On a connection between Naimark's dilation theorem, spectral representations, and characteristic functions

TL;DR

This paper presents a new proof of Naimark's dilation theorem using a Herglotz-type representation of spectral measures, and explores spectral properties of unitary rank-one perturbations of partial isometries.

Contribution

It introduces a novel proof technique for Naimark's dilation theorem and applies it to analyze spectra of rank-one perturbations of partial isometries.

Findings

New proof of Naimark's dilation theorem

Spectral description of unitary rank-one perturbations

Herglotz-type representation of spectral measures

Abstract

We give a Herglotz-type representation of an arbitrary generalized spectral measure. As an application, a new proof of the classical Naimark's dilation theorem is given. The same approach is used to describe the spectrum of all unitary rank-one perturbations of a given partial isometry.