Spectrum of Quasi-Class (,) Operators

An operator ?∈?(ℋ) is called quasi-class (?,?) if ?∗?(|?2|−|?|2)??≥0 for a positive integer ?, which is a common generalization of class A. In this paper, firstly we consider some spectral properties of quasi-class (?,?) operators; it is shown that if ? is a quasi-class (?,?) operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigenspaces corresponding to distinct eigenvalues of ? are mutually orthogonal, and the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. Secondly, we show that Putnam's theorems hold for class A operators. Particularly, we show that if ? is a class A operator and either ?(|?|) or ?(|?∗|) is not connected, then ? has a nontrivial invariant subspace.