On the adjoint of Hilbert space operators

Abstract

In general, it is a non-trivial task to determine the adjoint (Formula presented.) of an unbounded operator S acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator T to be identical with (Formula presented.). In our considerations, a central role is played by the operator matrix (Formula presented.). Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that (Formula presented.) always has a positive selfadjoint extension. © 2018 Informa UK Limited, trading as Taylor & Francis Group