Xt = Xo + where Xt , XO, u t
-b
t bt u(s,w)ds + 6 v(s,w)dB s
t m.n , v
c
Xt
on
the
(1.11)
m.nxm and u(s,w) is.1Jx 1-measurable,
u(s,w) ds < 00 a.s. An abbreviated version of (1.11) is (1. 12)
STOCHASTIC PROCESSES, INFINITESIMAL GENERA TORS AND FUNCTION THEORY
The Ito formula states that if in ]Rn and g t. CZ(]Rn, ]Rm) then Yt integral, given by
Xt =
143
is a stochastic integral g(X t ) is again a stochastic
(1. 13)
y(k) . h were t 1S coor d'1nate k
by the formulas dt'dB
(i) = t
. dB(i) = dt. i, and dB(i) t t We now turn to the Levy theorem. This result was first stated by P. Levy in 1948 [181 (in the complex plane), but - to the best of my knowledge - the first complete proof appeared in McKean (1969) [191. About 10 years later Bernard, Campbell and Davie [ 11 extended the result to higher dimensions. Both these proofs are based on stochastic integrals. j
,J.
T
1.2. Theorem (The McKean-Levy theorem).
Bernard-Campbell-Davie
extension
of
the
Let U C ]Rn be open and cp = (CP 1,;. ':" CPm) : U +]Rm be a C2-function. Let (Bt,~,PX) and (B t , ~,py) be Brownian motions in ]Rn and ]Rm, respectively. Then the following and (II» are equivalent:
«1)
(1)
cP (B t ) is - up to the exit time Brownian motion in ]Rm, except for More precisely, if we define
TU from U - again change in time scale.
=
T
a
(1.14)
then at is strictly increasing for a.a. wand cp*(w)
lim cp( Bt) ex i s t sa. e. on {w; a ( T)
< oo}
(1.15 )
t1T
And the process Mt(w,~),
(w,~) E~X~, defined by t
..(Xs)ds,
o
from
noW
151
STOCHASTIC PROCESSES, INFINITESIMAL GENERATORS AND FUNCTION THEORY
where ,,(x) ~ 0 is a continuous function and the set N = {x ; ,,(x) = 0 } has empty X-fine interior, i.e. 'N = inf {t > 0 ; Xt iN}= 0 a.s. QX for all x. Such a A will be called a time change rate (for Xt ). Note that for such A the function t -+ Ot(w) is strictly increasing -1 for a.a. w. Its inverse will be denoted by 0 (w). t 2.3. Definition. Let (Xt'~ ,QX), (Yt,~ ,QY) be Ito diffusions on open sets U c mn, V c mm and let ¢: U -+ V be continuous. Then we say that ¢ is Xt - Yt path-preserving if there exists a time change 0t such that if we define, for W open, 'IV C U and, = 'w
I
¢ (X -1 ) °t Y
t-o
,
, >,
o })l ds + EX[f(¢(X,»
. x{ t>o }l
,
or
EY[f(Zt)l
=
+ JE1Af)(Zs)dS
f(¢(x»
(2.20)
o
Similarly, Dynkin's formula applied to Yt gives directly A
t
A
EY[f(Yt)l = f(y) + fE~Af)(Ys)ds
(2.21)
o
So the two families of bounded linear functionals u t0) : CO(V) t ~ 0, i = 1, 2, defined by f
both satisfy the equation
E CO( v)
-+ JR,
154
B. (>KSENDAL
f
E CO(V)
(2.22)
So by uniqueness (see the next lenuna) u(l)(f) = u (2) (f) for all t t i. e. EY [f(Zt)] = EY[f(Y t )] for all
f E CO(V) f
E CO(V).
Similarly one proves by induction that
-
-
EY[fl(Ztl) ... fk(Ztk.:tt)] = EY[fl(Y tl ) ... fk(Ytk.t t )]
i
for all 0 tl < t2 < ... tk.,. t ~ 0 and fl, ... , fk E CO(V). That completes the proof that (Zt, QY) is identical in law with (Y t , QY), except for the uniqueness claimed above: This next lemma works in general if A is the infinitesimal generator of a contraction semi group {T t } of operators on Cb(V), the bounded continuous functions on V, equipped with the uniform topology. (See Lamperti [17], Ch. Then fJA denotes the (dense) set
n.
of functions f E Cb(V) such that Aft= {[Ttf - f] converges uniformlyon V as t -+ O. In our case the contraction semigroup is given by T t f = E'[f(Y t )], and it can be seen using the Dynkin formula (2.8) and the Lipscitz condition on band 0 in (2.5) that C2 c 2A
o
A'
2.5 Lenuna.
(Uniqueness Lenuna)
Let {w t } be a family Cb(G) intoN satisfying
of
bounded
for all b E
linear
functionals
from
.1);\
and for some constants C, m Wt f -+ 0
(ii)
for all
Then Wt g = 0 Proof: nt
Choose a
e- at
t -+ 0+
as
>m
for all f
E;
O.
and put
W't
-
A)-l (a I Let Ra Lamperti [171 , p. 142) . Then
Cb (V) -+ 5J A be the resolvent of A Choose g E Cb (V) and put f = Ra g E
(see
.1t.
ISS
STOCHASTIC PROCESSES,INFINITESIMAL GENERATORS AND FUNCTION THEORY
~(nt
f)
=
+ e- at Wt Af = nt(-aI + A)f
(-a)nt f
n t R-1 a f
n t R-1 a Ra g = - nt g.
Hence t
=-
f ns g ds
o
By
(0 nt f
+ 0
using (ii)
nt f
as
t +
co
and we conclude that for all a> m
This implies that Ws g
=0
Remark:
for all s
~
0, as claimed.
Note that it is a consequence of (2.11) that
Ah :: 0 in G (open in V)
~ A[ho 0 there exists a compact KC n such that PY(K) > 1 - e:. We refer to [ 7) for details.
B. ~KSENDAL
156
We end this section by showing that the Levy theorem as a special case: Choose Xt = Bt , Yt = Bt to be Brownian mm, respectively. Then the generators Laplacian operators ~~ inm n , mm and (2.11)
Theorem 2.4 contains motions in m n , of Xt , Yy are becomes
the
By comparing terms on both sides we easily see that this is equivalent to t~e conditions (1. 17) in the Levy theorem, with >.. (x) = IV¢i (x) I ; i = 1, ... , n. Finally, since the critical points of a (classical) harmonic morphism constitute a polar set (Fuglede [13), p. 116) the function>.. is a time change rate, and the Levy theorem follows from Theorems 2.4 and 2.6.
§3.
APPLICATIONS
Based on the operator characterization of path-preserving functions it is natural to adopt the following point of view: Given a function ¢ whose properties we want to investigate, we try to find semi-elliptic 2nd order partial differential operators A, A with continuous coefficients such that for f
E C2
o A,
for some continuous function >..(x) < O. If A, are "reasonable" we then conclude tha~ there exist Markov processes Xt , Yt (whose generators are A, A respectively) such that ¢ is Xt Yt path-preserving (with time change rate >..). Based on known behaviour of the paths of Xt , Yt we may then deduce properties about ¢, like one has done for analytic functions ¢: Uc a;n -;- a; in the case when Xt and Yt are Brownian motions. Here are 3 reasons why it is worthwhile to consider the general path-preserving theorem 2.4 rather than just the Levy theorem: I f ¢: uc: a;n -;- Ck with k > 1 is analytic, then ¢ is not necessarily BPP, not even if ¢ is biholomorphic. Thus the Levy theorem is not the right tool for studying these functions.
2)
Even in the case when the function ¢ is BPP, there may be other processes Xt , Yt as well such that ¢ is Xt - Yt path-preserving, and these processes may perhaps be chosen to serve our purpose better than Brownian motion. In a) below we give an example by considering the conditional Brownian motion. (However, if ¢ is analytic then such processes Xt , Yt must depend on ¢ , unless they are just time changed Brownian motions. See [ 7]).
STOCHASTIC PROCESSES, INFINITESIMAL GENERATORS AND FUNCTION THEORY
3)
a)
157
It is of interest to have an approach which is not limited to analytic functions. See b) below.
Analytic functions and conditional Brownian motion
Conditional Brownian motion was introduced by Ooob [101 and a detailed account of this process can be found in his latest book [ 111 • If h > 0 is a harmonic func tion on a domain U C lRn, then the h-conditional Browning motion, denoted by {B~\>O' is a Markov process in U (possibly with a whose generator is given by (I,
h f
= ;ti hf 1 =
(I,f
+
h
One way to construct
B~
lire
finite
time l;.)
Vb' 'Vi h
(3.1)
is as follows:
Let
Uj
be an increasing
O.
"*-* pair A o 1) "exists" and q < ., then (Ae ) - Ae " with as usual lIq + l/q' = I. q q The proof depends on the equi valence theorem ultra and some juggling with Hahn-Banach. On the basis is of course the fact already recorded (§ 4) that the K and J functionals are dual norms. _ Remarks. 1. One gets more general spaces A e ' K = (AO,A 1).e'K and A+' J = (A O,A 1) ~'J if one instead of + e uses'a general "f6nction patameter" +. But they are by far not that! important from the view of applications and the resulting theory is not that simple. (E.g. the equivalence theorem requires some further qualifications.) That is why we have here restricted attention to the special case + = + e • The special virtues of +e are of course connected with the fact {hat it relates both to the sp~e Lq and the (multiplicative) group structure of the interval (0,.). (Notice that dtlt is the Haar measure on~+ = (O,.)!) The definition of K space makes also sense if e = 0 or 1 provided q = ., and, similarly, the definition of J space makes sense if e = 0 or 1 provided q = 1. (One can show that e.g. (AO,At!Oc is the relative (Gagliardo) completion of AO in AO + Al etc.; see I:A"t].) One can also let 0 < q < 1 but then the resul ting spaces are not any longer Banach spaces, but quasi -Banach spaces. The duali t y theorem h~wever extends wi~\ proper modifications to this case too: the dual of As ' 0 < q < 1, is As.' It likewise e!Ctends to the case q = 00: the dual of 'tile closure of AO 11 At in Ae is Ae1. (a most useful result, which people are often ignorant of). 'fs for the equivalence theorem (in the quasi-normed case), the only restriction is that in the definition of the J space it is now necessary to use a "discrete" variable (see the following remark).
PARACOMMUTATORS AND MINIMAL SPACES
167
3. It is often technically advantageous (and, as we have seen, sometimes necessary) to use a "discrete" definition. In fact, one can put the definitions in the following form:
[~ (2 v9 K(2 v ,a»q]l/q v=-.
• v v=-. ,
3 u = (u)
< ••
a doubly infinite
sequence with values in AO n Al such that a =
~
v=-.
(convergence in AO + Al ) and
[~ (2 V9 J(2 v ,U v )q]l/q v=-·
u
v
< ••
Thus, formally speaking, the multiplicative group of positive reals.&... = (0,.) has been replaced by the additive group of integers b
o
Putting uk = a k - a O we get the desired decomposition, along with the estimate dke +Jiscre~e counterpart of (4» (4' )
k J (2 , uk)
~
k
4K (2 , a ).
#
The proof of the equivalence theorem is now easy. Indeed, the inclusion Aa. c: Aa .J is an immediate consequence of the fundamental lemma. Fo~'lhe con~"rse we have to involve inequality (3): From a = f~ u(t) dt/t we deduce that (5)
K(t,a)
~
00
J
min(l,t/s)J(s,u(s»
ds/s,
o
so if +aq(J(s,U(s» < 00 we readily conclude that +aq(K(t,a) < 00.#
171
PARACOMMUTATORS AND MINIMAL SPACES
Remark. The fundamental lemma, which has been along since the early 60's, has recently been strengthened by Brudnyr and Kruglyak [B8] who established the following remarkable result. FUNDAMENTAL LEMMA (strong form). Assume that A is "mutually closed" a = LP if lip = (1 - a)/po + a/pl' 0 < a < 1, PO Pl. This is relateWto the theorem by Marcinldewicz.
*
9. An important complement to the reiteration theorem again has been obtained by T. Wolff [W 1 • Let A l , A2, A3, A4 be Banach space all four continuously imbedded in one and the same (farge) Hausdorff topological vector space (so that (A 1,A 4 ) forms a compatible pair etc.). Let further a, 1'), A, ~ be parameters subject to the following conditions: 0 < a < I') < 1, a = AI'), I') = (1 -
~)a
+
~.
WOLFF'S THEOREM. Assume that A2 = (A l ,A 3 )Ap and A3 Then A2 = (A l ,A 4 )ap and A3 = (A l ,A 4 )l')p· Assigning weights 0, a, the following picture:
o
a
1'),
= (A2,A4)~q
1 to A l , A2, A3, A4 respectively we have 1
Thus, intuitively speaking, Wolff's theorem allows us to patch together a "large" scale out of "small" scales, in an analogous manner as one in Differential Geometry, say, manifactures a whole manifold out of small coordinate patche~. The proof is, surprisingly, quite straightforward. In one direction:
we readily deduce that
PARACOMMUTATORS AND MINIMAL SPACES
173
whence (A 1,A 4 )e1 c A2 • The other direction: If a 2 E A2. with II aU A = 1 then by the definition of the K functional we can write (for t fixed) ~ = a + a with 213
Similarly, we can write (for any s) a 3 = a' 2 + a 4 with II a 4 UA
f * gELqoo provided IIp = lIq + lIr - 1, no special restriction on e. For the proof let us write Young's inequality twice:
and interpolate: p
(L 0, L
p
1)
91
v
A
r r (L O,L 1)900
~
q q (L O,L 1)900.
If we work this out we obtain
Now we interpolate once more (keeping r fixed!) and the desired result follows. #
178
J. PEETRE
Notes for Lecture 1. There are now several excellent books (monographs) available which are in whole or in part devoted to the theory of interpolation spaces. Besides [Bll let us mention [Bl0], [ K3] , [ T2], [B91. There exists also a rather complete bibliography of interpolation spaces (until the year 1980> [C3]. Among more recent works of survey character let us mention [Oll, [03], [P5] (see also [P6]). The theory of interpolation spaces took its birth in the years around 1960 and among its fathers we mention Aronszajn, Lions, Gagliardo, Calderon, Krern. It has its origin in the classical interpolation theorems of Riesz-Thorin . More precisely, a function g on r is the trace of a function f in W lQ) P lIp k , this at least for 1 < P < 00. Later on Besov spaces have fcPund applications in many other branches of Analysis. The customary definition of Besov spaces makes use of finite n n differences. We shall here onl y deal with the case when Q = &, or Q= l,.. (periodic case), which makes it possible to exploit the underlying group structure, and so we shall use another more expedient route, making use of Fourier techniques. We follow by and large the exposition in [P3] where missing details can be found (compare the parallel treatment in [B2], chap. 6). Q
2. The guiding idea is to break up a given function (or distribution) f many pieces such that the Fourier transform of the \I th piece f is, roughly speaking, concentrated near the sphere {~ E Rn : I~ I = 2"} (\I E Z). This is known as the (a) dyadic decomposition
in
i!: into countabl y
~f.
.,....
N.B. - Strictly speaking, we have to deal with two spaces': in duality. We will not distinguish them in notation but we usually reserve Greek letters for the dual (phase) variables (~,1'), ••• ) using Latin ones (x,y, ••• ) for elements in the "direct" (configuration) space. The duality will be denoted by <x,~> = x1~1 + ••• + xn~n. The Fourier transform is (formally) given by A
£(~)
=
J e i<x
l £(x)
,~
dx.
....,
Rn
We also write Iff =
i.
In praxis, such a decomposition of f can be gotten by first picking a sequence of test functions {cp } ~ such taht each cp is in f (the v:v n:v Schwartz class of rapidly decreasin unctions in.a.) witH
PARACOMMUTATORS AND MINIMAL SPACES (~
-
181
n E R •
01
n EN). .".. 1\
We then put f = cp * f (* = convolution). Clearly supp f e: U too for each v E Z ~nd, l'f we also requir" that (*) 1: cp Yx) =v6 (delta function) or, equivalently, (**) 1:VE~ cpv(~) = 1 we 'h«iie aVtrue partition: f = 1: vEZ fv. However, it is sometimes not suitable to have such a strong conortion as (*) or (**) and then we have instead to impose a Tauberian condition m5rmander-Shapiro):
3. We can now define the Besov space Bsq = p real, 0 < p, q ~ 00:
rfpq (r:l1), ~
s arbitrary
Here {fv> vEZ is any dyadic partition associated with f and fO = 1:~Of. va isVeasy to see that Bsq is in general a quasi-Banach space (for the natural choice of the Pquasi -norm) and we have the continuous imbeddings :Ie: Bsq e:J' (tempered distributions). It is a Banach space if p ~ 1, q ~ 1 ada for simplicity we will in what follows usually make this assumption. (Sometimes it is wise to exclude the extremal cases p or q = 1 too.) Otherwise, the case 0 < p < 1 is quite subtle since then the space LP cannot be realized as a space of distributions (theorem of Day). 4. For technical reasons it is often convenient (and sometimes necessary) to use a related scale of spaces, namely the so-called q homogeneous Besov spaces (by contrast to the previous non-homogeneous ones). The idecfis to let the dyadic partition run all the way down to -00:
as
£ E
bsq ~){~ p
~Z
(2 vS R£V UL P )q}l/ q < 00.
w-J
The price we must pay in order to get (quasi-)Banach spaces here is that we have to reckon modulo polynomials. More precisely, let N be an int~er > s - nip (~ s - nip if we impose the restriction q = 1>. Then Ffq can be realized as a Banach space modulo pol ynomials of degree < N¥ (If N = 0 this means no extra qualification.) We say "homogeneous" because the spaces rfq turn out to be dilation p ...sq def ...sq invariant: If f E ts p then f). (x) = f()'x) E tsp and moreover
J. PEETRE
182
By contrast, the spaces Bsq do not possess this nice property. (Needless to point out, Ffq ~d all other spaces we deal with here are translation invariant too.~ Example. 6 (delta function) E B-n/p,oo for all 1 ~ p ~ p
00.
