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

 

Similarity of parts to the whole for certain multiplication operators


Author: Paul S. Bourdon
Journal: Proc. Amer. Math. Soc. 99 (1987), 563-567
MSC: Primary 47B38; Secondary 47B35
MathSciNet review: 875398
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Abstract: We show that the Bergman shift $ B$, multiplication by $ z$ on the Bergman space $ {A^2}$, is similar to its part $ B\left\vert {_N} \right.$ if and only if $ N = \varphi {A^2}$, where $ \varphi $ is a finite product of interpolating Blaschke products. In addition, we show that $ B$ is not unitarily equivalent to any of its parts. For the analytic Toeplitz operator $ {T_f}$ on $ {H^2}$, we obtain that $ {T_f}$ is similar to each of its parts if and only if $ {T_f}$ is unitarily equivalent to each of its parts if and only if $ f$ is a weak-star generator of $ {H^\infty }$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0875398-4
PII: S 0002-9939(1987)0875398-4
Article copyright: © Copyright 1987 American Mathematical Society