Invertibility Of 2×2 Operator Matrix

To begin with, the research of the invertibility and spectrum of 2 × 2 linear op-erator matrices is briefly outlined. Besides, the equivalent condition of invertibility of which is 2×2 closed linear operator matrices acting in Hilbert Space X × X is studied in this paper. In particular, the invertibility of infinite dimensional Hamiltonian operator is giv-en. Moreover, the description of the approximate point spectrum, detect spectrum and compressed spectrum of a kind of operator matrices which is 2 × 2 diagonal domain operator matrices is studied.