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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Double commutants of $ C\sb{\cdot 0}$ contractions


Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 69 (1978), 283-288
MSC: Primary 47A45
MathSciNet review: 0482301
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Abstract: D. Sarason has shown that an operator in the double commutant of a contraction T of class $ {C_0}(1)$ is interpolated by a function in $ {H^\infty }$, that is, $ \{ T\} ' = \{ \phi (T);\phi \in {H^\infty }\} $, [3]. Generally, in [4] Sz.-Nagy and C. Foiaş have shown that an operator in the double commutant of a contraction T of class $ {C_0}(n)$ is interpolated by a function in $ {N_T}$, that is, $ \{ T\} '' = \{ {\phi _1}{(T)^{ - 1}}{\phi _2}(T):{\phi _2}/{\phi _1} \in {N_T}\} $. In this note we shall show that an operator in the double commutant of a $ {C_{ \cdot 0}}$ contraction T with finite defect indices $ {\delta _{{T^ \ast }}} > {\delta _T}$ is interpolated by a function in $ {H^\infty }$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0482301-2
PII: S 0002-9939(1978)0482301-2
Keywords: $ {C_{ \cdot 0}}$ contraction, double commutant, inner matrix, lifting theorem
Article copyright: © Copyright 1978 American Mathematical Society