Spectral decompositions and the carleson condition

7.4.

SPECTRAL DECOMPOSITIONS

AND THE CARLESON CONDITION~

Completely nonunitary contractions T of a Hilbert space are contained in the Sz.-Nagy-Foias model [I]. Especially simple is the case when the spectrum o(T) does not cover the circle

c~SD

and d = d, < +~, where d = d i m ( I -- T*T)H,

indices of the operator T. its model-operator

If $ - ~

d, = d i m ( I -- TT*)H are the defect

, then such an operator T is unitarily

K=Ke=H~E)eeH~E) ,E=~(I-T*T)H

PZIK, where

equivalent

to

, e is a bounded and analytic

(E § E)-operator-function in ~ , unitary on the circumference T , H2(E) is the space of all E-valued H2-functions, Z is the multiplication operator f § zf, and P is the orthogonal projection onto K. In terms of this function 0 (the characteristic function of the operator), related simply with the resolvent [for example, |~(~)IXCl-l~lYr|0(~ , ~ e ~ ) ], one can investigate in detail the initial operator T: to find its spectrum, its point spectrum Op(T), the proper and the root vectors, to compute the angles between the spectral subspaces, etc. ([I-4]). In particular, the operator T is complete (E has a complete collection of proper and root vectors) if and only if det e is a Blaschke product. However, a more detailed spectral analysis assumes more: not only the selection and the description of the spectral subspaces, but also an indication of a method of reestablishing the operator T in terms of its restrictions to these subspaces. A very strong method of restoration is an unconditionally convergent spectral expansion, generated by the given partition of the spectrum. If T is a complete operator, then the problem is whether its root subs.paces K ~ , ~ ( T ) , form an unconditional basis. In the case of a simple point spectrum, necessary and sufficient conditions for such a "spectrality" (in this case similarity to a normal operator) can be found in [2, 3]; they consist of Carleson's vector condition

and the following

imbedding theorem: For any function

Z (l-,x0AxcItr

,

~,

~,~eH~(E),

~ cX)ll~E r (I-IXI)II~

we have (NC)

oo

Here A~ is the orthoprojection in E onto the subspace Ker e(~), A~ onto the subspace

O=0x.[&xEx+(l-Ex)] ,

is the factorization

~

-f-

uniform minimality

of the family

the sign inf is sin (K%, K%),

The geometric condition ~Kxlxr ~

and, moreover,

(NC)

to the eigenspace

(C) is equivalent

to the so-called

the number occurring

where K = V ( K ~ : ~ p ( T ) x [ k ] )

I, L. Carleson has shown that (C) ~ Problem I.

of the function O, corresponding

; see [2].

KevO(X~ ,

in (C) under

In the case d = d, =

(see [4]), but for d = d, = ~ this is not so [3].

Prove or disprove the implication

(C)

~

(NC), assuming that I < d = d, <

The case d = d, = I is nevertheless exceptional since for biorthogonal families of "general position" the distance between uniform minimality and (unconditional) basicity is very large. Nevertheless, for d = d, = I these conditions coincide not only for the families of eigenspaces but also for the root and (even!) for the more general spectral subspaces of a contraction T [5]. The existing equivalence proof [5, 4] has the character of an analytic trick and depends on the computation of the angle between the complementary spectral subspaces K~ , K~ ~d a K~,

, corresponding

to the divisor

~

of the function O.

Namely,

one

t v. I. VASYUNIN, N. K. NIKOL'SKII, and B. S. PAVLOV. V. A. Steklov Mathematical Institute, Leningrad Branch, Academy of Sciences of the USSR, Fontanka 27, Leningrad, 191011, USSR. Department of Physics, Leningrad State University, University Quay 7/9, Leningrad, 199164,USSR. 2152

succeeds

to show [5] (d = d i m E

= I) that for the family

{t~] of spectral divisors,

from (cv)

e~E,Ilell=4 ~ D

there follows ,

where

~

is the inner function corresponding

family of the family [ ~

to the subspace

V(K~:~r

, d being a sub-

The argument makes use of a lower estimate of the number ll@([)ellE

in terms of a quantity depending only on ll~(~eIIE,ll~(~)e}IE not possible (L. E. Isaev, oral communication). Problem 2. d, < ~ and that

(scv)

, which for d i m e

> I is simply

Prove or disprove the implication (CV) ~ (SCV), assuming that I < d = [{~ is an arbitrary family of spectral divisors of the function 0.

Finally we note that the character of the (C)- and (CV)-sets has been studied in a sufficiently detailed manner and that, apparently, the solution of Problems I-2 could clear up also other dark spots in the spectral theory on the model of Sz.-Nagy--Foias. LITERATURE I 9

2.

3.

o

5.

CITED

B. Sz.-Nagy and C. Foia~, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., Amsterdam (1970). N. K. Nikol'skii and B. S. Pavlov, "Bases of eigenvectors of completely nonunitary contractions and the characteristic function," Izv. Akad. Nauk SSSR, Ser. Mat., 34, Noo i, 90-133 (1970). N. K. Nikol'skii and B. S. Pavlov, "Eigenvector expansions of nonunitary operators and the characteristic function," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 11, 150-203 (1968). N. K. Nikol'skii, "Lectures on the shift operator. II," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 47, 90-119 (1974). V. I. Vasyunin, "Unconditionally convergent spectral expansions and interpolation problems," Tr. Mat. Inst. Steklov, 130, 5-49 (1978).

2153