Indeed, if f =
6 so that "f(~) :: 1 we have a dyadic partition {f } EZ of f such that IIfvllLP l¥ 2n p. By duality (see § 5) it follows that '{.,evliWVe the imbedding Bn/p,l l¥ C (continuous functions). Indeed, if 1 ~ p ~ 2 one hase even a Iltronger result in the form of the factorization:
This is essentially the Bernstein-Szasz theorem about absolutely convergent Fourier transforms. There is something special wi th the value s =nip! 5. As a first result on Besov spaces we mention the DUALITY THEOREM. (Bsq )* l¥ b-s:-q ' where' stands for the conjugate exponent; here we assumJ> p < 00 aRd q < 00. This is proven by some juggling with Hahn-Banach (alternatively one could have used the duality theorem for abstract interpolation; see § 8 where the link with interpolation is established). The (anti-)dua~ty e~ployed is of course the one coming from the inner product in L = L (Rn ): .",
(f,g) ~ =
J f(x)g(x)
--
dx.
Rn
Remark. One can also include the limiting case p = 00 or q = 00 if one replaces q by the closure of in the corresponding metric. (If p < 00 and q < 00 p is automatically dense in Ffq, as is readily seen.) Notice that the space Ffoo for s non-integer >PO is the well-known Lipschitz space (defined usT'ng first order differences), whereas for s integer it is the Zygmund ~ace (defined using second order differences). One often writes A = rf (even for s ~ 0>, and" for the corresponding closure s 00 s * -sl -sl* of j. Thus we have in particular "s l¥ Bl ' , (B l ' ) .. As so As is the second dual of" (a fact ignored by many). s
es
PARACOMMUTATORS AND MINIMAL SPACES
183
6. As we have already mentioned, a major problem in the theory of Sobolev and Besov spaces, which has attracted the attention of many workers (especially in the Soviet UnionD, is the characterization of the trace on smooth submanifolds. If we restrict attention to the limiting case when the dimension is precisely n, we have the Sobolev imbedding problem (that is, the study of relations between spaces with different parameters in any of our scales). Here is the result for (homogeneous) Besov spaces. IMBEDDING THEOREM. We have
provided s ~ sl' P S P1' q S q1 and (that is the important thing!> s nip = sl - n/p1· Let us indicate the proof in the special case q =q =00. Consider any d~adic decomposition {f } E~ a given function (distribution) f in Eroo , say, f = (f) * f whef1e )' r.} EZ is a family of p v v vv_
0\
test functions of the type considered in § 2. Let
{(f) v}
-
-
vEZ be another A
such sequence, "slightly larger" than (ell v } vEZ in the sense that eII(Z;) A
1 in the support of I'
v
(f)
v
•
=
Then we have the reproducing property
d~I ell v * I
=
(f)
v
*
(f)
v
*
I = ell v
*
Iv.
From Young's inequality for convolution (see Lecture 1) we have
Now it is easy to see that 1Ie11 IIL P = O(2vn o. Then one is lead to expect a formula of the type
N
U , N an
(2)
N
N-1
Here tat = tatta and tatf = f(x + t) - f(x). But in which range of s can we hope to be valid conclusion? By our previous "criterion", we must have
(ei~ - UN = O( I~ IN). This gives 0 ~ s ~ N. Indeed, a closer examination reveals that (2) is true without any further qualifications if only 0 < s < N. (To remove any doubts on this point, I suggest you verify (2) directly in the special case p = q = 2, by a brute (but instructive) calculation, just use Plancherel.) One can also prove such a result: Write s > 0 in the form s = k + 01, k integer, 0 < a ~ 1. Then
188
If P
J. PEETRE
=q =
00
we recognize here the definition of the Zygmund class A • a
" 2. Harmonic continuation. k(x) = lin· 11. Now u = u t = k t * f(x), viewed as a function of x and t, is harmonic . 2 n+1 . (6 lU = Q) 10 the upper halfplane R (upper halfspace l.. 1f n > 1); thn 1S why we say harmonic continu~ion. In this case we expect that
The requirement e - I ~ I = O( I ~ IS) gives s S 0 and, indeed, (3) turns out to be rigorously true if s < o. To do better we have to take derivatives. For instance, for s < 1, one can prove that
And so on for higher deri vati ves. Remark. Instead of the Poisson transform one ~n ~s well take ~he Gauss-Weierstrass transform (k(x) = 1I2,/n· exp(-x ), k(~) = exp(-~ "». Therefore our spaces also admit a description i~ ter~ of "temperatures" (solutions of the heat equation aulat = a u/ax ), rather than harmonic functions. 10*• The general question treated in § 9 can be given yet another twist. Let a "convolution kernel" k be given and set as before kt(x) =
= t -nk(x/t)
0h
(t > so that " k t (~) " = k(t~). If f is a given function (distribution) in.a,., set also u = k t * f; u will be though~+ff as a function of two variables. thus living In the upper halfspace.R:. n If X is any (quasi-)normed ~ace of functions or distribUtIons in l.and ~ a positive measure on we are interested in the inequality
t:
More generally, if X is only semi-(quasi-)normed but the semi(quasi-)norm induces a (quasi-)norm modulo polynomials of degree < a fixed integer N, it is natural to take instead the inequality
PARACOMMUT ATORS AND MINIMAL SPACES
(X,p,n)
{~
J IOau(x,t) IP
189
d~}l/P
la'r=N Rn+1
S cUfU X for f E X.
~
Let I be a cu~e in ~+1 of side length 1 = 1(1), with one of its faces contained in $-. We say that ~ is d-dimensioanl Un the sense of Carleson) if for all I holds the inequality (C)
~
(I)
S Cl
d
•
Example. Carleson measures in the ordinary sense have dimension n. The following two lemmata are left as an excercise for the reader. LEMMA 1. An equivalent condition, for d > n, is (C*)
~(I*) S C*ld
where 1* is the "upper" half of 1.# ooLEMMcf 2. Let f be a po:itive f~ctio~ of one variable such that dtlt < 00. Then J 0 f('; I x I + t'" d~ < 00 for d-dimensional ,measures. #
J0 f(Ut
One can now prove the following THEOREM. The following conditions, where we wri te X = Ssp and, similarly, use XL when s.p are replaced by sl' Pl' are equivalent? (i) ~ is (X,p)-Carleson for !!2m!!. reasonable kernel k. Ui) Same thing for all ditto. (Ui) ~ is (X1 ,Pl)-Carleson for all reasonable kernels k and any PI with sl = s + n(I/ P l - lip). Uv) ~ is d-dimensional where d = n - pes - N). What "reasonable" is will be disclosed in the proof (growth condition + mean value property). Proof. (after [HS1) For simplicity let N = o. (1) ~ (U>. Logic. (11) ~ U11>. Just apply the imbedding X c: Xl. (Ui) ~ 1 in the cube 10 of side 1 whose center is on the t-aXls too. Blow up fO and un in the scale 1, that is, consider fl = f o(x/l), u .. = kl * fl = u O(x/l,tll.). Then u 1 > 1 on I. The ,eX1,Pl)-inequalfty now gIves
(~(I»l/P S ClfllX
S Cl n/P l- s 1lfl 1
Xl
= Cl n/p - s = Cl d/P •
190
J. PEETRE
(iv) =>*(1). p > 1. At this stage at last "resonable" is needed. Let 1** denote I blown up in some fixed scale, say 11: 10. We require now the following MEAN VALUE PROPERTY: lu(x,t) I
~ C{t-nJ lul P dXdt/t}l/ P for (x,t)
E
I*.
I** Take this to power p and integrate: lu(x,t) I P ~ Cpt-nJ
lul P dxdt/t for (x,t) E
I**
I** J
I*
lul P dl-l ~ Cptd-nJ
lul P dxdt/t ~
I**
cP J
(t - 9 lu I)P dxdt/t,
I**
where we also used inequality (C *). With now loss of generality we may assume that I-l has compact support contained in a preascribed cube I. Now we make the same decomposition of I as in the proof of Lemma 1. (Surely you have done your home work!> We get the same inequality
. sets Iv* and Iv ** wlth bounded overlap of tne I
(.r* v
= 1,2, ••• ). Finally sum over v. Because of 's we get
But if we impose proper GROWTH CONDITIONS on our kernel
(see the
*
previous §) nf n d~f (J (t -s I k t f I P dxdtlt) lip is an equivalent norm in X. p ~ 1. This is easy. Under suitable GROWTH CONDITIONS on k we get p
lip
from Lemma 2: (J I k I dl-l) is as a p-normed a space is ("minimality"):
d-n Ca for a > O. On the other handd K generated by the functions k a -n a
~
any f E X has the representation f = 1: X k a n - d where 1: I X IP < 00.# v v av v
asP
is the largest space among all spaces COROLLARY. If d > n then X which have the d-dimeRsional measures as (X,N,p)-Carleson measures.# Remark. An even simpler way of doing it: Postulate (as a substitute for the mean value property) on the one hand that
for the functions u = k ... * f, on the other hand that
191
PARACOMMUTATORS AND MINIMAL SPACES
f
A(
X
+
t
S "C -, t) dl·d X,
t)
~
Ct
d
for d-dimensional measures ~. (Here A is a suitable positive kernel.) Then there is NOTHING to prove; in particular, the "geometry" gets eliminated. 11*. We discuss briefly the mean value property encountered in § 10. Gauss observed that if u is any harmonic function (A u = 0) then its value at a point P can be recaptured from its average over either a sphere S or a ball B about P: u(P) = 111 S I • f s....U = 11 I B I • f BU. This can be generalized to more general (hypo-)ellipticr>DE"2 To be specific, consider biharmonic equatio~ A u = O. Let G be the corresponding Green's function with pole P (A G = cS p ) and Dirichlet's boundary condition (G IS = BGIBn IS = 0; n is the exterior normal). Using twice Green's formula (1 B (ltv - uAv) 3v = 1S (BulBn e v -
uBv/Bn ) dS) I get u(P) = f (BAGIBn u - AGBulBn ) dV. Next apply this for4fuula to a family of c8ncentric b\lls B with rldii r between two r fixed numbers and form the average. Then I get a formula of the type u(P) = 1A au + 1A bBulBr where a and bare smooth functions supported by an "annulus" ~ about P, BIBr denoting radial derivation. By partial integration I get rid of the derivative BulBr and there results a formula of the same type with b = 0.# This procedure clearly is quite general. In partic1:f.\Sr, it applies to dilation invariant hypoelliptic equations in ~ of the form P(t,BIBt,BIBx)u = O. There results a mean value theorem of the desired type: lu(x,t) I ~ Ct-nf
lu(Z;,"C) I dZ;d"CI"C for (x,t) E 1*.
1** with I * and I ** as before (§ 10). Examples. 1) B2uIBt2 + Au = 0 (Laplace's equation). 2) 112t· BulBt - Au = 0 (heat equation in slight disguise). 3) B4 ulBt4 + 2B 2AulBt2 + A2u = 0 (biharmonic equation; the case we reall y have discussed here).
2 2 4) B ulBt + (2q + O/t· BulBt + Au = 0 Uhe singular GASPT equation of Alexander Weinstein; see e.g. Lions's (1 ) book [L31, chap. XII). In all these examples we can apply the result to the solution u of an appropriate boundary problem with boundary datum f given on~. This solution apparently is of the type u =,.kt * f. Below I list the Fourier transform k of k in each of these cases.
,.
1)
k(Z;) = exp(- I Z; I) (corresponding to the Poisson kernel).
192
J. PEETRE
2) k(~) kernel).
=
exp(-I~ 1~ (corresponding to the Gauss-Weierstrass
A
3)
k(~)
=
exp(-I~
1)(1 -
I~ I).
k(~) = const I (I ~ 1)/1 ~ 1q
the modified Bessel function; if q = q Apparently, the argument just given is confined to hypo-elliptic equations. 4)
Now apply this to the functions f 0 w , w E G, and the desired imbedding follows.# In the previous discussion we worked "modulo constants". If we require genuine norms, the supply of ~aces gets more restricted. In particular, the maximal space is then H • The minimal space is still~, for~ can be equipped with an invariant norm; take, for instance, II fll = II fll 10 + II fU' where UfU' is any of the invariant semi-norms. The two 10 spaces are of course not any longer each others duals. But H possesses an interesting MObius invariant pre-dual, which can be described as follows:
q
4. Another interesting Mobius invariant space is BMO. Here is a definition (due to Garcia) which is automatically MObius invariant: £ E BMO~)sup zEU
I
.... T
I£(~) - £(z) mz(~)
Id~1
.) Of course, there exists no dilation + translation invariant measure. 3) a itself. (A geometric argument shows that a acts simply transitively on each set E".) An invariant measure is gotten by transplanting the Haar measure on a to E". 4) a/{dilations}. (The maps which fix GO and one more point, say, the origin 0 are of t~e form z .. az (a real .. 0).) The invariant measure is an old friend: do i = I dl;l I I dl;2 1/1l;1 - l;2 I •
200
J. PEETRE
5) Again G itself. (The only transformation which fixes one point at the boundary, the pOint -1, say, and ~e an interior point 0, say, is the identity.) The invariant measure dO i is again gotten from the Haar measure. We describe now a general principle for obtaining invariant function spaces. Let 0 be any measure space on which an action of G is defined (usually via measure preserving transformations). Let further T be a map which to holomorphic functions on U associates measurable functions on 0 and in addition "intertwi!\fs" the two group actions (the one on 0 and the one on U>. Denote by T the formal adjoint of T (with respect to the natural G invariant pairings on U and 0 respectiv~ly, induced by the inner products in the Dirichlet space 6l and in L (0) respecti~ely). In the cases of interest to us T will be an isofetry from into L (0). Therefore we will havt, formally at least, ToT = id. This means dO i < 00. Similarly, if Y = LP we get the G invariant Besov spaces 0 , provided one imposes the restriction p > 1. In case 2) I know of ~o interesting spaces, except for the ."trivial" example of the space H • But in case 3) there arises a NEW description of
Here z and z2 are any two filled pOints in U with d(z ,z2) = " and we integra,"e witli respect to the Haar measure d6) on G. tase 4) is quite similar to the cases 1) and 3). For example, we get now back, from a more general point of view, one of the characterizations of (f:J in § 3. Finally, Garcia's definition of BMO (Ukewise § 3) obviously is connected with case 5).
201
PARACOMMUTATORS AND MINIMAL SPACES
5*. In this Section we discuss the questionrP.iven an invariant space X what can be said about the sequence z I X} >0 of norms of the "basis" {zn} n>O. n
{n
Example. If X =~ then IIznll SIll n and if X then IIznll the general case we expect zn to be somewhere in't,etween.
=13
SIll
1, so in
We introduce the quantity x(m,k)
dEf sup
I1 w
(=
X (k,
m) )
where we thus sup over all "matrix elements" of the representation C, being the M~bius invariant inner product induced by the norm of the Dirichlet space (§ 3). Remark. Indeed, the matrix elements can be expressed in terms of hypergeometric functions (alternatively Jacobi polynomials). They also satisfy a 2nd order linear DE. (lowe these two pieces of information to T. Koornwinder and U. Haagerup respectively.) One can do the same game with the more general actions C Il (see the corresponding remark in § 2). THEOREM. 1) Let X be a G invariant Banch space of holomorphic functions on U. Then
2) Conversely, if H(k) is a positive function on the positive integers (a "sequence") such that (**)
H(m)x(m,k)
~
mH(k),
then there exist a G invariant space X as above with Proof. This (t)
m liz II
(where II· II
i~-view
* \=
of the following general formula
m m/llz II
* ist the norm dual to
m k I1
w
II z k II = H(k).
~
m liz II
II •
II).
* liz k II
Therefore k m = mllz II/liz II,
whence the desired relati~ (*) by the definition of X. 2) Define X to be the 1 -hull of the elements C zm /H(m). It is clear that IIzmll ~ H(m). For the converse we use aga~n formula (t). Quite generally holds
202
J. PEETRE
I I
II/H(m), w
sup
m,w so by
(**)
I j
we obtain
m
k
suplI/H(k) k, w w This proves II z If X
m
II
supx(m,k) k
~
m/H(In)j
1 = H(m).#
=q then (*) gives (as now Ilzmll ,. m) x(m,k) ~ cm or, in view of
the symmetry, x(m,k) ~ c min(m,k). Thus (within equivalence) if (**)'
~
H(m) min(l,k/m)
~
(**)
is in particular fulfilled
cH(k).
