Weakly invertible elements in Bergman spaces

10.5.

WEAKLY INVERTIBLE

Definition

ELEMENTS IN BERGMAN SPACES~

is the Hilbert

I.

space of analytic functions

f in ~

with the norm

D Definition

2 [I].

Let ~

including a]_l single points,

be the set of all open, closed, and half-closed T

and ~.

A function

arcs

I,IcT,

is called a premeasure iff

.~ . ~ - ~

(~)~(I,uL) =~(I,)+#(I~) .h~. h,l~ eZ,

Definition

3.

A closed set

~, F c T ,

is called BeurlinF-Carleson

(n IFI--O;

~ ~+~

II l where In are the components Proposition

I [I, 2].

of

T\~

Let

}e~

and ~

and

(B.--C. set) iff

,

I.I denotes the linear Lebesgue measure. ~(Z)~0 (Z~--D) 9 Then the following properties

hold: (i) The limit

(2) ~--4-.o I exists for any arc

1,leT

;

(ii) the limit

~(I) = ~ #(I~) exists for any sequence of closed arcs arc ;

(In ) such that

(iii) p(I), defined by (3) for open arcs I , I c T , measure ; (iv) for any B.--C. set F, whose complementary solutely convergent;

(3) l~cI~

. and U I ~ =I , I being any open

admits a unique extension

arcs are In, the series

to a pre-

~(I~)

is ab-

(v) if we define

(F)= ~ ~@ IR0)I-~ ,~(I~),

(4)

for B.--C. sets F, then of admits a unique extension to a finite nonpositive every B.--C. set. Definition 4.

The measure of (defined on the set of all Borel sets contained

set) is called the x-singularmeasure associatedwith ~BORIS KORENBLUM. New York, 12222.

Borel measure on

Department

of Mathematics,

f, } ~

in a B.--C.

(it is assumed that }(Z)#0

State University

in

D ).

of New York at Albany, Albany,

2191

Proposition I follows immediately from the results of [I, 2], since ~ 6 ~ z

implies

(5) An element ~ , : ~ w _ . ' ~ l , is called

Definition 5. elos

weakly invertible (or cyclic) iff

{fg:~EH~}=~ ~..

Proposition 2. weakly invertible:

(a) ~(E) # 0

The following conditions are necessary for an element ~ , ~ E ~ ~ , to be

(s

(6) (7)

(b) af = 0. This proposition follows easily from the main theorem in [2] which gives a description of closed ideals in the topological algebra A -~ of analytic functions f satisfying

I~tz)l-~Of(t-IZI) -% (zr D), Conjecture I. vertible. Conjecture 2.

(8)

Conditions (6) and (7) are sufficient for an ~ i ~ E ~ 2, to be weakly inThe same conditions also describe weakly invertible elements in any Berg-

man space ~ P i ~ p < o o) of analytic functions f,with the norm

(9)

0 LITERATURE CITED I ~

2.

2192

B. Korenblum, "An extension of the Nevanlinna theory," Acta Math., 135, 187-219 (1975). B, Korenblum, "A Beurling-type theorem," Acta Math., 138, 265-293 (1977).