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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On $ p$-hyponormal contractions


Author: B. P. Duggal
Journal: Proc. Amer. Math. Soc. 123 (1995), 81-86
MSC: Primary 47B20; Secondary 47A10, 47B10
DOI: https://doi.org/10.1090/S0002-9939-1995-1264808-3
MathSciNet review: 1264808
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Abstract: The contraction A on a Hilbert space H is said to be p-hyponormal, $ 0 < p < 1$, if $ {({A^ \ast }A)^p} \geq {(A{A^ \ast })^p}$. Let A be an invertible p-hyponormal contraction. It is shown that A has $ {C_{.0}}$ completely nonunitary part. Now let H be separable. If A is pure and the defect operator $ {D_A} = {(1 - {A^ \ast }A)^{1/2}}$ is of Hilbert-Schmidt class, then $ A \in {C_{10}}$. Let $ {B^ \ast }$ be a contraction such that $ {B^\ast}$ has $ {C_{.0}}$ completely nonunitary part, $ {D_{{B^ \ast }}}$ is of Hilbert-Schmidt class, and $ {B^ \ast }$ satisfies the property that if the restriction of $ {B^ \ast }$ to an invariant subspace is normal, then the subspace reduces $ {B^ \ast }$. It is shown that if $ AX = XB$ for some quasi-affinity X, then A and B are unitarily equivalent normal contractions.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1264808-3
Keywords: p-hyponormal contraction, Hilbert-Schmidt class, quasi-similar, $ {C_0}$-contraction
Article copyright: © Copyright 1995 American Mathematical Society