Therefore we get COROLLARY. If H(m) is equivalent to a concave function then there exists a G-invariant space X such that IIzmll ,. H(m).# This again leads to the question whether there is an inequality in the opposite direction: x(m,k) ~ c'min(m,k) (c' > 0>. However, a somewhat heuristic calculation of Haagerup's based on the WKB (or GreenLiouville) approximation points to that this is not so. So maybe there are "exotic" M6bius invariant spaces which cannot be obtained by "interpolation" even in a very weak sense. 6. As an application of the previous theory, more precisel y, the idea of the minimal space (see § 3), let us know sketch a simple proof of the trace ideal criterion for Hankel operators, usually associated with the name of Peller (see [P6]), in the case 1 ~ P < 00. (The case 0 < P < 1 is much harder; see [P7], [S11.) It is expedient to consider Hankel forms, rather than operators. By a Hankel form with symbol . But this
Ur..
implies that the norm + .. IIS 1 is G-invariant. The analogue for C- 1 of the Arazy-Fisher theorem ("A51 (see § 3) then gives
or, using the above relation between the two "symbols" cp and +,
This is one endpoint for the interpolation. The other endpoint is gotten from Nehari's theorem
Consequently, the interpolation gives half of Peller's theorem; the converse follows by a duality argument (see [P61, [P3]).' Remark. The prime virtue of the above argument is that it can be adapted to a variety of similar situations (typical abstract nonsense). For instance, it extends to Hankel forms in several variables (cf. next §) and to "paracommutators" in Rn (cf. Lecture 4). Notice that in the latter case the group under consideration is not any longer semisimple. Indeed, even in the present case (unit disk) invariance under the full Mabius group is not that important; what is essential is invariance just under its "parabolic" subgroup (dilations + translations).
1...:.. * Most of what we have been doing until now extends mutatis mutandis to the case of holomorphic functions over the unit ball B = B in n, But there is one initial difficulty. Naively, one would 2think o"l the following analogue of the Dirichlet integral: ~ II df(z).. do. (z), where II. stands for the invariant norm . duce~ _ by the lBerdman metric on tile cotangent bundle of B {explicitly: df Ui 222 .. (1 - II z. )( adf a - I Rf I ) where a· II is the corresponding Euclidean norm, Rf =1: z.8f/8z i being the radial derivative of f) and do. is the 1 1
r.
a·
=.
J. PEETRE
204
2 n+1
corresponding volume element (explicitely: da i = datU - UzU) , da being the Euclidean ditto). The drawback is that for f holomorphic the integral is divergent (unless f == const). So one has to proceed differently. One possibility is first to consider the "norm"
J 1f ( z) 12 (1
-
II z H2 ) a
da ( z ) ,
B
a a complex parameter, which is invariant if we let f transform according to a suitable multiplier representation, and subsequently to use analytic continuation; this is the technique of M. Riesz. N. B. - "Invariant" in the above discussion refers, of course, to the group of holomorphic selfmaps of B, which is isomorphic to the projective group PU 1>. At!, ,p)-Carleson measure (cf. Lecture 2, § 10) corresponds then to the li~illng case when all mass sits on the diagonal.
L* The ultimate setting for the theory is perhaps something like a Siegel domain. Here 2 we content ourselves with a few words ab~ut the case of the bidisk U = UX U = «z,w) : Iz 1 < 1, Iw 1 < 1} in £.. (The formally more general case of the polydisk is of course quite analogous.) The group 02' say, of all automorphlms of U2 is generated by transformations of the following two types: 10 the maps
where each of the "partial" maps z -+ (a l z + b 1)/(c l z + d ) and w -+ (a~~ + b 2)/(c2w + d 2 ) belongs to the previous "l-dlmensional J group 0. ""2 the map (z,w) -+ (w,z) ("symmetry"). We inquire2 whether there exist Hilbert spaces of holomorphic functions on U which are invariant under 02. Here is the result:
'8
THEOREM. the only 02-invariant Hilbert spaces of holomorphic functions on U are the following ones: 10 41= (1) 20 1tconsists of all holomorphic functions f of the type f = fl (z) + f 2 (w), with
"',s,:
U£U
2
=
~ L..
A 2 ~ A 2 n 1£1 (n) 1 + m 1£2(m) 1 < n=1 m=1
L..
00;
to get a genuine Hilbert space we have to work modulo constants, as usual. 3°1(consists of all holomorphic functions f wi th
If 12
A = ~ nm I£(n, m) 12 < n= ,m=1
or, equivalently,
00
207
PARACOMMUTATORS AND MINIMAL SPACES
bow we have to reckon modulo functions of the type f~z) + f 2(w).# The interesting case, apparently, is only the case"""3°. In "this case we can likewise define a maximal space and a minimal one. In particular, the maximal space ("Bloch space") is defined by the condition sup (1 z,v
Iz I
2
) (1 -
Iv I
2
2
) I a f (z, v) I azav I
norm of A. 5*. At this junction it is convenient to take the step to infinite dimensional spaces. If A is a 3-linear form over a HUbert space H say taht A is "l-bounded" if P (defined as in § 4) is bounded in the usual sense. Define "2-bounded" and "3-bounded" in an analogous way. l-bouned is however trivial (;;; bounded) and so is 3-bounded (;;; HUbert-Schmidt>. So what about 2-bounded? Clearly, this is the same as tR. say that A extends to a continuous map on (H H) & H where 81 = e(projective tensor product> and ~ is the Hil'bert· tensor proauct (remark by Jonathan Arazy). Let us consider a concrete example. My problem thus now concerns the form
e.
(1)
A(x,y,z) =
~ ;(i
+ j
+ k)xiyjz k
in H = 12. Alternatively (another observation of Jonathan Arazy's) the form
J. PEETRE
218
(1')
A(£,g,h)
=
f
;P£gh
Idzl/2n
J... 2 2 2 on H = H (JJ (Hardy class). In this case H~ H = H (leT). Thus the product fgh in -1) are the weighted Bergman spaces; from our point of view th:Y are just Besov spaces: Aocp = BSP (,I? for s = -(oc + 1) Ip. By general results (cf. notably Lecture 3) p
(OC +
~)/2
and also
(AOC,l(T»* ~ Al (T) .. +oc ....
(Lipschitz space)
Combining all this information we get THEOREM.lhe form 1
[A.}
and the norm estimates on the individual terms in (2.3)
1.
a new complication shows up.
The summability of
are not enough to insure that the sum (2.3) converges.
One way to
establish the convergence is to dominate the sum by an integral. That is If
Lemma 2.4:
p> l ,
a> 2,
then the map of
f
to
Tf
given
by
Tf (z)
is a bounded map of
L (U,dxdy)
to itself.
This lemma is an immediate consequence of the following two lemmas. Lemma 2. 5:
If
a> 2
Lemma 2.6:
Suppose
q=p/(p-l)
+ ex,
then
1< P
1 .
ing par t a for
The proof of the theorem for pattern.
to norm estimates when prov-
cx," 0
follows the same general
The local modulus of continuity estimates are almost
unchanged because powers of
yare roughly constant on each
Bi
Lemmas 2.5 and 2.6 are sufficient to establish an analog of Lemma 2.4 for the space LP (U,ycx, dxdy) • C.
Variations 1.
Other'domains.
proof of Theorem 2.2:
There were two basic tools used in the first, a reproducing formula which gives a
bounded operator even after absolute values are brought inside the integral sign; second, an underlying geometry with respect to which the kernel of the reproducing formula (and hence also the functions being reproduced) satisfy convenient modulus of continuity estimates. To some extent both of these tools are available for spaces of holomorphic functions defined on symmetric domains in Cn (such domains have a transitive group of conformal automorphisms) and which are integrable with respect to (possibly weighted)Euclidean vohnne.
(These are the general Bergman spaces.)
The appropriate
reproducing formulas involve integration against the Bergman kernel function (or its powers).
Fourier transform techniques can be
used to obtain an analog of Lemma 2.5.
The analog of Lemma 2.6
is true for any measure space and hence a version of Lemma 2.4 can be obtained.
The transitive group of automorphisms can be used
to reduce local questions about the modulus of continuity to a
236
R. ROCHBERG
fixed base point.
Local questions at a fixed base point can then
be analyzed directly. For instance, here is the version of Theorem 2.2 for functions defined on the unit disk in in
en.
or, more generaly, the unit ball
The Bergman kernel for the ball is B(z,w)=c(1-z.;)-(n+I).
We consider spaces
Ilflli
C
I
pr
Iz
A'
of holomorphic functions for which
pr
J r< I
If(z)IPB:z,z)-r dV(z)
(The Bergman kernel for
U
- -2 c (z - w) ,
is
hence
r
in
Apr
). A lattice is defined as before pr in terms of the invariant distance and Theorem 2.2 is true as corresponds to
r/2
A'
in
stated with the basic decomposition formula (2.3) replaced by B (z , z.)
f (z)
r:
f... • 1
B(z., z.)b - (l+r) /p 1
with
b
1
1
b>(I+r)max(1, lip). In this formulation the theorem holds for other domains also,
see [CR] (but also [CR2 D.
The decomposition given in (2.7)
involves powers of the Bergman kernel evaluated at points of a lattice.
(However, other analogs of these results suggest that
the basic building blocks can more profitably be thought of as derivatives (or certain directional derivatives) of the Bergman kernel, rather than as powers of the kernel.) The Hardy space H2. For 0 < P < co, the (holomorphic) Hardy space HP • is that space of functions which are holomorphic 2.
in
U and for which
When convenient, these functions will be identified with their
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
boundary values on
~.
237
(Unfortunately, the space we called
earlier is not the space we just defined with
p=1
HI
Precisely,
the space defined before is the complexification of the space of real parts of the boundary values of the functions in the space just defined, see [CWJ).
A function is in is supported on AP~
H2
(0,00)
if and only if its Fourier transform 2
and is in
L «O,oo),dt).
need not have pointwise boundary values on
however, have boundary distributions.
Functions in the
R.
They do, A function is in A20 if
and only if its boundary distribution has Fourier transform sup2 -1 ported on (0,00) and in L «0,00), t dt). Thus integration of order one-half (which, by definition, means dividing the Fourier transform by t 1l2 ) is an isomorphism of to H2. We can
iO
use this fact to obtain a decomposition theorem for functions in H2 . Theorem 2.8: isa
Fix
c=c(b,d) (a)
if
b > 3/2.
d
If
so that for any f
f (z) = I: A.
1
is in
H2
is sufficiently small then there d-lattice
[Zi} ,
then
(2. 9)
b (z-z. ) 1
is finite then the sum in
Conversely, if
(b)
(2. 9) converges in norm and uniformly on compact sets to a function f
in
H2
which satisfies
Ilfll22 -; cI: 1Ail2 H
This is a direct consequence of Theorem 2.2, of
A20
to
H2,
~f
the relation
and of a Fourier transform calculation which
shows that half-order differentiation and integration have the -a expected effect on the functions (z - z.) • The requirement 1
238
R.ROCHBERG
b> 3/2 for
is an artifact of the proof.
b> I
as is shown by Luecking [L3] using a different proof.
The relation between the when
p=2
for other 3.
The result is actually true
A20
and
HP
is this simple only
In fact, the tempting analog of the previous theorem p
is false [RS].
Bloch, Besov, and BMO.
In this section we discuss the
the decomposition theorems for certain (diagonal) Besov spaces, for the Bloch space, and the space mean oscillation.
BMO
of functions of bounded
The Besov spaces we consider are obtained from
Bergman spaces by integration and hence the decomposition theorems will follow from Theorem 2.2 for
O.
The Bloch space will
be seen as a limiting case of these results for it requires a separate proof.)
The space
BMO
Bloch space and the Besov spaces we consider.
p = ex:>
(although
fits between the It also can be
regarded as a limiting case of the Besov spaces but in a more de lica te way. We have been considering (interchangeably) holomorphic functions defined on
R R
U and their boundary distributions defined on
There are analogous results for spaces of distributions on which are not of holomorphic type.
The difference in the decom-
position theorems is that two lattices must be used •.. one in
U
and one in the lower half plane
L
will involve the derivatives in
U of the holomorphic extension
to
The definitions we will give
U of distributions of holomorphic type.
The definition of
the corresponding general spaces of distributions uses the harmonic extension of the distribution to For
O
select
to be those functions
f
m>l/p
U and partial derivatives. and define the Besov space
which are holomorphic in
U
BP
for which
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
239
It is a consequence of the general theory of Besov spaces, or of Theorem 2.4, that the space so defined does not depend on For any
m> I
m.
the space of Bloch functions is defined to be
the space of
f
for which
sup U
Again, the definition can be shown to be independent of different
m
To define measure.
m
with
giving equivalent norms. BMO
A measure
we first recall the definition of a Carleson u
U
on
is said to be a Carleson measure
if
sup { I a xElR a>O
I~!((x-a,
x+a)x(O,a))}
A holomorphic function is said to be in the space ylf' (z)!2 dxdy llfl\BMo
by
is a Carleson measure.
BMO
We define the
if
BMO
norm,
Although this definition
I\fl\BMO
will be convenient later,
it is not the traditional defintion.
Traditionally a function
f
is said to be in
BMO
of the line
if
sup xER a>O
I (inf 2a c
x+a
J
!f(t) - cldt)
x-a
With this definition it is relatively easy to see that ,the dual of the space given earlier.
HI
The fact that the two definitions of
for holornorphic functions is not so obvious. various descriptions of
BMO
is
for which the atomic decomposition was
BMO
see [G],
It is easy to check that i f
p< r
[Ko]. then
BMO
agree
For more about the
240
R.ROCHBERG
BP
C
Br c
BMO
Bloch.
C
Here are the decomposition theorems for these function spaces. Theorem 2.10: (b> 1
Suppose
Omax(O, 1- lip) • If
d
is sufficiently
so that for any
d-lattice
{zi} (a)
if
f (z)
f
is in
l:"A.
1
B
-
(z - z" )
then
b
1
1"- .I P -s;
l:
with .
1
cUfllP
BP
-
.
f
If
is in the Bloch space then sup£! A)} ;; cllfllBloch
can be represented as in (2.11) with f
is in
BMO
then
f
f If
can be represented as in (2. 11) and the
numbers satisfy a quadratic Carleson measure condition
(Here
6
is the point mass at
z"1
(b)
Conversely, i f
converges in f
in
BP
BP
l: ILIP 1
is finite then the sum (2.11)
norm and uniformly on compact sets to a function
which satisfies
then the sum (2. 11) fies
z1".)
llf ll : p ;; cL:I"-i IP .
c~nverges
IIfllBloch;; c sup £lAiD.
If
sup£1AJ}
to a function in Bloch which satisIn this case the convergence is
weak* convergence with the Bloch space realized as the dual space 10 2 of A • If L: A" y" 6 is a Carleson measure then the sum in 1 1 zi
(2.11) converges to a function in
BMO
which satisfies
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
IIflliMO -;; cllL: 1Ai\2Yi 6z
.lI cM
·
241
In this case the convergence is
~
in the weak* topology with
BMO
regarded as the dual of
In these last two cases, Bloch and need not converge pointwise.
BMO
HI.
the series (2.11)
However, if an appropriate constant
is subtracted from each term then it is possible to obtain convergence which is uniform on compact subsets of
U.
HI
as the dual of
have mean zero.
Hence if we regard
then the elements of constants. space.
BMO
(Functions in HI
BMO
are only well defined modulo additive A similar observation holds for AlO and the Bloch
This is why we only get pointwise convergence only after
subtracting appropriate constants.) Note that for any
b> 0 ,
have maximum modulus of
1
As we mentioned, the
on BP
b --b y. (z-z.) ,
the individual terms
~
~
U. case of the theorem follows from
Theorem 2.2 by term by term integration.
To obtain the Bloch
version of the theorem, first prove an analog of Theorem 2.2 for holomoprhic functions
g
which have
y\g(z)1
(The proof extends without difficulty.)
bounded in
Since such
g
U
are exactly
the derivatives of Bloch functions, the required representation for Bloch functions now also follows from term by term integration. (The proof in [eR] is more awkward. ) The decomposition of a function in starting with the derivative of
f,
BMO
is also obtained by
following the pattern of
the proof of Theorem 2.2 and then integrating the resulting sum term by term.
The proof that such a sum is in
BMO
is obtained
by a direct estimate of the Carleson measure norm of
y 1f'I2 dxdy .
This is given in detail in [RS] where there is also a description of how the individual coefficients
Ai
in the decomposition can
be estimated in ways that emphasize the behavior of
f
near
z
There is natural control at the first step of the approximation
242
R.ROCHBERG
process.
The issue is keeping control as the approximation pro-
cess is iterated.
That type of local control is useful in the
applications given in the last section. 4.
Other spaces.
We have been concentrating on decomposi-
tions of Bergman spaces of holomorphic functions.
Similar tech-
niques can be used to give decomposition theorems for spaces of harmonic functions, including mixed norm spaces with norms such as
IIfll
co
co
o
-co
(J (J
I
\f(x,y)\rdx)syi3dy)rs
That theory, including the application to the description of certain mean oscillation spaces, is given in full by Ricci and Taibleson [RT] for R and by Bui for Rn [B]. Similar results for various mean oscillation, Beurling, and Hardy spaces have been given by Chao, Gilbert, and Tomas [CGT], [CGT2], and by Merryfield [M]. decomposition of tion of 5.
HI
Their results include the atomic
(based on the square function characteriza-
HI.) Other proofs.
Our analysis was based on reproducing
formulas which used the Bergman kernel function.
There is another
type of reproducing formula which was introduced by A.P. Calderon and which has been very useful in obtaining decomposition theorems for various function spaces.
In particular it is at the heart of
Uchiyama's decomposition for
BMO
the atomic decomposition of Chang and R. Fefferman [CF].
HI
[U] and
~
Wilson's proof of
[W] as well as earlier work of
Applications of that point of view
to the Besov spaces are given by M. Frazier and B. Jawerth [FJ]. Related work has been done by G. Cohen [C]. Luecking has developed techniques for analysis of functions in the Bergman spaces
whi~h
yield
theorems we described [LI,2,3].
many of the decomposition
Although his approach is less
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
243
constructive (since he uses a duality argument from functional analysis at a critical stage) it gives a clearer insight into the role of the local geometry and the dual nature of the conditions that a sequence be dense and that it be separated.
