Operator Inequalities and Spectral Analysis of Non-normal operators

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My research is to characterize the classes of operators which are defined by operator inequalities or operator norm inequalities. The following two results are main results in this period :(1) We show that for a (p,k)-pquasihyponormal operator T and a non-zero isolated point λ of its spectrum, λis an eigenvalue of T and the Riesz idempotent E is self-adjoint whose image is equal to the kernel of T-λ. As a corollary, we obtain that every (p,k)-quasihyponormal operator satisfies Weyl's theorem,(2) We define three properties (I), (I'), (II) of (approximate point) spectrum of operators and show that every operator with at least one of these properties has Bishop's property (β) and (SVEP), moreover, we show that every paranormal operator has Bishop's property (β) and (SVEP). By using Birkoff-James orthogonality, we extend property (II) and show that every operator with this property has Bishop's property (β) and (SVEP). As a corollary, we obtain that every hereditarily normaloid operator has Bishop's property (β) and (SVEP).