Two problems on spectral synthesis

7.5.

TWO PROBLEMS ON SPECTRAL SYNTHESIS~

I.

Synthesis Is Not Possible.

We consider the synthesis of the invariant subspaces

for the operator Z* , conjugate to the multiplication ~ :f § zf by the independent variable z in some space of analytic functions. More precisely, let X be a Banach space of functions, defined and holomorphic in the circle

D

these functions, and such that z X c X

, embedded topologically in the space if ~ e X

zero of the function f at the point ~ , ~ ; integral function in the circle

~

, then ~p(~)de~

X~[~:I~X~

~>I ~]

H0~C~)

of all

the multiplicity of the , where k is a nonnegative,

A closed subspace E, invariant relative to Z , is said

to be divisorial in E = Xk for some k [necessarily, ~(g)=~E(~) &el ~ u ~ ~(g),

g6~

].

Conjecture I. In any space X of the described- type, there exist nondivisorial, invar iant Subspaces.

Z-

The statement which is dual to the property of being divisorial consists in the possibility of spectral synthesis for the operator Z ~ : If the space Y, conjugate (or preconju-

gate) with X, is Cauchy realizable [ i . e . , = implies

(~)~(nO on polynomials], then Z ' F c F "

E=VCC~-Xz)

z

:~-~-r

(I)

where k = kEi. In other words, the possibility of synthesis for the subspace E means that it is reestablished (in the topology defined by the duality <X, Y>) from the root vectors of the operator Z ~ which are contained in it. All the known results on F -invariance (see [I]) confirm Conjecture I. The principle requirement for the space X is that it be a Banach space and the problem becomes nontrivial if , for example, the polynomials

~A

are contained densely in X and

[~ " ~

sP~~

I pO;)l

•

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