A Note on the Left Essential Spectra of Operator Matrices

When A∈ B(H) and B∈ B(K) are given, we denote by M_C the operator matrix acting on the in.nite dimensional separable Hilbert space H⊕K of the form M_C( A C 0 B ). In this paper, a necessary and sufficient condition for M_C to be left Fredholm for some C∈ F(K, H) (C∈ Inv(K, H)) is given, where F(K, H) and Inv(K, H) denote respectively, the set of Fredholm operators and the set of invertible operators of B(K, H). In addition, we give a necessary and sufficient condition for M_C to be left Fredholm for all C∈ Inv(K, H).