Some questions of the spectral theory of operator-matrices (Q1178096)

The author studies the spectral properties of the selfadjoint operator \(T: E_ 1\oplus E_ 2\to E_ 1\oplus E_ 2\), represented by the matrix form \(T={{T_{11} T_{12}} \choose {T_{21} T_{22}}}\), where \(E_ j\) are Hilbert spaces, and the operators \(T_{ii}: E_ i\to E_ i\) are selfadjoint. In particular, he proves that if \(\alpha=\sup \sigma(T_{11})<\beta=\inf \sigma(T_{22})\) and \(D(T_{ii})\subset D(T_{ji})\) for \(i\neq j\), then the interval \((\alpha,\beta)\) contains no points of the spectrum of \(T\).