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A characterization of k-hyponormality via weak subnormality
Journal article   Open access

A characterization of k-hyponormality via weak subnormality

Raúl E Curto, Il Bong Jung and Sang Soo Park
Journal of Mathematical Analysis and Applications, Vol.279(2), pp.556-568
03/01/2003
DOI: 10.1016/s0022-247x(03)00034-9
url
https://doi.org/10.1016/s0022-247x(03)00034-9View
Published (Version of record) Open Access

Abstract

An operator T acting on a Hilbert space is said to be weakly subnormal if there exists an extension acting on such that for all . When such partially normal extensions exist, we denote by m.p.n.e.(T) the minimal one. On the other hand, for k⩾1, T is said to be k-hyponormal if the operator matrix is positive. We prove that a 2-hyponormal operator T always satisfies the inequality , and as a result T is automatically weakly subnormal. Thus, a hyponormal operator T is 2-hyponormal if and only if there exists B such that and is hyponormal, where . More generally, we prove that T is (k+1)-hyponormal if and and only if T is weakly subnormal and m.p.n.e.(T) is k-hyponormal. As an application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix.
 Applied Mathematics Analysis

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