Spectral Properties Of A Class2×2Operator Matrices

The necessary and sufficient conditions are given for the upper triangular operator matrix (AC0B) to be injective and to have dense range, where A, B, C are given. Combing with descriptions for the surjectiveness and closed range property, we characterize the spectral properties of (AC0B) based on the properties of its operator entries A. B and C. Also some characterizations on the spectra such as the point spectrum, contimous spce-trum. resid lepectrum and approximate point spectrum are further obtained Besicps the analogues for the operator matrix (ACDB) with closed R(C) are given by using space decomposition method.