Spectral Theory of Self-adjoint Finitely Cyclic Operators and Introduction to Matrix Measure $L^2$-spaces

TL;DR

This paper develops a spectral theory for finitely cyclic self-adjoint operators, introducing matrix measure $L^2$ spaces and proving a representation theorem linking these operators to multiplication operators in such spaces.

Contribution

It provides a detailed introduction to matrix measure $L^2$ spaces and establishes a new representation theorem for finitely cyclic self-adjoint operators.

Findings

Representation of finitely cyclic self-adjoint operators as multiplication operators.

Introduction of matrix measure $L^2$ spaces for spectral analysis.

Analysis of spectral properties like absolute continuity.

Abstract

We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space. One of the main goals here is to obtain the representation theorem for such operators in a form analogous to the one well-known in the cyclic self-adjoint operators case. To do this, we present here a detailed introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the multiplication by scalar functions operators in such spaces. This allows us to formulate and prove in all the details the less known representation result, saying that the finitely cyclic self-adjoint operator is unitary equivalent to the multiplication by the identity function on $\mathbb{R}$ in the appropriate matrix measure $L^2$ space. We study also some detailed spectral problems for finitely cyclic self-adjoint operators, like the absolute continuity.