Operator Inequalities and Spectral Analysis of Non-normal operators

我的研究是描述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)-拟次正规算子T及其谱上的非零孤立点λ， λ是T的特征值，Riesz幂等E是自伴随，其像等于T-λ的核。(2)我们定义了算子（近似点）谱的三个性质(I), (I'), (II)，并证明了每一个至少具有这些性质之一的算子都具有Bishop's性质（β）和（SVEP），此外，我们还证明了每一个超常算子都具有Bishop's性质（β）和（SVEP）。利用Birkoff-James正交，我们扩展了性质（II），并证明了每个算子都具有Bishop的性质（β）和（SVEP）。作为推论，我们得到了每个遗传正则算子都具有Bishop的性质（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).