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 H. Firstly, by introducing the generalized notion of left invertibility we show that if T ? B(H) is skew (A,m)-symmetric, then eisT is left (A,m)-invertible for every s ? R. Moreover, we examine some conditions for skew (A,m)- symmetric operators to be skew (A,m?1)-symmetric. The connection between c0-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 some perturbation by nilpotent operators commuting with T. In addition, we show that if T is a skew (A,m)-symmetric operator, then Tn 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.