He also gets
decomposition theorems for weighted space---a context in which conformal automorphisms are less useful. 6.
Atomic and molecular decompositions of other spaces.
Decompositions of the type we have been describing were first used systematically for the Hardy spaces
[CW].
A good way to
find out about recent activity in that area would be to scan the conference proceedings [BCFJ]. Various other function spaces have also had their members dissected in similar ways recently.
A decomposition theorem for
a Sobolev space, together with various applications is given by Jodiet in [J].
The spaces of functions made of blocks (not the
Bloch space) was first introduced by Taibleson and Weiss via an atomic decomposition.
For more on these spaces see [TW] and [So].
The tent spaces of Coifman, Meyer, and Stein are closely related to the Hardy spaces and have similar decompositions.
(See
[CMSI,2]. ) APPLICATIONS Many applications of atomic and molecular decompositions use the fact that the individual building blocks satisfy size, localization and cancellation estimates.
Although some of our applica-
tions are of this sort,most of them exploit the explicit form of the terms in Theorem 2.2. Some of the results we describe could be obtained by working directly with the reproducing formula. proofs using quite different ideas.
Others have alternative
Our main theme here is that,
once the decomposition theorem is available, many results follow quiet naturally and easily.
To emphasize the simplicity of the
method we present some results in less than full generality.
R.ROCHBERG
244
However, a virtue of the method is that once the proof is given for the simple case, the more general proof is often identical. (For instance, many of the proofs extend directly to Bergman spaces of functions of several complex variables. ) A.
Inclusion Theorems, Rational Approximation and Interpolation For fixed
p
p
and variable
or for fixed
~
and variable
~
there are no inclusion relations between these spaces.
However,
if both parameters vary at once then there is a family of inclusions.
This is an immediate consequence of the form of the
individual terms in (2.3), the fact that for fixed
p
Theorem 2.2 can be used with any sufficiently large inclusion relations between the sequence spaces t P Theorem 3.1:
~
and
a,
and the
That is,
Suppose
o< P < pi,
(2
+ ~) / p =
(2
+ ~I
)
/
p,
(3.2)
•
Then
the inclusion is continuous, and if
Al~1 in
pi = 1
of the convex hull of the unit ball of
then the closure in
AP~
contains a ball
Al~'
Thus, informally, taining
A~.
Corollary 3.3:
Al~I
is the smallest Banach space con-
Precisely The spaces
AP~
and
Al (2 +~) /p
have the same
duals. We have not defined Bergman spaces with
~
= -1.
With
appropriate normalization that limiting case correspond to having norms defined by intergation along the real axis.
This suggests
that we might interpret the Hardy space (which, for holomorphic functions can be defined in terms of the LP(R,dx) norm of their boundary values) as the spaces AP - 1. Although not all the
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
245
conjectures suggested by this analogy are correct, in this case we have Theorem 3.4:
Ol
For
p< 1
that is innnediate.
For
it is still true as can be seen by going through the proof
of Theorem 2.2.
(The crucial fact here is that the kernel in
Lemma 2.4 is positive and hence cancellation is not being used when combining the estimates for the individual terms. ) Corollary 3.6: Bl is an algebra. Tha tis, i f F,G are in then FG is in Bl and IIFGIl 1 < cllFlI 1 IIGII 1 B B B
Bl
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
247
Note that by the definition of F = I2f for some f in AlO •
Proof:
ros
we can write
in
F
Similarly we write
2 G = I g.
Thus we must show that
n2 (FG)
=
fG + 2 If Ig + Fg
is in
AlO .
A20 •
Hence the product is in
By the previous theorem,
note that, by Theorem 2. 10,
AlO
If
and
Ig
To see that
are both in fG
is in
AlO
G can be written as a sum of uniformly
bounded functions with summable coefficients.
Thus
Multiplication by a bounded function clearly takes The term Fg is in AlO for the same reason.
G f
is b ounde d. into AlO •
The individual terms in (2.11) are rational functions, all of the same degree.
Hence Theorem 2. 10 can be used to study questions
of rational approximation. on
For a holomorphic function
U we can measure how well
f
f
defined
can be approximated by rational
functions using the approximation numbers Rn (f) = inf{l!f - riiBloch ; r n
is a rational function of degree with poles in the lower half-plane}
or r
is a rational function of degree
n
with poles in the lower half-plane} •
In fact, either of these two sets of numbers characterize the spaces
BP Theorem 3.7: (a) space
f
is holomorphic in
.t P
U
,
O
f
is in
BP
i f and only i f
{R } n
is in the sequence
f
is in
BP
if and only if
{r n }
is in the sequence
.t P (b)
space
Suppose
248
R. ROCHBERG
Proof discussion:
f
If
BP
is in
then, by Theorem 2.10,
can be written in the form (2. 11) for some integer
b
be the non-increasing rearrangement of the numbers
{Ai}
number
~k
r
of
Since the Bloch norm of
to
f
the
k
maximum of the remaining ~k ~
the estimate sequence
Let
* {Ai} The
can be estimated by selecting as the rational appro-
ximation Ai.
f
[A k }
terms in (2.11) with the largest values
Ak
cA k* ·
can be estimated by the
(using Theorem 2.10 again) we get
Since the
tP
is in
f - r ~k
the sequence
are decreasing and the
~
is in
tP
The proof of the analogous estimate for the numbers
rk
goes
the same way as soon as we know which terms in (2.11) to use in the rational approximation. have small
BMO
The goal is that the remainder should
norm (as estimated using Theorem 2. 10).
This
requires the following Lemma 3.8: in k
U and
Suppose [A.}
0< p < ...
Suppose
{zi}
is a fixed
tP
is a sequence of numbers in
~
d-1attice
There is a
so that it is possible to renumber these sequences so that the
sequence with n-th term
II is in Proof discussion:
Let
indexed by
IJ.
be the measure
2 I: A. y. 6
Regard zi the points of the lattice as indexing a set of Car1eson boxes in
U •
Let
times the
be
IJ.
~
measure of the Car1eson box
• That is
It is fairly direct to show that the sequence the sequence
~
[Ai}
is.
[oil
is in
t P
if
Renumber the lattice so that the function
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
of
i
given by
249
max {ok : d (zi' zk) < I}
is non-increasing.
A
stopping time argument can then be used to show that the sequence with terms given by (3.9) is in t P • Details are given in [RS]. The proofs in the other direction, that functions with good rational approximation are in BP , requires the theory of Hankel operators.
It is completed by an appeal to Theorem 3.14.
(It would be interesting to free the second part of the proof from the theory of Hankel operators.
A related problem is to find
an appropriate analog of this theorem for harmonic functions in
Rn+ 1
+
.
)
Theorems 2.2 and 2.10 can be used to study the intermediate spaces (in the sense of real or complex interpolation theory) between various Besov and Bergman spaces. results in any detail.
We won't discuss those
The general idea is quite similar to that
in the previous theorem and it's proof.
The heart of the study
of intermediate spaces is splitting a function into two pieces with good control of the norm of the pieces in different normed spaces.
The previous proof was exactly of that type except that
we measured the "norm" of a rational function by it's degree. Standard interpolation theoretic techniques for the sequence spaces t P can be combined with Theorem 2.2 and Theorem 2.10 to yield most standard results about interpolation of the Bergman and the diagonal Besov spaces.
For instance, it follows easily from
Theorem 2.10 that the complex interpolation spaces between the Bloch space and BP are other BP spaces. USing the previous lemma one can also obtain the (slightly more subtle) result that the real intermediate spaces between BMO and BP are BP spaces. B.
Hankel Operators
In this section we use the decomposition theorems for the spaces BP to describe the Schatten ideal theory for Hankel
250
R. ROCHBERG
operators. [Po].
We will use SOme of the introductory material from
A different approach to these topics has been developed by
Peller,
[PI, 2, 3], [PH].
These references together with the survey
by Peetre [Pe2] give a good introduction to the theory of Hankel operators and Schatten ideals. Suppose
b
is a holomorphic function on
Hankel operator with symbol H2
H2
P
the
L2 (R,dx) ) given by
= Q( b f)
Here we write to
~,
is the map from the Hardy space
to its orthogonal complement (in ~ (f)
L2
b,
U.
(3. 10)
for the Cauchy (
Q= I - P
and write
= orthogonal)
projection of
for the complementary projection
onto conjugate analytic functions.
We will use the same name and 2 notation for the operator defined on L «O,oo),dt) by 00
~ (f) (t)
-
J b (s + t)f (s )ds
(Here, and throughout this section, form.)
(3.11)
o
denotes the Fourier trans-
These two operators are unitarily equivalent.
The equiva-
lence is implemented by the Fourier transform followed by the change of variables sending
t
to
-t.
(The equivalence of various points of view about Hankel operators uses the fact that the orthogonal complement of L2 H2.
H2
in
is the same as the space of complex conjugates of functions in For instance,
(3.10) to (3.11). variables,
that equivalence is used in the passage from In other contexts, such as functions of several
there is no natural substitute for this equivalence.
This causes problems in trying to decide what would be the "natural" extensions of the theory of Hankel operators to other contexts. ) The basic boundedness result for Hankel operators is that
~
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
251
will be a bounded operator if and only if more the
BMO
norm of
b
b
is in
BMO
Further-
is equivalent to the operator norm of
l1,. The general theme we will be developing is that small exactly if ~
is that
b
is smooth.
n
will be
A classical result of that sort
will be an operator of rank
rational function with
~
n
exactly if
b
is a
poles, all in the lower half plane.
Another result of the same general sort is that compact operator exactly if
b
l1,
will be a
has vanishing mean oscillation.
(Fqnctions of vanishing mean oscillation form a subset of consisting of relatively smooth functions. for instance, as those functions in
BMO
BMO
They can be described, for which the decomposi-
tion series (2.11) converges IN NOR~ ) The Schatten ideals (also called trace ideals and Schattenvon Neumann ideals) are classes of operators on Hilbert spaces with small ranges.
They are intermediate between the finite rank oper-
ators and the compact operators. operator
A
More precisely, for any bounded
(on a Hilbert space) and any non-negative integer
n
let s
n
=s (A)=inf[IIA-BIl ; n
B
is an operator of rank at most n}.
These numbers are called the singular values (singular numbers) of
A
If
A
is a compact positive operator, they are exactly
the eigenvalues of A
A
in decreasing order.
I
they are the eigenvalues of
the Schatten ideal
S
p
For a general compact
IA = (A*A) 1/2.
For
0< p
0) .
1 1 1
Thus the Hankel operator corresponding to an individual term is the operator which takes
to
f
H. f : 1
S y.e izi(t+s) f (s ) ds a>
o
1
This is the one dimensional operator which takes -iz.t 1
with
f
Now note that the norm of
to
(f,ei>;i
and hence
also the operator norm of this one dimensional operator, is independent of
Thus the sum (2.11) generates a sum of one
z.. 1
d~ensional operators with individual norms at most
the operator norm and the operators we have
IIHblll =
S
p
cIA.I. 1
Since
norm are the same for one-dimensional
II !:AiHi ll
-;;!:
lA)
IIHilll -;; c!: IAil -;; cllbll 1 B
which is the desired estimate. When if
b
p< 1
is in
BP
the lack of convexity works with us.
That is,
then we can argue the same way and then use (3.12)
in place of the triangle inequality to conclude that
~
is in
S
p
Suppose now that d-lattice
[zi}
~
with small
is given. d
and let
the corresponding building blocks for
H2
fj =
Pick and fix a 3/2 - - 2 be yj (z - Zj)
in the sense of Theorem
R.ROCHBERG
254
2. B.
The functions
fj
are not an orthonormal set but they are (That follows
the image of an orthonormal set under a bounded map. easily from Theorem 2.B.)
Hence (since the
estimate (3.13) also holds for the work directly with the picture.
f.
fj •
S
p
are ideals) the
However, rather than
we will work with the Fourier transform
J
This has the advantage that the domain and range space
of the operator are the same.
It also gives us an opportunity to
correct some arithmetical errors which occur in [Rl].
We wish to
To compute this we first compute that
(t> 0) •
Thus A
~
A
(R. f., f.> =c -0 J
J
3
II b(s+t)y.J tse
-iz.t -iz.s J e J dsdt
When we change the variables
t
w= s + t
integral this gives
and evaluate the 00_
I
C
t
and
s
to new variables
t
and
•
3 -1Z'W b (w)w e J dw. A
o
This is the Fourier transform formula for computing the third derivative, hence
A...
3 J
(Ii f. , f. > = c y. b -lb J
J
(3)
(3.15)
(z . ) J
Thus the sum we are considering is a Riemann sum for the integral we wish to estimate.
f. , f.) I. -0 J J
c t I (R. all
Specifically
t ly~b3(z.)! J
J
Taking the supremum of the left hand side over
d-lattices gives the required estimate:
~
IIbllB 1
cI: I(~fj' fj>! ~ cll~lIl' CRl] for
&
= 1/4.
be replaced by
(This is the argument on pg. 917 of
Severa 1 places in tha t argument,
2&.)
&
should
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
255
This argument extends unchanged to all
p> 1.
The only new
ingredient we need is to note that (3.13) is valid for those Suppose now tha t
p> 1
Hb
want to show that
and the symbol
is in
Sp
b
is in
p.
BP •
We
This follows from interpolation
theory applied to the map from symbols to Hankel operators. know that that map takes functions in
Bl
We
to operators in
Sl
(from the first part of the proof) and we know that the map takes functions in
BMO
to bounded operators.
Interpolation theory
then insures that the map takes the (appropriately defined) spaces between
Bl
and
BMO
to the corresponding spaces between
and bounded operators.
Sl
It is a result from the theory of Schatten
ideals that these latter spaces can be taken to be the spaces
S
P
We noted in the previous section that Lemma 3.8 can be used to show that the spaces intermediate between are the spaces in
BP •
BP
1< P
P
(3.16)
If this matrix were diagonal then (3.16) would be a consequence of the definition of
What is true is that the matrix is
S
p
nearly diagonal in a useful sense. where
D is the diagonal part.
Write the matrix as
D+R
If we could show that the
norm of the off diagonal remainder
R
S
p
satisfied (3.17)
for a very small constant
IIbli p B
IIDli s
c
,
then we would have
p
which gives the required control of
b.
(3.17) is true if the lattice constant, The reason is that estimates for the
S
p
d,
is very large.
norm of the matrix can
be obtained by combining the norm estimates for the one dimensional operators corresponding to the individual matrix entries. this proceedure is carried out for particular choices of
When b,
the
resulting estimate is a Riemann sum for an integral similar to the one estimate. in Lemma 2.5.
If attention is restricted to the off
diagonal elements of the matrix then the corresponding integral is of the same general sort but the domain of integration is restricted to the set of
z
integral controls
with c
,
d (z,,) > d.
Since the value of that we may make c' sma 11 by making d
large. We cannot choose the lattice constant
d
to be large in
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
Theorem 2.B.
257
However, we can take a lattice with small
express it as a union of lattices with very large this has the effect of splitting many subs paces.
2
H
d.
d
and
Roughly,
as a direct sum of finitely
The Hankel operator which is being studied can,
in effect, be realized as a direct sum of operators of the same general sort acting on each of these subspaces.
The considerations
of the previous paragraph apply to each of those operators.
We
can then combine the estimates without loss because the operator norm estimate for a direct sum of operators is the maximum of the individual estimates. The details of this proof are in [R1], [S], and, in an improved form, in [RS].
Alternative proofs are given by Peller [P1,2,3]
whose proof of Theorem 3.14 for
p= 1
(by quite different tech-
niques) was the inspiration for much of the recent work in this area. Analogs of Theorem 3.14 hold for some operators closely related to Hankel operators. we define the operator defined on
(0,=)
~~f(X)
=
For instance, for any complex
~,~
~~ to be the operator acting on functions
given by
J= s~t~(s+t)-~-~ o
b(s+t)f(t)dt
D~ + ~ c = b. These opera tors can c also be viewed as Hankel type operators acting between potential This opera tor is
D~H D~ wi th
spaces (i.e. spaces of the form
D~2 ).
Alternatively, these operators can be regarded as Hankel type operators on the Bergman spaces A2Y • That is, one can define operators on the A2Y using formulas similar to (3.10) but starting with
f
in the Bergman space and using the Bergman projection.
(Such operators were studied in [CRW].) Fractional integration 2Y 2 gives a unitary equivalence of A and H and hence can be used to pull these operators over to .~.
When this is done (by
258
R.ROCHBERG
straightforward Fourier transform calculation) the resulting operators are of the form
~~.
The techniques we have been discussing (and also those of Peller) extend fairly directly to these more general operators if some restrictions are put on ~
and
a
~
and
(What happens for
a
outside that range is a bit of a mystery. )
Theorem 3.18:
Suppose
min(l/Z,l/p»O.
~~
0< p < <Xl is in
and that
min (Re a, Re
i f and only if
Sp
b
~)
+
is in
B
P
Furthermore the norms are equivalent. A class of operators which has been studied a lot recently are the operators defined on plication by a
LZ (lRn)
as commutators between multi-
(relatively) smooth function and operators given by
(possibly singular) intergral kernels.
The basic philosophy is
that the smoothness of the function used in multiplication helps balance the singularities of the kernel and the resulting operator will be tamer than the
integral operator alone. LZ (lR) ) the projection
In one dimension (i. e. on onto
HZ
P
LZ
of
can be taken as the fundamental singular integral operator.
This leads to a close relation between Hankel operators (and, more generally, the
~~)and
commutators.
