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By Dale E. Alspach, William B. Johnson (auth.), Ron C. Blei, Stuart J. Sidney (eds.)

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7] [8] Drury, of 4 (1951) finite Shifts Harmonic 258-281. S. Analysis of O p e r a t o r s on H i l b e r t 1970. , a n d A. operators, T. A n d o , Space, J. r e i n e angew Math 208 London Math Varopoulos, cation theory, 512, On a c l a s s (1975) Varopoulos, J. Func. Soc. 7 inequality (1975) contractions, of B a n a c h to the for H i l b e r t 49-50. Acta Algebras, Sci. Lecture Math 24 Notes 180-184. On a n i n e q u a l i t y of the m e t r i c inequality 300-304. Bull On a p a i r of c o m m u t a t i v e No.

N a + IK = 1 2 K _ k n=l n Proof. • Exercise. Corollary. We may write Nj ~(z) where degree mn K. is a m o n o m i a l = ~ mn(Z) ~n(Z) n=l of d e g r e e K-k and ~n is h o m o g e n o u s of Further 2 [I[~nll2 ~ -1/211~ll ~j Splitting using up the inner product the e s t i m a t e on l[sll op according we have to this the decomposition required and result. I should l i k e to take this oppo~uni~y to thank the U~versity of Connecticut for i t s hospitality d~ing the special year in Harmonic Analysis.

To p r o v e will it f o l l o w s the t h e o r e m o f the m a i n our main j u s t be c a l l e d IIBIIC V, that 16 > ~ - Since ~ e > 0 -- follows. lemma lemma we lemmas shall in the need some following. auxiliary First we lemmas shall that introduce some notations. For notational the additional convenience assumption tion we are given finite we that first prove A = B(H) subsets the main while {ai }N i=l C A h lemma B = B(K). By assump- {b i} ~ = i C 8 and under , such that a2 i ~ IH (i H being the identity operator on H) and 7.

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