Operator Inequalities and Spectral Analysis of Non-normal operators

我的研究是刻画由算子不等式或算子范数不等式定义的算子类。（1）证明了对于（p，k）-p拟亚正规算子T及其谱的非零孤立点λ，λ是T的特征值，Riesz幂等元E是自伴的，其像等于T-λ的核。（2）定义了算子的（逼近点）谱的三个性质（I），（I '），（II），证明了至少具有其中一个性质的算子都具有Bishop性质（β）和（SVEP），并证明了每个超正规算子都具有Bishop性质（β）和（SVEP）。利用Birkoff-James正交性，推广了性质（II），证明了具有该性质的算子都具有Bishop性质（β）和（SVEP）.作为推论，我们得到了每个遗传正规类算子都具有Bishop性质（β）和（SVEP）.

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).