Spectrum of Quasi-Class (A,k) 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.