We will say that a function P(b)
b
BP
is in the space real
and the complex conjugate of
(I-P)(b) BP •
holomorphic type) are in the space
if
(which are both of
We will use the same
notation for a function and the operator of multiplication by that function. where
D
We are interested in the commutators is differentiation and
the so called Calderon commutator. bounded on LZ if b is in BMO
DB = b.
[b,P)
and
[B,P)D
The second opera tor is
Both of these operators are (see
[CMl, Z)).
e. g.
We discusS
the boundedness in Section D. Theorem 3.19: if
b
is
Suppose
in the real
O
=c
([B, P]D)f(x)
J
b (u) - B (x, u)
x-u
-0>
with
Integration by parts gives
B(x,u) = (B(x) - B(u»/(x - u)
parameter
b=2
f (u)du
(3.28)
If we use Theorem 2.10 with
to decompose the function
b,
we find (using
partial fractions) that the kernel of (3.28) can be written
b (u) - B (x, u)
c I:
x-u
A. (x - z. ) ~
(u - z.)
~
2
~
Thus [B,P]Df = c I: A.(f,g.) f. ~
~
~
and
with the ftinctions be in
L2
By Theorem 2.8
are weakly orthonormal, that is
if the
ai
are square summable.
same property we would only need that
Ai
If
I: a.g. ~
{f i }
will
~
had the
are bounded to conclude
that the operator given by (3.29) is bounded.
However
not that nice (and that conclusion would be wrong).
{f.}
is
~
What is true
is Lemma 3 30: sequence
Suppose
{Ai}
{zi}
is a
is such that
d-lattice for some
~ y.A:6 ~
[Aif i }
~:
and compute
I:
\LI 2 ~
g
Since
y.g(z.) ~ ~
I: 1Ai\2 Yi6z.
L2
g
in
is a Carleson measure
~
this last quantity is dominated by for all
If the
zi is weakly orthonormal.
~
then the set of functions Take a
d.
is a Carleson measure
H2
cllgll 2 •
Having this estimate
is equivalent to the required estimate.
case of the theorem follows from this. We now consider the
BMO
boundedness.
It is enough to
The
R.ROCHBERG
268
consider the case when
B
is conjugate analytic.
In that case
h = [B, P ]Df
Thus
h
is conjugate analytic and is determined by
show
h
is in
Pick
g
with
BMO g
P(f).
by showing it pairs with functions in HI.
in
< c!: II.1 . I Yi If' (z. ) I I g (z. ) I 1 1
I (h, g) I
The first factor is controlled using the Carleson measure condition on the
Ai
and the fact that
controlled by the of
!:
I
H
norm of
y~If'(z.)126 1 1 z.
g
is in g
HI.
The second factor is
and the Carleson measure norm
That measure is a discrete version of
1
If' (z) 12 ydxdy
which is a Carleson measure because
f
is in
BMO.
The operators just considered are the linear terms in the multilinear expansion (i.e. Taylor series) of the weighted norm inequalities for the Cauchy projection and of the Cauchy integral on chord-arc curves.
With more effort, the same techniques---
decomposition of the symbol of the operator, partial fraction analysis of the kernel, and weak orthonormality results obtained using Carleson measure estimates---can be used on the quadratic terms in both
e~pansions.
That is, the same ingredients can be
used to establish the boundedness on L2 with b in BMO and of [A, [A,PD 2 ]] if
of the operator
[b, [b,P]]
A'
Other
is bounded.
trace ideal results for operators related to the Cauchy integral are in [S2]. In the proof of the decomposition theorems there is a close
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
relation between the value of (2.3) or (2.11).
f
269
at
zi
and the numbers
Ai
in
Similarly in the proof of the theorem on inter-
polation of values there is a close relation between the value which the function
f
is to take at
constructed function near
zi'
z.
1
In the previous sections we didn't
try to make this local control precise. norm control.
and the size of the Instead we settled for
It is possible however to extract the local estimates
from the proof.
The key is to estimate the spreading effect when
the crucial approximation step is iterated.
One way to do this
is to get good estimates on the high powers of the operator in Lemma 2.4.
T
Here is an example of the type of control which is
possible. Lemma 3.31:
Suppose
is given.
&>0
possible to choose the numbers BMO
function
f
so that
Ai
In Theorem 2.10 it is
in the decomposi tion of the
IA.I O.
Suppose
Suppose a sequence of values
Solve the interpolation problem
,where of center
b.
1
f (zi)
{zi}
is an
s-scattered
is given and we wish to
=b i '
Let
b (z)
=I: b iXi (z)
is the characteristic function of the hyperbolic disk and radius
1 •
If
s
is sufficiently large then
270
R.ROCHBERG
a sufficient condition that the interpolation problem have a solution is that for some (and hence any) finite.
In that case the function
f
z
in
U,
Gb(z) be
can be choosen to satisfy
If (z) I < c Gb (z) • Theorem 3.26 is a special case of this because operator
G
the
(or other operators for which similar results hold)
is bounded on the various
AP~.
The point of both proofs is that for small G dominates the operator tion when iterates of
~ C~n.
Thus
T are combined.
C the operator
G controls the situaThe fact that
G has
the required properties is obtained by direct computational estimates. We now describe a singular value estimate for products of Hankel operators.
We will only indicate some main themes of the
proof. We consider Hankel operators as maps from of functions in
H2.
~
That is,
that we have two such operators, and
b2
H2
to conjugates
is given by (3.10).
HI
and
H2
Suppose
with symbols
bl
2 * (which maps H HlH2 In addition to its intrinsic interest, this operator
and we wish to study the composite
to itself).
arises in the study of Toeplitz operators as the semi-commutator HI*H2 = T"b b - T"b Tb 1 2
1
where the
TI s are Toeplitz operators.
2
With Hankel operators and with commutators a basic theme was that smoothness in the symbol led to smallness of the operator (i.e. small singular values).
Here the basic theme is that the
smallness of the product operator is controled by the product of measures of smoothness for the two factors. [zi}
U• b (x)
by
is a
d-lattice in
The derivatives of
U and b
at
(the boundary values of Yi.
b
That is, suppose
is a function holomorphic in measure the smoothness of near
and on a scale given
This informal reasoning suggests that we try to control
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
271
the product operator with the numbers
b' (zl)b' (z2) •
Although this general philosophy is correct, in order to get the proof to work we use a different measure of smoothness, one based on mean oscillation.
Also, the estimates on the singular
numbers involve the products of smoothness estimates summed over Carleson boxes. Suppose p>2 U
and
and
z
d
is a positive integer and
n>pd+ I and
w
nomial of degree
For a function in
d
U of
we let b
b
nand
satisfy
which is holomorphic in
P(b)(z)(w)
at the point
(P(b)(z)(w)=b(z)+b'(z)(w-z)+ •.• ).
p
z
be the Taylor polyevaluated at
For a point
w
z=x+iy
in
U we measure the smoothness of the boundary value function at the point
x
and the
sc~le
y
by the number
osc(b,z)
b(t) given
by CZ>
osc (b, z)
(J -CZ>
n - I Ib (t) - P(b) (z) (t) P _Y'---_ It -
I
z,n
dt) IIp •
Roughly, this quantity measures the oscillation of Over the Carleson box centered at
z
b(w)
Various spaces of distri-
butions on the line which are characterized by mean oscillation conditions can also be characterized by the size of these numbers as
z
varies over an
0< p < co
r-Iattice.
In particular the spaces
can be characterized that way.
Suppose tha t
{z.} 1
is a
BP
(See (11.9) of [RT].)
r-Iattice in
U.
Define
D.1
Let
*
tD.}
~eorem
1
be the non-increasing rearrangment of
3.33:
If the lattice constant
there are constants
c
and
K
so that
r
tD.}. 1
is small enough then
R.ROCHBERG
272
n=1,2,3, ••.• To understand this theorem a bit better we first restrict attention to the case osc (b, zk)
2
•
b
If
b1 = b2 •
D n
is then a sum involving
is assumed to be in
ation numbers are bounded.
BMO
then these oscill-
However, the fact that the oscillation
to be fini teo n Hence the theorem does not give the sharp boundedness criterion. numbers are bounded is not enough to force
D
On the other hand, if b is in BP then the oscillation numbers will be in t P and that is enough to force the D to be in 2 t P/ This gives the conclusion that is in nsp This
Eb
argument works for all
p
and can be used to replace the inter-
po1ution argument in the proof of Theorem 3. 14. Suppose now that there are two different operators.
Just
as we could not recover the boundedness criterion for a single operator, we do not recover the very nice results ofAx1er, Chang, Sarason, [ACS] and Vo1berg [V] which give a necessary and sufficient condition for
H*H2 to be compact. There are, however, 1 conclusions to be drawn. If b i is in l i then osc (bi' zk) is in
t
P. and hence the product operator is in
1.
S
P
l/p =
with
We didn't really need the theorem to get this result (because an analog of Holder's inequality holds for the Schatten ideals) but we can also get a localized version.
That
is, suppose th8t each point of the (extended) line has a neighborhood in which the boundary values of those of
b2
in
opera tor is in
B
S
P2
are locally in
l/p = 1/P1 + 1/P2
with
P1
and
P2
and
then the product
(Of course one must define "locally" and
P
the various estimates must be un form. ) that the indices
b1
The innovation here is
may vary from point to point.
273
DECOMPOSITION THEOREMS FOR BERGMAN SPACES
Another consequence of the theorem is that it gives boundedness and Schatten ideal criteria which apply even when one of the operators is unbounded. p
For instance, if
then the oscillation numbers of
a way that can be estimated.
If
bl b2
bl
is in
LP
for some
will be unbounded, but in is smooth enough then its
oscillation numbers will be small and can balance this unboundednesS.
The theorem allows this to be made precise and, as before,
to be localized. Here are some of the ideas of the proof of the theorem. ~ =
suppose tha t we only had one opera tor, that f
b
is conjugate analytic.
H
explicitly
1
to each term.
in
H2
and decompose
1
We can
By direct calculation
b(x) -P(b)(z.)(x)
m - 1/2 Yi
~(
f
We may suppose
m - 1/2 --m f = I: A. y . (z - z i ) .
according to Theorem 2.8 as
apply
Pick
H.
First
1
Now the point is that this vector is the number
osc (b, zk)
times a vector with good orthogonality properties (more precisely, the same type of orthogonality properties as the functions in the proof of Lemma 3.30. tion shows up.)
f.1
That's why a Carleson measure condi-
The exponent
p
in the definition of
osc(p,z)
is to allow for the use of Holder's inequality in the demonstration of this orthogonality.
p>2
is required so that the conjugate
exponent will be less than two.
That allows quadratic estimates
on other terms to be obtained using a maximal function. To use the estimates on the oscillation numbers we partition the lattice into two sets.
A finite set of controlled size gives
rise to the finite rank operator which does the approximation. norm estimate for the oscillation numbers associated with the complement gives an operator norm estimate for approximation.
H
minus the
To do the analysis of the product operator, the
A
R.ROCHBERG
274
second operator is applied to the coefficient functional which sends
to
f
A. (f) 1
(Note that this functional is in the domain
HI* . ) I t is in the analysis of that coefficient functional that estimates are needed of the spreading effect produced by
of
iteration of the approximation operator.
Those estimates are
similar to the ones needed for the preceding two theorems. 1.
Supported in part by the N. S.F.
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Manuscript, 1984.
)PERATOR-THEORETIC ASPECTS OF THE NEVANLINNA-PICK INTERPOLATION PROBLEM
Donald Sarason Department of Mathematics Universiry of California Berkeley, CA 94720, U.S.A. The Nevanlinna-Pick problem is the problem of interpolating prescribed values on a given set of points in the unit disk by means of a holomorphic function obeying a prescribed bound. These lectures are intended to bring out certain operator-theoretic aspects of that problem. Two approaches to the problem will be discussed, the original function-theoretic one of R.Nevanlinna and a recent operator-theoretic one due to J. A. Ball and J. W. Helton. The latter approach will be employed to study the extension problem for Hankel operators.
1.
Lecture 1:
Introduction; Preliminaries on Krein Spaces
Lecture 2 :
The Schur Algorithm
Lecture 3:
The Nevanlinna-Pick Problem as a Problem on Extensions of Operators
Lecture 4:
Extensions of Hankel Operators
Lecture 5:
Extensions of Hankel Operators (Continued)
INTRODUCTION; PRELIMINARIES ON KREIN SPACES These talks concern a basic problem in function theory first
studied nearly eighty years ago by C.Caratheodory (7).
The focus
here will be on a variant of Caratheodory's problem introduced by G.Pick (17) and R.Nevanlinna (15) which is conveniently stated in two parts.
Let Hoo denote, as usual, the space of bounded holo279
s. C. Power (ed.), Operators and Function Theory, 279-314. e 1985 by D. Reidel Publishing Company.
280
D. SARASON
morphic functions in the open unit disk, The closed unit ball in Hoo
D,
of the complex plane.
(relative to the supremum norm) will
be denoted by ball Hoo .
(A)
If
zl"",zn
are distinct points of
D and w1, ... ,wn
are
complex numbers, under what conditions does there exist a function (B)
(where
denotes the given inner product on H). The spectrum of J
is {l,-l}.
are commonly denoted by H+ and H_,
The corresponding eigenspaces and the orthogonal projections
onto these eigenspaces are denoted by P+ and P . J = P+ - P _,
Thus
and
What follows is a review of the basic notions and facts from the theory of Krein spaces that are needed in the sequel. proofs will be given.
No
The standard references on the subject
are (4),(6),(13). The vector x [x,x] ~ 0
in the Krein space H is called positive if
and negative i f
[x,x] ;;; O.
A subspace of H is called
positive if it consists of positive vectors and negative if it consists of negative vectors.
(Convention:
subspaces of a Hilbert
space are assumed to be closed.) If
T is a contraction operator whose domain is a subspace
of H
and whose range is contained in H ,
of T,
is a positive subspace of H.
+
-
then G(T),
the graph
Conversely, each positive
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
283
subspace of H is the graph of such a contraction, called the angular operator of the subspace.
A positive subspace is maximal
positive, that is, not properly contained in another positive subspace, if and only if the domain of its angular operator is all of H+.
A positive subspace is called uniformly positive if
the norm of its angular operator is less than 1.
(In particular,
the trivial subspace, {O}, is uniformly positive.)
There are
analogous connections between negative subspaces and contractions whose domains are in H
and whose ranges are in H+.
A number of basic Hilbert space notions have J-analogues. For instance, two vectors x and y in H are called J-orthogonal, written x [1] y,
if
[x,y] = O.
The J-complement of the subspace M
of H, written M[1], is the set of vectors in H that are J-orthogonal to every vector in M; it is related to the usual orthogonal complement of M by the equality
M[1] = JM1.
the operator T on H, written T[*], [Tx,y] = [x, T[*]y]; the equality
are assumed to be bounded.) [x,x] for all x,
is defined by the relation
it is related to the usual adjoint of T by
T[*] = JT*J.
adjoint i f T=T[*].
The J-adjoint of
(Convention: Hilbert space operators The operator T is called J-self-
I t is called a J-contraction i f [Tx,Tx];;;
a J-isometry i f [Tx, Tx] = [x,x] for all x,
J-unitary if it is both invertible and a J-isometry. operators are those satisfying
T[*]T = TT[*] = 1.
and
The J-unitary An operator on
H is called a J-projection if it is a J-self-adjoint idempotent. The subspace M of H is called regular if it can be written as the J-orthogonal sum of a uniformly positive subspace, M+, and a uniformly negative subspace, M_.
One can make such a subspace
M into a Krein space in its own right by retaining the J-inner product but introducing a new positive definite inner product. The new inner product is the one that agrees with [ :find with - [ ,
on M.
] on M+
It makes M into a Hilbert space with a
norm equivalent to the one inherited from H.
One consequence is
. that a linear manifold in the regular subspace M is dense in M if and only if no nonzero vector in M is J-orthogonal to it.
It
284
D. SARASON
can be shown that an equivalent condition for the regularity of M is that M be the range of a J-projection; another equivalent condition is the equality
H = M +M[l].
In particular, the regu-
lari ty of M implies that of M[ 1] . The simplest example of a Krein space is the two-dimensional space «::2 with the symmetry and y = y
+
@
y
-
=(~ _~).
J
Thus, for x = x+
@
x_
in C 2 we have (x,y) [x,y]
If
T = (:
~)
is an invertible 2 x 2 matrix, then T corresponds
both to an operator on C 2 and to the linear fractional transforma-----'"" az+b One easily tion z --,. cz + d on C, which we also denote by T. verifies that the linear fractional transformation T maps the unit disk into itself if and only if the operator T transforms negative vectors to negative vectors.
Further, one can show that the
preceding condition holds i f and only i f T is a scalar multiple of a J-contraction.
Similarly, the linear fractional transforma-
tion T maps the unit disk onto itself if and only if the operator T is a scalar multiple of a J-unitary operator. is J-uni tary i f and only i f ab=cd.
=
Id I 2 - Ib 12
I t is a J-contraction i f and only i f
Id 12 - Ib I2
;;; 1,
= 1
and
lal2-lcl2 < 1,
and
Iab 2.
Ia 12 - IC 12
The operator T
cd 12
~
(1 -
Ia I 2 + I c I 2)( Id I2 - Ib I 2 -
1) .
THE SCHUR ALGORITHM The Schur algorithm effectively handles part (B) of the
Nevanlinna-Pick problem.
It is based on the Schwarz-Pick lemma,
which is the conformally invariant form of the well-known Schwarz lemma.
