Invariant subspaces of a shift operator in certain spaces of analytic functions

~1.5.

INVARIANT

SUBSPACES OF A SHIFT OPERATOR IN CERTAIN SPACES OF

ANALYTIC FUNCTIONS*

I.

Let X be a Banach algebra of analytic functions

tions of pointwise multiplication (~) _ _ X CCA ~ <+e,o. Let ~ ) (.~)--~[~

and addition) .

in the circle

~

We assume that X C ~ As

~6]['~ ~(i)(~)----O,

(with the operaand 0 4 ~ = ~ [ ~

:

O~j ~ l -

Definition. Let E be a closed subset of the circumference ~ ; we say that E is a D nset relative to X if for any function f, f ~ X, such that E(n)(f)m E, t h e r e e x i s t s a s e q u e n c e {fk}k>~z of functions from X having the properties: .,+4

-

nx

=o.

It is proved in [1] that for a series of standard algebras of analytic functions, ing--Carleson set~ is a Dn-set. In particular, this holds also for the algebras H~+~=[~:~ (~+O ~ there arises the following problem. Problem

I.

Is every Beurling--Carleson

HP]

set on the circumference

any Beurl-

, I < p < +~.

Here

a Dn-set for the algebra

H~+l, n ~ 0? Remark.

In the case when [ = ~ 4

~

' where mE k = 0, k = I, 2,...,m and the complementary

intervals of the sets Ek on the circumference decrease as the terms Of a geometric progression, it has been proved in [2] that E is a Dn-set for H~+ l and the required sequence {fk}k~l can be chosen independently of f. 2. of class

Let

~0)

L~),

be the Bergman space in the circle

analytic

Problem 2.

Let G, 6 c ~ P , be an invariant

G is generated by a single function, G = V(zng:n ~ 0)?

polycylinder "diagonal

Problems in

~z

operator"

It is known that

and let

~',

, ~(~) ~ C

S.

to the oper-

We assume that

Does it follow from here that

i.e., that there exists a function g, g s

be a Hardy space in ~

in H P ~ O =) , i.e., ~2~(~)=~(<,~), < ~

exist no finitely generated

common zeros in (see [6]).

subspace of the operator

2 and 3 arise from the following considerations.

D,(HP(D~)):AP ,0

exists an invariant

, invariant relative

, which is not finitely generated?

for each point ~, ~ ~ ~ , there exists f, ~ G

Remark.

[i.e., the space of all functions

~ ].

Is there a closed subspace G, G c A P , G # A P

ator S, 5 ~ ) ( ~ j ~ ( % ) Problem 3.

in

~

< p <+~(see

Sl,2-invariant

(see [3]).

Let

, for which D ~ be the unit

We denote by D2 the

, and by S i the operator defined by

H~(D~)

[4, 5]), and that in the spaces

subspaces

(see [3]).

subspace which is generated by two functions

In addition, from H ~ ( ~ )

there

in H~(O ~) there , having no

and which, on the other hand is not generated by any of its functions

*F. A. SHAMOYAN. Institute of Mathematics, Academy of Sciences of the Armenian Barekamutyan 24b, Erevan 19, 375200, USSR. ~For the definition see, e.g., p. 2265 of the present volume -- Editors.

SSR, UI.

2193

LITERATURE CITED I.

2.

.

4. 5. 6.

2194

F. A. Shamoyan, "The structure of closed ideals in certain algebras of functions which are analytic in a circle and smooth up to its boundary," Dokl. Akad. Nauk Arm. SSR, 60, No. 3, 133-136 (1975). F. A. Shamoyan, "The construction of a certain special sequence and the structure of the closed ideals in certain algebras of analytic functions," Izv. Akad. Nauk Arm. SSR, Ser. Mat., ~, No. 6, 440-470 (1972). W. Rudin, Function Theory in Polydiscs, Benjamin, New York (1969). C. Horowitz and D. M. Oberlin, "Restrictions of HP functions to the diagonal of un, '' Indiana Univ. Math. J., 24, No. 7, 767-772 (1975). F. A. Shamoyan, "An embedding theorem in spaces of n-harmonic functions and some applications," Dokl. Akad. Nauk Arm. SSR, 62, No. I, 10-14 (1976). C. A. Jacewicz, "A nonprincipal invariant subspace of the Hardy space on the torus," Proc. Am. Math. Soc., 31, 127-129 (1972).