W.\xa0Stenger\u2019s and M.A.\xa0Nudelman\u2019s results and resolvent formulas involving compressions

Abstract

In the first part of this note we give a rather short proof of a generalization of Stenger’s lemma about the compression A_0 to {{mathfrak {H}}}_0 of a self-adjoint operator A in some Hilbert space {{mathfrak {H}}}={{mathfrak {H}}}_0oplus {{mathfrak {H}}}_1. In this situation, S:=Acap A_0 is a symmetry in {{mathfrak {H}}}_0 with the canonical self-adjoint extension A_0 and the self-adjoint extension A with exit into {{mathfrak {H}}}. In the second part we consider relations between the resolvents of A and A_0 like M.G. Krein’s resolvent formula, and corresponding operator models.