Operator-valued bounded analytic functions

8.4.

OPERATOR-VALUED BOUNDED ANALYTIC FUNCTIONS~

Let ~ mapping

~

i~ *

be two Hilbert spaces and ~

be the space of all bounded linear operators

into ~* . The following was proved in [I].

THEOREM. Suppose 0 is a bounded ~(~,~)*) -valued function analytic in the unit disc The following assertions are equivalent: (a) there exists a bounded ~ ( ~ )

-valued function ~ analytic in ~

,['],(~)O(~)=l~ [~e~)

and satisfying

(1)

;

(b) the Kernel function Ks:

is positive definite, i.e., (2) for anyfinitesystems {%l,...,Xn}, {dl, .... an} , where

~

, dj~O

Condition (I) obviously implies that

lec~)&i~slgl (j~l~4),

(3)

The question is whether (3) implies (2) with the same s or at least with some, possibly different, positive constant. In the special case when dim ~=~ and a l m ~ the equivalence of (I) and (3), and thus the equivalence of (2) and (3), follows from the Corona theorem of Carleson (cf. [2]). A proof of the equivalence of (2) and (3) in the general case, and possibly with operator theoretic arguments, would be an important achievement. LITERATURE CITED 1 9

2 ~.

B. Sz.-Nagy and C. Foia~, "On contractions similar to isometries and Toeplitz operators," Ann. Acad. Sci. Fenn., Ser. AI: Math. Phys., No. 2, 553-564 (1976). W. Arveson, "Interpolation problems in nest algebras," J. Funct. Anal., 20, 208-233 (1:925).

~B. SZOKEFALVI-NAGY. Hungary.

2154

Bolyai Institute of Mathematics, 6720 Szeged Aradi V~rtanuk tere I,