For
a
in D we let b a
factor for D vanishing at
a:
denote the normalized Blaschke
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
~(a-_z)
b (z) a (b o(z) = z).
1-
az
This is the unique conformal automorphism of D that
to 0 and 0
sends a
285
derivative at 0,
to the positive real axis (or has a positive
in the case a = 0) •
Schwarz-Pick Lemma (9, p. 2) :
If the function cp is in
ball HOD and is not a unimodular constant, then, for any
a
in D,
The preceding inequali ty is strict at all points of D other than a unless
cp
is a conformal automorphism of D.
Turning to the Nevanlinna-Pick problem, we assume given n distinct points zl' ... ,zn in D and n
values wI' ... ,wn which
are to be interpolated along zl' ..• ' zn by a function in ball Hoo • We assume the interpolation is possible, and we ask for a description of the most general function performing it. we shall write bj
for b zj .
Suppose the function cp in ball Hoo satisfies j = 1, ... ,no
cases:
We look first at the point zl
IwI I = 1
and
IwI I < 1.
the maximum principle that cP in particular, w2
For simplicity
' •••• wn
CP(Zj) = Wj ,
and distinguish two
In the former case we conclude from is identically equal to wI
all coincide with wI).
(and so,
In the latter
case, the Schwarz-Pick lemma tells us that the function
(4)-:1 )
bl
l-w l 4>
1
is in ball HOD.
'1'
Solving the preceding equation for
we find that b 1 = h(zj)
Mb
bH2;
k j (z)
=
(1- zjz)
-1
,
is the kernel function for the point
for h in H2.
The shift operator on H2 will be denoted by S: zh(z).
it is a sub-
(Sh) (z) =
The subspace Mb is S*-invariant, being the orthogonal
complement of the S-invariant subspace bH 2
* k j are eigenvectors of S:
In fact, the functions
S*k j = -zjKjo
On Mb we define an operator A by setting
The
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
operator A*
293
obviously commutes with S* 1Mb.
We ask whether A*
can be extended to an operator on H2 which commutes with S*. An operator on H2
which commutes with S*
an operator which commutes with S,
is the adjoint of
and, as is well known and
easily shown, the operators commuting with S are the multiplica00 tion operators induced by the functions in H. Suppose cJ> is a function in Hoo ,
the adjoint of whose induced multiplication
operator extends A*. (k i , A*k j ) equals
Then
cJ>(Zj) = Wj
for each j,
because
Wj (k i , k j ) on the one hand, while on the other
hand i t equals (<j>k i , k j ), and (<j>k.,k.) 1
J
The same argument shows that, conversely, if cJ> oo
H
such that
cJ>(Zj) = Wj
is a function in
for each j, then the adjoint of the
multiplication operator induced by cJ> extends A*. norm of the multiplication operator on H2
Moreover, the
induced by cJ> is 1IcJ>1100.
We see therefore that part (A) of the Nevanlinna-Pick problem can be reformulated thus:
Under what conditions can A*
to an operator on H2
of norm at most
be extended
1 which commutes with S*?
An obvious necessary condition for the existence of the
desired extension of A*
(and hence for the solvability of the
associated interpolation problem) is the inequality If
is a typical vector in Mb ,
I i,j and
then
cic. (k., k. ) J 1 J
.
I
i,j The inequality
IIA*II ~ 1 .
wiCiWjC j (ki,kj )
IIA*II,;;; 1
definiteness of the matrix
is thus equivalent to the positive semi-
294
D. SARASON
the so-called Pick matrix associated with our interpolation problem.
That the positive semidefiniteness of this matrix is in
fact equivalent to the solvability of the interpolation problem goes back essentially to Pick in his original paper on the subject. Recently J. A. Ball and J. W. Helton (5) had the very nice idea of putting the preceding extension problem into a Krein space context. That device replaces the operator extension problem with a subspace extension problem.
Here is a sketch of the reasoning.
We deal first with the case space
H
= H2
@ H2,
the symmetry
J
=
II A*II < 1.
We form the Hilbert
which we regard as a Krein space relative to
(~ _~).
The shift on H is the operator S @S,
but we shall be sloppy and for convenience denote it simply by S. It is a J-isometry as well as an isometry, and The graph of A*, h E~},
that is, the subspace
=
S[*] G(A*)
S*.
{h @ A*h:
=
is S*-invariant because A* commutes with s*IMb'
also uniformly positive because of our assumption that
It is
IIA* II < 1.
The problem of finding an extension of A* which commutes with S* and has norm at most 1 is equivalent to the problem of finding an S*-invariant subspace of H which contains G (A*) maximal positive.
and is
To establish the existence of such a subspace
we analyze the subspace
N
= G(A*)[l]
The subspace N is S-invariant.
We apply to it the J-analog ue
of a method used by P. R. Halmos (10) to analyze shift-invariant subspaces.
Because N is the J-complement of a regular subspace
it is regular. IIA *11 < 1.) regular.
(It was to obtain this property that we assumed
Since S is a J-isometry the subspace SN is also Therefore N is the vector sum of the two mutually
J-orthogonal, regular subspaces SN and SN is the vector sum of S2N and SL, so
L
=
N () (SN) [1].
N = L+ SL+ S2N,
three subspaces on the right being mutually J-orthogonal.
But then all Iterat-
ing this reasoning we find that, for any positive integer n, N
L + SL + .. ,
all subspaces on the right being mutually J-orthogonal.
This
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
295
implies (because L and its images under powers of
S are regular)
that any vector in N which is J-orthogonal to SnL for all n ~ 0
n SnN and so mus t be O. Because of the regulari ty n>O 2 it follows that N is spanned by the subspaces L, SL, S L,
mus t be in of N,
The subspace L is neither positive nor negative.
In fact,
one easily sees that if L had either of these properties, then N would have the same property. because its J-complement,
But obviously N is not positive,
G(A*),
is positive.
And if N were
negative then its J-complement, being positive, would have to be maximal positive, which it obviously is not. Since L is neither positive nor negative, it contains a pair of vectors x I ,x 2 such that
[xI,x l ]
=
I
=
-[x 2 ,x 2 ]
and
2
Let N+ be the subspace spanned by xl' SX I ' S Xl'
[x I ,x 2 ]
=
O.
One easily
checks that N+ is a positive subspace, so the subspace G(A*) + N+, being the J-orthogonal sum of two positive subspaces, is itself positive.
It is also S*-invariant, because S*xI
is in G(A*)
(being J-orthogonal to N). We shall show that the subspace positive.
G (A*) + N+
is maximal
For that we need only to show that the image of
G(A*) +N+ under
P+ (the orthogonal projection operator from H
to its first summand) is all of H2.
Since that image is closed
and obviously contains Mb' we need only to show that it contains Snb for n~ O. Now because S * xl lies in G(A*), the components are sent by S*
in Mb + Cb.
into Mb , which means these components lie Hence the image under P+ of G(A*) + CX I is contained
in ~ + Cb.
Because P+
of xl
of G(A*)
is one-to-one on
under P+ has codimension I
The former image is Mb , the image of
G (A*) + N+,
the image
in the image of G(A*) + Cx l .
so the latter one must be Mb
G(A*) +N+ under P+ contains
b.
+ Cb.
Hence
The obvious
iteration of this reasoning shows that the image of G(A*) + N+ under P+ contains Snb
for n=I,2, ••. ,
maximal positivity of G(A*) + N+.
thus establishing the
296
D. SARASON
We have now produced a maximal-positive S*-invariant subspace of H containing G (A*) ,
so we have shown that A* has an extension
of the desired kind and thus that the condition
IIA*" < 1
is
sufficient for the solvability of our interpolation problem. The sufficiency of the weaker condition "A*" ;:;; 1 follows now by a simple limit argument.
We have thus responded to part (A) of
the Nevan1inna-Pick problem. We proceed to analyze the situation in greater detail in order to answer part (B) from an operator-theoretic viewpoint, maintaining the assumption "A*" < 1.
The space H2,
originally introduced
as a space of ho10morphic functions in D, will be identified in the usual way with the corresponding space of boundary functions on
aD
(a subspace of the L 2 space of normalized Lebesgue measure). In this way H becomes identified with a subspace of the C 2-va1ued L2 space of normalized Lebesgue measure on aD. We think of C2 as a Krein space in the way mentioned at the end of Lecture 1 and used in Lecture 2. First we observe that the equalities and
aD,
[xl ,x 2 ] = 0 also hold pointwise on
[xI(ei8),xI(ei8)] = 1 = -[x2(ei8),x2(ei8)] for all e i8 in as n
aD.
[xI,x l ] = 1 = -[x 2 ,x 2 ] in other words, and [xI(ei8),x2(ei8)]=O
In fact, since [xl,Sn xl ] is 1 or 0 according
is 0 or positive, and since 1f
1
21f
f
-1f
we see that the function [xl(e
[xl (e
i8
i8
), xl(e
) 'Xl (e i8
)]
i8
)]
e- in8 d8
has the same Fourier
series as the constant function 1, and thus it equals the constant function 1, at least almost everywhere. the components of xl
lie in Mb + Cb
Moreover, we know that
and so are rational func-
tions. In particular, they are continuous, so actually "8"8 i8 [Xl (e 1 ) 'Xl (e 1 )] = 1 for all e on aD. The other two relations are established by similar reasoning. We can now see that the dimension of. L is exactly 2.
In
fact, if x is a vector in L which is J-orthogona1 to both xl
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
then by the reasoning above x(e ie )
and x 2 ' both xl
297
·e )
(e 1
is J-orthogonal to
which means x = 0 because, by the result in the last paragraph, xl(e ie ) and x 2 (e 1·e ) form for each e a basis for «:2. Thus the vectors xl and x 2 form a basis for
and
·e )
x/e 1
L.
We write matrix function of xl
xl
p Ell rand
(~ ~).
x 2 = q Ell s,
and we let U denote the
It was noted above that the components
are rational functions, and the same is clearly true of the
components of x 2 . in H 2;
for all e,
The entries of U are thus rational functions
in particular, they lie in Hoo
J-unitary at each point of the relations
aD
The matrix function U is
due to the pointwise validity of
[xl'x l ]=1=-[x 2 ,x 2 ],
[x l ,x 2 ]=O.
It follows that
U acts via multiplication as a bounded operator and as a J-isometry from H to H.
Also, the range of U is closed, because U
acts via multiplication as an invertible operator from
L2 Ell L2
onto itself (its inverse being its pointwise J-adjoint). is clear that UH is contained in N and contains for all nonnegative integers n,
snxI
As it and snx2
we can conclude that UH = N .
We are now in a position to rederive the description, obtained in Lecture 2 by means of the Schur algorithm, of the most general solution of our interpolation problem.
In operator-
theoretic terms, we want to describe the most general operator on H2
that commutes with S,
adjoint extends A*.
has norm at most 1, and whose
Equivalently, we want to describe the most
general maximal-negative S-invariant subspace of H contained in N. If N'
is such a subspace then, by virtue of the properties of U -1
mentioned in the last paragraph, the subspace U
N
,
is maximal
negative; it is also obviously S-invariant (since U commutes with S).
Hence, every subspace of the kind we want is the image
under U of a maximal-negative S-invariant subspace of H. The converse of the last statement is also true but it takes a bit of work to establish it. S-invariant subspace of H,
If N"
is a maximal-negative
then UN" is clearly negative and
298
D. SARASON
S-invariant; it is also clearly maximal negative in N, clearly maximal negative in H. is uniformly negative.
but not so
The situation is simplest when Nil
Then H is the J-orthogona1 sum of Nil
and
the positive subspace
N'" = (N") [1] ,
sum of UN"
Consequently H is the J-orthogona1 sum of
and UN'" •
the negative subspace UN"
so N is the J-orthogona1
and the positive subspace
G (A*) + UN'" ,
which implies the maximal negativity of UN". Applying the preceding observation to the special case where N" -- {a} m w H2,
the vector x 2 under P
. we see t h at t h e S' -1nvar1ant sub space generate d b y
is maximal negative.
The image of that subspace
(the orthogonal projection operator from H to its
second summand) is the S-invariant subspace generated by s (the second component of x ), and hence
s
must be an outer
function. Consider now any maximal-negative S-invariant subspace N" of H.
Such a subspace is the graph of the multiplication operator in ball Hoo :
induced by a function tjJ thus have
N"
{tjJh GO h: h E H2}.
UN" = {(ptjJ+ q)h ~ (rtjJ+ s)h: h E H2}.
maximal nega ti ve amounts to showing that tion.
We do that by writing
outer, and so is sum of
1
1 + tjJr / s
is an outer func-
rtjJ+s=s(l+tjJr/s).
because tjJr / s
is
We know s i s
lies in ball Hoo (and the
and a function in ball Hoo is always outer (12, p .117) .
Hence rtjJ + s
is outer, being the product of two outer functions,
and the maximal negativity of UN" see that UN" by UtjJ.
rtjJ + s
To show UN"
We
is established.
Moreover, we
is the graph of the multiplication operator induced
We can conclude that the general solution of our interpo00
1ation problem has the form UtjJ with tjJ
in ball H
At this point we have recaptured the essence of what was established in the last lecture by means of the Schur algorithm. The analysis can be further refined; we mention a few facts but do not pursue them in detail. been established about
First of all, from what has already
U it is not too hard to show that its
determinant 1S a constant multiple of b,
so, rep1acinl!, xl'
by a constant mUltiple of itself, we can assume
det U
=
b.
say, One
THE NEVANLINNA-PICK INTERPOLATION PROBLEM
then obtains the equali ties last lecture. xl
and
Second,
299
p = bs and
r = bq on
aD
as in the
U obviously depends upon the choice of
x 2 ' but only to within multiplication from the right by a
constant J-unitary matrix, and that in fact is the extent of the arbitrariness of U.
In particular, a matrix function U constructed
by the present methods and one constructed by the Schur algorithm for the same interpolation problem are the same to within a constant J-unitary factor on the right. Third, it is natural to ask if one can somehow express the entries of the matrix function U in terms of the operator A. We shall go into that in a little detail. concerned wi th q
and s,
We need only to be
the componen ts of x 2' because, as
mentioned in the preceding paragraph,
p
and
r
can be expressed
in terms of them. The subspace N is easily seen to be the orthogonal sum of bH 2 E9 bH 2 and
{Af E9 f: f
Mb },
E
the graph of A.
We have noted
+ Cb. Conse(Af @ f) + (ab E9 Sb)
that the components of a vector in L belong to Mb quently, any vector x for some f
in Mb
in L has the form
and scalars a, S.
x =
That x
is in L means that
S*x belongs to G (A*), in other words, that (S*Af E9 S*f) for some g
in Mb.
S*Af
+
+ (as*b E9 SS*b)
g E9 A*g
Equating components, one finds
as*b
S*f
g ,
+ SS*b
A*g
which combine to give S*f
+ SS*b
Thus the function f
S*A*Af
vector
aA*S*b
and scalars a, S must satisfy the equality S*(l- A*A)f
for x
+
to belong to L. x = (Af E9 f)
+
(aA* - S)S*b
The reasoning is reversible, so the (ab E9 I3b)
will belong to L i f f, a, 13
300
D" SARASON
satisfy the equation above.
The object now is to determine a
simple solution of that equation which will yield a strictly negative vector x in L and thus, after normalization, one possible choice of x 2 ,
explicitly expressed.
It turns out one
arrives at such a solution starting from the premise ex = O.
The
reasoning is slightly different for the cases b (0) of 0 and b (0)
= 0,
corresponding, respectively, to the invertibility and noninvertibility of S* 1Mb. function s
We omit further details.
c[(l-A*A)-lu + b(O)b]
S
where
The particular
one obtains is
u = l-b(O)b
(the projection of the constant function 1
on Mb ) and
The corresponding function
q
is
cA(l - A*A)-l u.
Final"ly, a few words are in order concerning the case IIAII
=
1. oo
In that case we know there is at least one function
b > O.
¢a is cyclic for all
a > b.
PROOF. above. Now let Then
Let
¢a
Similarly, c
~
b
be cyclic. ¢na
Then
¢2a
is cyclic by (iii)
is cyclic for all positive integers
be given, and choose
n
such that
n.
na - c > b.
¢na = ¢c¢na-c and both factors are cyclic by (iii) above.
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
COROLLARY 2.
Let
cp e H(G)
and let
some number
G
be simply connected, let
have no zeros in
a > 0,
325
cpa
then
G.
If
cpa
is cyclic for all
M(E)
00
=
H (G) •
is cyclic for a >
o.
We now pose two test questions about cyclic vectors in a general Banach space QUESTION 3. > Ig(z) 1
E
of analytic functions in a region G (see [49]) •
If
for all
f,g e E,
z e G,
if
g
then must
is cyclic, and if f
If(z)
1
be cyclic?
Since the constant function 1 is always a cyclic vector, we have the following special case of this question. QUESTION 3'. must
f
If
feE
and
If(z)
> c > 0
1
for all
z e G,
be cyclic?
QUESTION 4.
f, f- l e E
If
must
f
be cyclic?
This question (for the Bergman space) was posed in [47] Question 25' on p. 114).
Harold S. Shapiro [41] used the term
"weakly invertible" in place of cyclic. then be rephrased as follows: tibility?
(see
The above question could
does invertibility imply weak inver-
From Proposition 8(iii) we see that the answer is affirm-
ative if, in addition, we assume that 1 = ff- l is always cyclic .
f
is a multiplier.
Indeed,
..,
Recently F. A. Samoyan [40] gave the first example where Question 4 has a negative answer.
No examples are known where
Question 3 has a negative answer.
