Narrow Results

In this paper, we introduce a new class of operators called the class of quasi-dominant operators. This class includes hyponormal, $M$-hyponormal and dominant operators. Our results are motivated by several known properties of $k$-quasi-$M$-hyponormal operators [F. Zuo and S. Mecheri, Spectral properties of $k$-quasi-$M$-hyponormal operators, Complex Anal. Oper. Theory12 (2018) 1877–1887], then by those of normal operators. Besides we study the local spectral properties for this class of operators.

In this paper, we define new variant of Weyl type theorems property (Sw) and the Browder version of property (Sw). We also obtain the inclusion relation among these new properties.

We introduce the class of $n$-quasi-$m$-isometric operators on Hilbert space. This generalizes the class of $m$-isometric operators on Hilbert space introduced by Agler and Stankus. An operator $T\in B(H)$ is said to be $n$-quasi-$m$-isometric if

A bounded operator $T$ in a Banach space $X$ is said to satisfy the descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the descent spectrum of $T$ as an element of the algebra $B(X)$ of all bounded linear operators on $X$. In this paper, we give some conditions under which the equality ${\sigma }_{\mathrm{\text{desc}}}(T)={\sigma }_{\mathrm{\text{desc}}}(T,B(X))$ holds for a single operator $T$.