On Cayley Identity for Self-Adjoint Operators in Hilbert Spaces

TL;DR

This paper extends the Cayley identity to self-adjoint operators in Hilbert spaces and introduces new characterizations of vectors in the singular spectral subspace using resolvent properties.

Contribution

It provides an analogue of the Cayley identity for arbitrary self-adjoint operators and new methods to characterize vectors in the singular spectral subspace.

Findings

Established a Cayley identity analogue for self-adjoint operators.

Developed two new resolvent-based characterizations of singular spectral subspace vectors.

Enhanced understanding of spectral properties in Hilbert space operators.

Abstract

We prove an analogue to the Cayley identity for an arbitrary self-adjoint operator in a Hilbert space. We also provide two new ways to characterize vectors belonging to the singular spectral subspace in terms of the analytic properties of the resolvent of the operator, computed on these vectors. The latter are analogous to those used routinely in the scattering theory for the absolutely continuous subspace.