The Orthogonal Complement Of The Range For A Class Of Operator Matrices

Le X and Y be Hilbert spaces,and let T:D(T)(?)X→ Y and S:D(S)(?)Y→X be densely defined closed linear operators.Assume that(?) where a and b are complex numbers.In this paper,we characterize the orthogonal com-plement of the range for operator matrices A by the graphs of T and S,and then some spectral properties of TS and ST are obtained.