Non-self-adjoint resolutions of the identity and associated operators

TL;DR

This paper investigates non-self-adjoint resolutions of the identity in Hilbert spaces and characterizes when such operators have spectral representations similar to self-adjoint operators, using bounded transformations.

Contribution

It provides a characterization of closed operators with spectral representations akin to self-adjoint operators via bounded invertible transformations.

Findings

Operators have spectral representations if and only if they are similar to self-adjoint operators.

The study extends spectral theory to certain non-self-adjoint operators.

Conditions for resolutions of the identity to generate such operators are established.

Abstract

Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a closed operator $B$ has a spectral representation analogous to the familiar one for self-adjoint operators if and only if $B=TAT^{-1}$ where $A$ is self-adjoint and $T$ is a bounded operator with bounded inverse.