Operators and approximation

6.4.

OPERATORS AND APPROXIMATION~

I.

What Is a "Blaschke Product"?

The answer to this question is well known if one con-

siders scalar functions, analytic in the unit circle following equivalent ~X

statements: A.

Ikl) < ~

];

B.

The function B can be represented as a product

gx=(x-z)C -Xzi~

of elementary factors

[k is an integer-valued function i n 0

The function B is an inner function

the shift operator

D , and it is contained in any of the

,

B =D x heD ~Ck)(4-

(in the Beurling sense) and the part of

~ I K , K =Ks-~-H ~ e ~ H z, has a family of root subspaces, complete in K; here

H 2 is the known Hardy class, z, B, respectively;

C.

Z, ~

are the operators of multiplications by the functions

The same for the operator

orthoprojection onto K); D.

TB~i--PKzIK

, conjugate to ~ I K

(Pk is the

The function B is an inner function and

The spectral interpretation of the definitions A and D plays an important investigation of the operators in the language of the characteristic function, lems discussed in this section reduce to a "proper" choice of the corresponding the general case when one talks about the inner operator-valued functions [the

role in the and the probconcept in equality

]B(~)l = I a.e. ~, ~ e T , is replaced by the unitary property of the operator B(~) in the Space E of the coefficients, H 2 is replaced by H2(E), etc.]. Under appropriate modification the statements A-D coincide for the operators T B having a determinant [(I -- T~TB) is a nuclear operator]. Apart from this case, the most natural definitions of the "Blaschke product" are B and (or) C. Question

I.

In terms of the function B, how does "and" differ from "or"?

The accepted definition implies the presence of a metric criterion to determine whether the characteristic function of B belongs to the class of "Blaschke products" [i.e., a criterion for the completeness of the operators T B and (or) T~]. Question 2.

Are the conditions

~/I

T

T

such criteria? If we restrict ourselves to operators reduces to the following one. Question 3.

~=Z

IK with a simple spectrum, then Question 2

How can one describe in terms of the function B the subspace KB, generated

by the eigenfunctions

of the operator ~

, i.e., the subspace of the form V C ~

:X ~0)

?

is the family of orthoprojections in E and a question of no small imHere ~-[~:~e~] portance is the one regarding the coincidence of the subspace of the previous form with all of of HZ(E), i.e., the following question. Question 4.

For which families

Clearly, for AI = ~ Ikl) =+oo)

[Ax:~eD]

does

~6H2(~)

, ~X~(~)=~ ( ~ D )

or I, everything reduces to the scalar uniqueness

imply f z ~ ?

theorem ( ~ o ( 4 -

and it is known (Yu. P. Ginzburg) that a necessary condition for completeness

is

%N. K. NIKOL'SKII. V. A. Steklov Mathematical Institute, Leningrad Branch, Academy of Sciences of the USSR, Fontanka 27, Leningrad, 191011, USSR.

2150

~(4-I~I~1%(~k~ ~=+~ .

Apparently, the answer to Question 4 is "

k~ ~

=~

for some

k~ 0

e, e e E." Regarding Question 3, for dim E = 1 [i.e., f o r H2(E) = H 2 ] , i t s a n s w e r c a n b e formulated also in terms of the so-called pseudo-extensions of the functions f r o m V (M. M. Dzhrbashyan, g . T s . T u m a r k i n , R. D o u g l a s , H. S h a p i r o , A. S h i e l d s , etc.) and it is not excluded that this setting is suitable a l s o f o r d i m E > 1. 2. tion,

Weak G e n e r a t o r s ~ (w)

is

the

of the

weakly

Algebra

closed

algebra

' ~ ( T e) .

In this

of operators,

It is known (D. Sarason) that ~e~tTe)~=~a~(Te),~eH

section generated

~ .

0 is

a scalar

by the

inner

operators

The operator q(Te)

func-

* and I.

acts in the

space K = Ke according to the rule ~(Te) ~ =~Kq~, ~ 6 K

. The description of the weak genera-

tors of the algebra

with the property ~ ( ~ ) = H ~ ] is also

~(Z)=H ~

[i.e., the functions ~

known (Sarason) and can be expressed in a geometric terminology by the properties of the (necessarily univalent) image ~(~) . Since the algebra ~(~8) is isometrically isomorphic to the algebra H~/eH ~, we can say that Sarason's theorem "admits projection." Question 5. Is it true that ~(q(Te)) =~(T~) tor of the algebra H~176 Question 6.

Which operators

q(Ts)

if and only if q + @ H ~

contains a genera-

have a simple spectrum, i.e., there exists f, f

If ~ is a generator, then the cyclic vector f from the problem 6 exists automatically and all these vectors can be easily described. In the special case ~ = expa[(z + 1)/(z -- I)], Question 6 reduces (at least for certain ~ ) to the problem of the unicellularity of the operator ~(Te) , or of the operator ~ - - ' ~J (0 ~ ( 5 ) ~ ( ~ - ~ 5

in the space L2(0, a) (G. ~. Kisilev-

skii). A paper by J. Ginsberg and D. Newman [J. Approx. Theory, 24, No. 4 (1970)] and problem 14.5 of this collection refer to this circle of problems. Other references, history, and discussions can be found in the papers of N. K. Nikol'skii [in: Itogi Nauki, Matematicheskii Analiz, Vol. 12 (1974); The Theory of Operators in Functional Spaces, Novosibirsk (1977)].

2151