8.4.
OPERATOR-VALUED BOUNDED ANALYTIC FUNCTIONS~
Let ~ mapping
~
i~ *
be two Hilbert spaces and ~
be the space of ...
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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,