ON SKEW (A;m)-SYMMETRIC OPERATORS IN A HILBERT SPACE

Abstract

In this paper, we study skew $(A,m)$-symmetric operators in a complex Hilbert space. Firstly, by introducing the generalized notion of left invertiblity we show that if $T \in \mathcal{B}(H)$ is a skew $(A,m)$-symmetric, then $e^{isT}$ is left $(A,m)$-invertible for every $s\in \mathbb{R}$. Moreover, we examine some conditions for skew $(A,m)$-symmetric operators to be skew $(A,m-1)$-symmetric. The connection between $c_0$-semigroups of $(A,m)$-isometries and skew $(A,m)$-symmetries is also described. Next, we investigate the stability of a skew $(A,m)$-symmetric operator under perturbation by nilpotent
operators commuting with $T$. In addition, we show that if $T$ is a skew $(A,m)$-symmetric operator, then $T^n$ is also skew $(A,m)$-symmetric for odd $n.$ Finally, we consider a generalization of skew $(A;m)$-symmetric operators to the multivariable setting. We introduce the class of skew $(A,m)$-symmetric tuples of operators and characterize the joint approximate point spectrum of such a family.