Balanced Operators and Operator Domains

TL;DR

This paper introduces the concept of balanced operators on Hilbert spaces, characterizes their properties via phase and modulus, and provides spectral measure-based criteria for domain equality of positive self-adjoint operators.

Contribution

It defines balanced operators, explores their structure, and offers new spectral measure characterizations for domain equality of certain self-adjoint operators.

Findings

Balanced operators are characterized by phase and modulus.

Examples of balanced operators are constructed.

Domain equality for positive self-adjoint operators is characterized spectrally.

Abstract

We shall say that a densely defined closed operator $T$ on a Hilbert space is balanced if $\cD(T)=\cD(T^*)$. Balanced operators are described in terms of their phase operators abnd their moduli. Examples of balanced operators are developed. A characterization of the domain equality $\cD(A)=\cD(B)$ for positive self-adjoint operators $A$ and $B$ with bounded inverses is given in terms of their spectral measures.