There is however one common
situation where Question 3 has an affirmative answer. PROPOSITION 9. >
Ig(z)
1
for all
PROOF. Hence
Since
g = CPf e
[f)
If
M(E) = HOO(G) ,
z e G,
cp
=
g/f
and if
g
if
f,g e E
is cyclic, then
> 1,
f
If(z)
by Axiom 7, and so
f
then, as remarked earlier,
f
1
is cyclic.
is bounded it is a multiplier on
E.
is cyclic.
We now consider Questions 3 and 4 for the spaces 0.
with
D. 0.
When
is cyclic if and only if it
has no zeros in the closed unit disc (recall that the functions are continuous on the closed disc).
It follows easily that both ques-
tions can be answered in the affirmative.
A. L. SHIELDS
326
0 < a < 1; see Chapter Three
Both questions are open when for partial results when
a = 1,
see also [9]
(Theorem 1 and
Corollary 1, as well as Corollary 2 to Theorem 2) . For
a < 0
Question 3.
Proposition 9 gives an affirmative answer to
For
a
Question 4 can also be answered in the affirmative: i f f and f- l are both in H2 then they are both outer functions (indeed, i f either f or f- l had a non-trivial =
0
inner factor, then so would
a < O.
f-lf = 1) .
Question 4 is open for
This leads to another interesting problem, where for sim-
plicity we specialize to the Bergman space then for
If
feB
r = Izl < 1:
2 If(z) 12= IIf(n) (n+l)-l/2 (n+l) 1/2 z n 1 Hence i f
B (a = -1).
f- l e B
then
.::.lI f ll 2
II f-lll- l
1f (z) 1 .::.
(1_r2)-2.
(1_r 2 ).
This suggests
the following question, which was first posed by H.S. Shapiro (see the Remark following Theorem 5 in [43]). QUESTION 5. c, k > 0
If
(and all
feB
and if
~),
z e
If(z) 1 > c(l-Izl)k
then must
f
for some
be cyclic?
As noted above, an affirmative answer to this question would imply an affirmative answer to Question 4 for the Bergman space (see Chapter Two for further information) • Along these lines one can pose similar questions for any Banach space of analytic functions on a bounded plane domain sider continuous functions
cp
on
is positive on
G
QUESTION 6.
Let
G.
feE
cp
on
and zero on E
(the closure of
G
G)
such that
aGo
be a Banach space of analytic functions
cp,
Does there exist a function satisfies
We con-
G.
If(z) 1 > CP(z)
for all z e G,
Such functions exist for the by Proposition lOb. below, i f
D
a
f e D
a
fies a Lipschitz condition of order has a zero on the boundary, say
as above, such that if then
f
spaces when
a > 1.
(1 < a < 3)
then
b = (a - 1)/2.
fell = 0, then
If(z)1 = If(z) - f(l)I.::.clz - lib,
Izl .::. 1.
is cyclic? Indeed, f
sat is-
Hence, if
f
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
¢(z)
Hence if satisfies
(1 _ Izl)a
=
I fez) I .:: ¢ (z)
o
where 6,
in
D
a
then it also works for
a < b,
a , since
Iz I
do not exist for the
(5)
< 1), ~,
and
c
is a posi-
Also, one can construct singular measures
w(8)/8
tends to infinity arbitrarily slowly.
and [44, p. 265].)
Thus, given any function
there exists a singular inner function for
¢
is the modulus of continuity of
tive constant.
§5]
works for
~,then
w
which
¢
is a singular inner function with associated singular
(r =
where
f e Da
Indeed, H.S. Shapiro has shown (see [41], p. 164) that
S(z)
measure
and if
has no zeros on the
In contrast to this, such functions space
327
S
¢
such that
~
for
(See [l8; as above, IS(z) I .:: ¢(z)
I z I < 1. Question 6 is open for the spaces
intervals:
~
0 < a
1,
and
PROPOSITION lO.a. I g (z) I where
c b.
~ c
If
a
c
then
8,
not on
I I < 1),
(z
g.
(1 < a < 3), then
If(z)-f(w)1 < cllfll where
a < O.
Izl) -(1-8)/2
II g II 8 (1 -
feD
in the remaining
a
g e D8 (8 < 1)
depends only on If
D
a
depends only on
Iz_ w l(a-l)/2 (Izl < 1, Iwl < 1), a,
not on
f.
For the proof see 19, p. 278]. CHAPTER TWO:
THE BERGMAN SPACE.
As noted in Chapter 1, B,
the Bergman space
M(B),
the space of multipliers on
is just the space
analytic functions in the disc.
00
H
of all bounded
Then from Proposition 8(iii) we
00
see that if if both
¢
¢ e H and
f
and
feB,
are cyclic.
then
¢f
is cylic if and only
Hence it is of interest to learn
328
A. L. SHIELDS 00
which
H
functions are cyclic in
Proposition 6.
Hoo
However
B.
We would like to apply
is not separable and hence has no
cyclic vectors (in its norm topology). Shapiro [43, p. 325], now follows from
If
e
f
inner function, then 5
is contained in
B.
The next result
Proposition 6 and 8(iii).
COROLLARY.
and only if
Hl
However as noted by H.S.
f
Hl
is outer and if
is cyclic in
5
B, and
is a singular fS
is cyclic if
is cyclic.
Before discussing singular inner functions we introduce a class of "thin" subsets of the unit circle that playa basic role in the theory. By a BCH set (sometimes called a Carleson set) we mean a compact subset
K
of
such that
a~
K
has Lebesgue measure
0,
I
II I (-log II I) < 00. Here {I} are the disjoint open n n n arcs in the complement of K, and I· I denotes normalized and
Lebesgue measure.
These sets were introduced by Beurling [5] in
1940, studied by Carleson [11] in 1952, and rediscovered in a completely different context by Hayman [19] in 1953. NOTATION. in
00
A
will denote the class of analytic functions
all of whose derivatives are continuous on the closed disc.
~
PROPOSITION 11. a) an outer function tives vanish on
f K;
in
KC
If 00
A
a~
is a BCH set then there is
such that
in addition,
f
f
and all its deriva-
has no other zeros in the
closed disc. b)
If
f
satisfies a Lipschitz condition of some positive
order in the closed disc (and is analytic in the open disc), then the boundary zero set of
f
is a BCH set.
Part b) was observed by Beurling
[5] in 1940.
Part a) was
proved, for functions with a prescribed finite number of derivatives continuous on the closed disc, by Carleson Ill] in 1952. The result was later extended to
00
A
by several authors [27],
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
[35], and [52].
329
The last of these papers also describes the
zeros inside the disc. COROLLARY.
The union of two BCH sets is a BCH set.
It can also be shown that any closed subset of a BCH set is a BCH set (this follows fairly easily from the definition of BCH set) • We turn now to singular inner NOTATION.
5
functions.
denotes the singular inner function coming
].J
from the positive singular measure THEOREM 1.
a~.
on
].J
A necessary and sufficient condition that the
singular inner function
5
be cyclic in
].J
B
is that
put no
].J
mass on any BCH set. The necessity was proved by H.S. Shapiro [43] in 1967.
The
sufficiency was obtained independently by B.I. Korenb1um [31] and J. Roberts [38] about 1979.
Roberts' proof was unpublished for
several years but was available in an unpublished exposition by Joel Shapiro [46].
Korenb1um showed that the result is an easy
consequence of the theory developed in his earlier papers [29] and [30].
\ie shall give
H.S. Shapiro's proof of the necessity
since we shall need the argument again in what follows. PROOF OF NECESSITY.
Let
g
00
be a
function on
C
E (Inl
generally, we only need to require that
_00
+ 1)
a~
(more
Ig(n)1
2 0
and all
PROOF. ment that
f
If
f € N
n B,
Izl < 1,
The lower bound on -1
-co
eA.
if then f
If(z) I > c(l f
is cyclic in
Izl)k for B.
is equivalent to the state-
If
f = SlF l /S 2 F 2 , then Sl is cyclic, as in seen by applying Proposition 16 to l/f. The result now follows from the previous corollary. Note that this gives an affirmative answer to Question 5 in
case
f € B
is also in the Nevanlinna class.
This result was
also known to Korenblumi a special case had been proved earlier [1], where the hypothesis condition that the range of
feN f
was replaced by the stronger omits a set of values of positive
338
A. L. SHIELDS
logarithmic capacity. f,f- l e B,
if
Berman, Brown, and Cohn 14J showed that
and if
feN,
then
f
PROPOSITION 17 (Shapiro [42J). Borel measure on ~,
~
Let
f
and satisfy, for some
{p} n
PROOF. tiples of if
f
P f n
Let
[f]
in
L2(~).
, f
r}. 2
L
in
m(r)
-s
dp
0,
m(r) = rnin{lf(z) I: such that
~
Let
B.
be holomorphic and non-vanishing in
c(l
-
-
I z I) k
has
feB
and
o.
This would prove H.S. Shapiro's conjecture: if
f
for some
c,k,
Indeed the hypothesis is equivalent to:
if
then f is cyclic in B. -00 f- l e A Such an f
_00
is cyclic in
A
and hence
of = 0
by Korenblum's result.
Korenblum has informed us that he can prove the above conjecture in case if
f
is in a smaller Hilbert space, more precisely,
Illfl2(1 - r)-E
O.
This generalizes the
It is not clear, however, if this f
is in a smaller
1
AP
2
state a specific question we note that AO c:: A2 lation shows that ~ 2 Ilfll~,2= Illfl2(l - r)2 < cElf(n)1 /(n If(n) Irn 2 Ml(f,r)
Also, on
rdr
we have:
by Cauchy's inequality.
If(n) I/(n + 2) 21Iflll,0.
ex
space.
To
Indeed, a calcu-
Integrating
By the Fejer-Hardy-
Littlewood inequality (sometimes called Hardy's inequality; see [14, p. 48], and for some historical remarks see [50]), we have ~
n
E If(n) Ir /(n + 1) 2 nM1(f,r).
340
A. L. SHIELDS
Integrating
rdr: l:lf(n) I/(n + 1)2 2. c Ilflll,o.
the previous result we see that gUESTION 7. must
f
If
e
f
1
II f II
Using this and
2. c II fill , 0 as required.
2,2
has no zeros in
AO
and if
/',
f
-1
-00
eA
,
2? A2 ·
be cyclic in
We conclude with a few remarks about the invariant subspaces of
B.
Those are the closed subspaces
M CB
such that
zMCM.
The collection of all such subspaces forms a lattice (partially ordered by inclusion), which we denote by
Lat(M ). z it was an open question whether two such subspaces
o.
intersect only in
At one time
(i {O})
could
It follows from Beurling's theorem that 2
this cannot happen in
However C. Horowitz showed in his dis-
H •
sertation [21] that there are two Bergman zero sets whose union is not a zero set, and thus two non-trivial invariant subspaces can indeed intersect only in
O.
More recently, Bercovici, Foias, and
Pearcy have shown that the Bergman lattice is much more pathological than anyone had suspected (see [3, Chapter 10]). positions contain
some of their results.
lattice of all closed subspaces of PROPOSITION 18.
Let
The next two proLat B
denote the
B.
¢: Lat B + Lat M z that is injective, increasing, preserves closed spans, and has the
following property:
There is a function
if
if
{E} C:Lat H, n
nE
n
=
then
{O},
n¢(E ) = {O}. n
They derive a number of corollaries including the following. COROLLARY. Lat Mz
such that
There exists Ea
PROPOSITION 19. exist that
E,F C Lat M, z zE CF. COROLLARY.
codimensions > 1
n Eb
=
x
{O}
for
a
Given an integer with
FeE,
There exists in
{E} (-00 < x < (0)
E,
E
and
i k,
and hence
M IE z
b. 1 < k 2.
dim(E
e Lat Mz
contained in
e
00,
F) = k,
such that
zE
there such
has
has no cyclic vectors.
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
341
This answers questions raised by various authors. Recently Stefan Richter has shown how to lift some of this pathology into the lattice of invariant subspaces of CHAPTER THREE:
H2
of the polydisc.
THE DIRICHLET SHIFT
In this chapter we consider the Dirichlet space
D.
We recall
that from the corollary to Proposition 6, a cyclic vector must be an outer function.
The converse is not valid; in 1952 Carleson
[11] gave examples of outer functions in
Dk , for any integer given in advance, that are not cyclic in D. For a more
k > 1
general result see Theorem 4 below.
First we recall some facts
about the boundary values of functions in In 1913 Fejer proved that if holomorphic and univalent for f
converges uniformly on
Izl
is continuous for
< 1,
Izl = 1.
(since a univalent function is in finite area).
f
D. Izl ~ 1,
then the power series for
Such a function is in D
if and only if
~(~)
D has
Landau pointed out that Fejer's method proved a
more general result:
for any
feD,
the power series converges
at each boundary point where the radial limit exists, and the series converges uniformly on any set where the radial limit exists uniformly (see [32, §13, pp. 65-67]). valid by Abel's theorem:
Of course the converse is
the radial limit exists at any boundary
point where the power series converges.
In fact, the non-tangential
limit exists at all such points (and uniformly on any set where the power series converges uniformly) • Also, the non-tangential limit exists almost everywhere by the Fatou theory.
This result was improved substantially by
Beurling [5] in 1940:
the non-tangential limit exists except for
a boundary set of logarithmic capacity zero.
Salem and Zygmund
[39] in 1946 gave a new proof of this result, and extended it to the
Da
spaces
(0 < a
~
1),
replaced by a related capacity.
with the logarithmic capacity In 1950, Carleson, in his disser-
tation [10, Chap. III, §3], showed that for bounded functions in D a stronger result is true:
the non-tangential limit exists except
perhaps for a set of logarithmic length zero.
A. L. SHIELDS
342
~
In non-tangential convergence one considers angles in with vertex at a boundary point Stolz angle the limit of
(Stolz angles).
For every
is required to exist as
z
J. Kinney [24] in 1963.
feD
wo
~,
He replaces the Stolz angles by a family
tangent to the boundary at
wO'
and making
an arbitrary finite order of contact with the boundary. that if
~
An important new result was obtained by
inside that angle.
of subregions of
f(z)
w.
He shows
then at almost every boundary point the limit
exists inside any such region.
He also gives information about
the capacities of the exceptional sets where the limits do not exist.
Finally in 1982 definitive results were obtained by Nagel,
Rudin, and J. Shapiro [33, Thm. 1], [34, Thm. A]. if
feD
They show that
then the boundary limit exists almost everywhere inside
subregions making exponential order of contact with the boundary. They show the precise connection between the kind of subregions in which approach to the boundary is permitted, and the size of the exceptional set where the limit fails to exist. As regards cyclic vectors, we consider first the two test questions: 1)
If
f ,g e D,
cyclic, then must 2)
If
f
f and
If(z)
1
> Ig(z)
be cyclice? f- l are in
D,
for
1
z e~,
then must
f
and
g
is
be cyclic?
In [9, Cor. 1 to Thm. 1, p. 281] L. Brown and the author show that 1) has an affirmative answer if, in addition, one assumes that g e M(D).
The also raise the following problem (p. 282).
~UESTION 8.
If
g e 0
there exist a sequence IIPngll""
2
const.,
{Pn}
I]PngIID
2
nH""
is cyclic in
0
then does
of poly?omials such that: const., and
(Pn g ) ~ 1
in
~?
An affirmative answer would improve the above partial answer 00
to the first question: (instead of
the result is correct if merely
g e M(D».
g eon H
In the proof of Theorem 2 of [9] an
affirmation answer to Question 8 is obtained under the additional hypothesis that
igl
is Dini continuous on
a~
(and therefore
343
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
g e A,
the disc algebra).
It is not known whether the answer is
still affirmative if merely
g e DnA.
It is easy to see that if h e D
if and only if
f,g e D
nH
oo
f,g e D
h' e B).
nH
oo
then
fg e D (recall
Assume for the moment that
In [9, Proposition 11] it is shown that if
•
cyclic then both
f
and
g
are cyclic.
fg
is
An affirmative answer to
Question 8 would imply the converse result.
More generally one
has the following problem. QUESTION 9.
If
f,g,fg e D,
cyclic if and only if both
f
then is it true that
and
g
fg
is
are cyclic?
As regards the second test question above, we have the following result: if f,f- l e D Hoo then each is cyclic in D.
n
This follows from one of the results mentioned above, since their product is cyclic. Also, it was shown in [48] that if merely f- l e Hoo , and feD, then f is cyclic. However it is unknown f- l
where
must be cyclic in this case.
QUESTION 10.
If
feD,
If I > c > 0
in
~,
then must
f- l
be cyclic? Note that
f- l
must be in
that the derivative is in
D
(one sees this by showing
B).
The following is a combination of [9, Proposition 13] with one of the results above. PROPOSITION 19. a) and
g
is cyclic, then b)
zeros in
If
h
~,
if
If f
g e M(D),
feD,
If I > Igl
in
~,
is cyclic.
is analytic on Re el > 0,
then
(the closed disc) and has no
~
hel
is cyclic and is in
M(D) •
\'1e turn now to another approach to the problem of classifying the cyclic vectors in NOTATION.
D.
Z (f)
CONJECTURE ([9, p. 292]). is an outer function and
feD
is cyclic if and only if f
Z(f) has logarithmic capacity zero.
344
A. L. SHIELDS
If correct this would immediately imply an affirmative answer to our first test question, where is cyclic. question.
If I > Igl
~
in
and
g
It would also imply an affirmative answer to our second Indeed, as noted at the beginning of this chapter,
Beurling showed that if
feD
then
f
has a (finite) radial
limit except for a set of capacity zero. on a set of positive capacity then
l/f
If this limit were zero would have an infinite
radial limit on this set, and hence could not be in
D.
