Putnam's inequality for quasi-*-class A operators

Rashid, M. H. M. (2023) Putnam's inequality for quasi-*-class A operators. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 34 (1). pp. 65-77. ISSN 1786-0091

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Abstract

An operator T ∈ B(H ) is called quasi-∗-class (A, k) (abbreviation, T ∈ Q∗ (A, k)) if T ∗k (|T 2 | − |T ∗ | 2 )T k ≥ 0 for a positive integer k, which is a generalization of ∗-class A. In this paper, firstly we consider some spectral properties of quasi-∗-class (A, k) operators; it is shown that if T ∈ Q∗ (A, k), then the nonzero points of its point spectrum and the joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of T are mutually orthogonal and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Also, we consider the Putnam’s inequality for quasi-∗-class (A, k) operators. Moreover, we prove that two quasisimilar quasi-∗-class (A, k) operators have equal essential spectra.

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