The intersection of left (right) spectra of 2×2 upper triangular operator matrices

AbstractWhen A∈B(H) and B∈B(K) are given, we denote by MC the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form MC=AC0B. In this paper, for given A and B, the sets ⋂C∈Bl(K,H)σl(MC),⋂C∈Inv(K,H)σl(MC) and ⋃C∈Inv(K,H)σl(MC) are determined, where σl(T),Bl(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H