Finally,
if the conjecture is correct it would yield an affirmative answer to Question 9, since the union of two sets of capacity zero has capacity zero.
The conjecture has been proved in one direction.
THEOREM 4 ([9, Theorem 5]).
If
positive logarithmic capacity, then
feD f
and if
Z(f)
has
is not cyclic.
The following results, which we state without proof, are also relevant.
Recall from Chapter One that if
f' e H2
Also, it follows from the Cauchy inequality that lutely converger.t power series and hence PROPOSITION 20.
a)
[8].
If
E C
b)
D,
such that
[9, Theorem 3].
If
has an abso-
a~
is a closed set of feD (\A
that is
Z(f) = E. If
is at most countable, then NOTATION.
f
f e M(D).
f e A.
logarithmic capacity zero, then there is an cyclic in
then
f f
E C a~
is outer,
f' e H2,
is cyclic in
and if
Z(f)
D.
is a Borel set of positive capacity,
let D = E
Here q.e.
1£
e D: lim f(re
:i8
)=0
(rtl), q.e.
in
EL
(quasi-everywhere) means except for a set of capacity
zero. THEOREM 5.
DE
is a closed subspace of
D.
This result was pointed out to us by J. Shapiro.
It is a
consequence of a result in his paper [34] with Nagel and Rudin. See [9, p. 295] for more details.
In 1952 Car1eson [11, p. 335]
345
CYCLIC VECTORS IN BANACH SPACES OF ANALYTIC FUNCTIONS
proved this in the special case that Two) of positive capacity.
E
is a BCH set (see Chapter
(Actually, he proved a slightly dif-
ferent but equivalent result.)
Theorem 4 above is an immediate
corollary to Theorem 5, though it is simpler to prove Theorem 4 directly. QUESTION 11.
If
E,PC:d~
are Borel sets, when is
DE
Clearly this will happen if the symmetric difference of and
F
E
has logarithmic capacity zero, but this is not the only
case.
Indeed, Carleson [II, Theorem 5J has given a sufficient
condition for
DE
to equal
{O},
a set of uniqueness for the space Let
[f]
multiples of
in other words, for
E
to be
D.
denote the closure in
D
of the polynomial
f.
QUESTION 12.
E
Dp 7
=
If
feD
is an outer function, is
[f]
DE'
If
MC D
is a (closed) invariant subspace
Z(f)7 QUESTION 13.
(that is,
zM CM),
must there exist
feD
such that
M
[fJ?
=
More generally, one might ask for a chacterization of all invariant subspaces, perhaps along the lines of Korenblum's description [28] of the invariant subspaces of the space (those functions whose first derivative is in cally,
let
and let
E
M(E,~)
d~
be a Borel set, let
~
2
H ).
be an inner function,
denote the set of all those functions in
whose inner factor is divisible by M(E,~)
is easy to show that
~.
2
HI More specifi-
DE
In view of Theorem 5 it
is a closed subspace of
D,
and it
is clearly invariant. 9UESTION 14.
Is every invariant subspace of the form
M(E,~)?
Even if this has an affirmative answer, there will still remain the problem of deciding when two of these subspaces coincide (see Question 11).
Also, this raises the question of describing
those inner functions that can occur as divisors of functions in D.
346
A. L. SHIELDS
One answer to this question was given by Carleson [12] who gave a formula expressing the Dirichlet integral in terms of the canonical factors:
outer, singular inner, and Blaschke product.
However
it still seems very difficult to give a complete description of the zero sets, in
L,
of functions in
D, for example.
Carleson
showed that a Blaschke sequence with just one limit point on the boundary may fail to be a zero set for in [13]).
D
(the proof is presented
On the other hand, there are zero sets such that every
boundary point is a limit point of zeros.
For a "best possible"
sufficient condition that a sequence
{z} be such a zero set see n [45]; the condition is in terms of the moduli {Iz I} alone. n
Finally, we state three more questions about the space If
1¢
L,
in
(z) 1 < 1
9UESTION 15. D?
vllien
C¢
let
For which
a mUltiplier on
a bounded ogerator on
for some
E: > 0, must
¢
be
Is a random power series from the Dirichlet . 1 1 lp "2 ?
More precisely, i f
12
¢.
D?
space almost surely in
< 00,
if
for almost every
t?
n
is
¢ € D nLipE:
If
QUESTION 17.
Znla
¢
0
is bounded must it preserve cyclic vectors?
QUESTION 16.
if
f
D.
{r (tl} n
1
(z) = Zr (t)a zn, then is f t in lip 2 n n See Duren [14, Chap. 5] for information f
t
It follows from [22, Chap. VII, Thm. 2]
about Lipschitz classes. that, for almost every
are the Rademacher functions,
t,
ft
e
Lip a
for all
a
1985 by D. Reidel Publishing Company.
while,
N.J. YOUNG
352
more recently, J.W. Helton's far-reaching application of nonEuclidean functional analysis to electronics also centres around this problem [4].
However, the engineering slant generally calls
for something slightly different from the old results.
In the
first place, one is generally aiming eventually at a method for the practical computation of solutions, and so numerical considerations such as stability, conditioning and storage requirements playa role.
A second major difference is that in most applic-
ations the functions one has to deal with are matrix-valued rather than scalar.
This is essentially because the states of most
interesting engineering systems are described by vectors rather than scalars:
solving a system of linear differential equations
with constant coefficients using the Laplace transform introduces a matrix of rational functions.
Now numerical complex analysis
and the theory of analytic matrix functions are both substantial mathematical topics which have received plenty of attention from mathematicians independently of any practical implications.
Still.
this contact with engineering deserves to be heartily welcomed by the pure mathematicians in the field.
It is surely desirable
that our subject be in contact with other branches of science and, more concretely. it provides an orientation and a public in an otherwise large and diffuse area. In these lectures I shall show how the two factors I have mentioned - matrix valued functions and the search for efficient numerical algorithms - affect the Nevanlinna-Pick interpolation
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
problem.
353
Let us begin with a modification of the versions pre-
sented in the lectures of D. Sarason.
Let
be
distinct points in the open unit disc D and let Wl ' complex numbers.
.(z.) = 1.
w., 1.
in D satisfying the interpolation
• 1 ~
i
to belong to the unit ball of functions in
D
~
n,
r
but instead of asking for -. (the space of bounded analytic
••
show using nOl'llal families that the infimum of
f
11.11 Hco
with supremum norm). we ask that
minimised over all interpolating functions
ed.
be
It is easy to
11.11
Bco
is attain-
We shall re-formulate this problem somewhat. Suppose that aD is any B function satisfying the interpolation conditions -
for example, the Lagrange interpolating polynomial. second function
• - f
I
is divisible by the polynomial
- 8.8
J
Then a
• E HaD satisfies those conditions if and only if
or in other words, if and only i f As
be
As in Sarason I s problem (A) we look for a
bounded analytic function conditions
••• , •.• wn
• E
(z -zl)(z -z2) ... (z - 3 n ),
f +
is a unit in the ring BaD, (z - 8
1
)
...
where (as in [8]) b(z)
=
n
TT
= bB aD,
z - z.
j=l 1 -
the finite Blaschke product with zeros arrive at the following.
we have
(z - 8 )B-
n
(z -8n )B-.
(8 -zl)
i1
,
8.3
J
8 1,
... ,
8
n'
We thus
354
N. J. YOUNG
Problem (C). find
Given f € Hoo and a finite Blaschke product
• € f ... bH oo ~ ~
11.11 H-
b,
is minimised.
As was indicated in Sarason's lectures, there are numerous approaches to this problem which, from a purely mathematical viewpoint, seem to be of roughly equal power.
For computation,
though, some are more convenient than others.
The Krein space
method gives the theoretical results in a most elegant fashion, but its prescription for obtaining the extremal
•
contains the
step "Pick a maximal negative z-invariant subspace of containing the negative subspace ••• ".
H2
®
H2
While one could doubtless
develop methods of handling such injunctions on a computer, for the present a more straightforward option is to use an approach based on familiar, concrete entities like singular values of matrices.
For this reason my own computer programs [2] for the
numerical solution of Problem (C) do not use the methods modestlypresented by the earlier speaker, but rather Sarason's own operator-theoretic approach [7].
Here is his construction.
Corresponding to the data in problem (C) above we define an operator
from the Hardy space
T
H2
to its subspace
g2
e
bH2
by (1.1)
where and
Mf
P :
Note that
:
H2 _H2
_ 82
H2
T
8
is the operation of multiplication by f
bH2
is the orthogonal projection operator.
depends onJ.y on the coset
f ...
bHa:
not on
t
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
itself, for if
.:
355
f + bh is a typical element of this coset,
z E
bhz E bH2
P(bb): 0
and
Sarason proved that the opposite inequality is also true:
the
we have, for any
H2,
so that
hence
It follows that, for any
• E
f + bH flO ,
lip 1/
:
00
F + BB2.x2 instructive~
< 2; conversely, any such
-
of minimal norm.
This trivial
it shows that the supremum norm is
too weak an indicator of the "size" of a matrix-valued function for us to expect any useful uniqueness results in problem (D). One natural response to the profusion of solutions to problem (D) is to describe all of them.
This was done (in
th~
equivalent block Hankel matrix formulation) by V.M. Adamyan, D.Z. Arov and M.G. Krein in [1].
Let us examine their strategy
and see how it can be used to compute a solution of problem (D). This reformulation of the relevant part of (1] is taken from
361
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
F. B. Yeh's thesis [10].
For simplicity we take
We start from the observation that if problem (D) and vI' v 2
Hence, if B2 2
V
t
m: n
= 2.
is an extremal for
is any maximising vector for
T then
tv
= Tv.
are linearly independent maximising vectors in
we have the following relation between 2 )( 2 matrices:
=
t(s)
(1.3)
Actually, we need a stronger property than linear independence for
V
I
and
V •
2'
we require them to be
ent, in the sense that 4: 2
for almost all
vIes), V2 (s)
e: aD.
s
are linearly independent in
The vectors 8 22 ,
are linearly independent in independent.
pointtuise UnearZy independ-
[1
s]T and
[.3
s2]T
but not pointwise linearly
If we can choose the maximising vectors with this
stronger independence property then we can solve (1.3) to obtain
a.e. on aD, so that
•
is uniquely determined in this case.
Since we already know that
•
is not in general unique, on the
face of it it appears that all we can conclude is that there do not always exist pointwise linearly independent maximising vectors for
T.
Nevertheless, Adamyan et aZ. contrive to reduce the
general case to the special one in which such a exist. B
They effect this by modifying the given functions
in a subtle manner.
function
VI and V 2
F:
F + BFo
Observe that, for any lies in the coset
Foe: t2 x2 ,
do
F and the
F + B8;:2' and hence
N. J. YOUNG
362
-
, + zBEfO
2x2
c:
=
-F
+ BBI» 2x2
F
+ BBI»
2X2
11.11 I» over the left hand coset is thus no less
The infimum of
than its infimum over the right hand coset, the quantity called for in problem (D).
If we can choose
Fa
so that the two
infima coincide then any extremal function in the left hand coset will be a solution of problem (D).
And as the left hand coset
is smaller it can in principle contain fewer functions of minimal norm. which
F + zBB CIO 2X2
P
for
contains a unique function of minimal norm,
and moreover, the Sara son operator corresponding to
where
Fa
Adamyan et at. show that there is a choice of
B2 GaBB2 222
8 2 _
this coset,
is the orthogonal projection,
does admit two pointwise linearly independent maximising vectors. Hence if we can find
Fa E
¢2X2
(necessarily non-unique in
general) such that and
(1)
(2)
T has
two pointwise linearly independent maximising vectors,
then we can use the idea outlined above to compute a solution of problem CD).
The operator
T
is a "one step extension" of
T:
T
in terms of block Hankel operators, passing from
T
corresponds to the addition of an extra column.
I f this lIethod
to
is applied to the example given above it entails passing from
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
363
the coset
+
sH
co
2x2
to anyone of the smaller cosets
with
IAI:
The latter coset contains a unique function of
1.
minimal norm, to wit
[: It is interesting that no choice of "natural" solution Algorithms
A (i.e. of
Fo) gives us the
t . diag{2, I}.
for the solution of problem (D) or its equivalent
by the method of one step extensions have been implemented by S.Y. Kung and D. Lin [5] and by F.B. Yeh [10].
Fo
sired matrix find one.
Although the de-
is known to exist, it is no trivial matter to
Kung and Lin achieve it by solving an algebraic
Riccati equation for matrices: ive procedure.
this involves at least one iterat-
Yeh's method is essentially rational, but is still
a substantial numerical step, and there is evidence that it may be rather unstable.
Both methods must inescapably require the pro-
gram to make an arbitrary choice at some point.
364
N. J. YOUNG
Here is the method used in Yeh's implementation for finding the desired one step extension. and (2) above are satisfied: such an
Po
Let
F
o
E
be such that (1)
by the result of Adamyan et aZ ••
The codomain of
does exist.
a:2 x2
T
admits the orthog-
onal decomposition ::
consists of the constant functions in identify this subspace with gonal projection of its codomain
~t
and denote by
Po
H2. 2 •
let us
the ortho-
With respect to this decomposition
-
T can be written as a block operator matrix
::
where the multiplication operator here is regarded as acting from
Hl
Ll.
into
II Til::
p
B*F + Fo
and let
V
being non-analytic in general.
be a maximising vector for T (there are
such vectors in the rational case or. more generally, when is the sum of a continuous and an compact) •
Let
The requirement that
B*F
H oo function, as T is then
II Til:: IITII
clearly entails
that
::
(1.4)
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
to be satisfied by
FO.
365
The second requirement is that
pointwise linearly independent maximising vectors. the pointwise orthogonal complement of
V in
Let
82
p
exactly.
V
be
We require
2
to have norm
T have
Thus, if we write
we require that
and furthermoroe, the left hand side is singular. operator
Q : V -
8 2 .
so that
=
g(O>
(1.5)
= 0, we have (1.6 )
N.J. YOUNG
366
On the other hand, if
g E V and g(O)
= v(O)
we have, from
(1.5) and (1.4),
= Po (B*Fgl + Fog(O)
x Qg
=
(1. 7)
Po(B*Fg) + Po(B*PV).
Conversely, one can show that if X : H22
G
BHl
.. c2
(1.6) and (1.7) and has norm 1 then the operator R form
PoMB*F +F
o
IV
for some
In the rational case
H22
8
Fo BHl
= XQ
satisfies has the
having the desired properties. has finite dimension and the
relations (1.6) and (1.7) reduce to a finite set of linear constraints
where the vectors
Zj' Vj can be computed.
Finding X of norm 1
(when there is one) satisfying these conditions is a straightforward piece of linear algebra..
X's:
it is at this point that the arbitrary choice is made in
Q is also a known operator, and hence
Yeh's implementation.
R
=XQ
FO'
Typically there will be many such
can be calculated.
From this point it is not hard to find
Library routines can be used to find a pair of independent
P,
maximising vectors for struction of function
•
and formula (1.3) completes the conof minimal norm in
F + BH;:2.
INTERPOLATION BY ANALYTIC MATRIX FUNCTIONS
2.
367
Strengthened minimisation and the restoration of uniqueness. In the last lecture I dropped a few hints which were intended
to suggest that the results of Adamyan et
at~,
and successors who
have used one step extension techniques, are not the last word on the matrix Nevanlinna-Pick problem, particularly for those interested in the computation of solutions. the specialization
Given t E F
F E H
CD
m~
+ BHmn CD X
Recall problem (D), with
C. I: and an innel' function
such that
II til GO
B E H CD
m~
,
find
is minimised.
The procedure outlined above involves constructing ....
FEF1'BH
CD
mxn
unique 1IIember
such that the smaller coset t
F + aBH CD contains a mxn
-
of minimal norm, and that this
of the original problem.
is a solution
The most concrete objection to this
method is that the calculation of numerical stability.
t
F is lengthy and of uncertain
However, my main reason for putting forward
an alternative approach is rather mathematical intuition. is not inconsistent with having an eye to applications.
This Esthetic
considerations have long been inexplicably effective in theoretical physics, and I have faith that the same will be true in engineering. See the final two pages of [4] for poetic thoughts on this topic. A less tangible objection to the method of one step extensions is the untidiness of making an arbitrary choice, which is necessitated by the non-uniqueness of the solution of problem (D).
368
N. J. YOUNG
Another objection is that the solutions obtained by this method do not seem to be fully consonant with the spirit of the Nevanlinna-Pick problem.
To explain what I mean, let me return to
the earlier example: m
= n = 2,
F
= diag{2,
l},
B
= sI.
The solutions of problem (D) in this case are all functions of the form
• =
diag{2, g}
IIgIIH .... s.2.
where g(O) = land
The solutions obtainable by
the method of one step extensions satisfy on
aD.
t
identically
U.II ....
equal to values of
.(3)
aD.
almost everywhere on
.(3)
are constant and That is, the singular
are actually as Large as they can be, consistently
being a solution of problem (D).
Nevanlinna-Pick problem as being to minimise available sense, over a coset ask for solutions minimised.
any
of problem (D) obtained in this way is such that all
the singular values (or a-numbers) of
•
=2
This illustrates a general property of the method:
solution
with
Ig(s)1
•
F
+ BB
If we think of the
.,
in the strongest
....
mxn ,then it seems natural to
for which all singular values of
.(3)
are
This is practically the opposite of the one step ex-
tension approach.
To make the formulation precise, let us write 8
0 (A) ~ Sl(A) ~
for the singular values (eigenvalues of