On normal operator logarithms

TL;DR

This paper investigates the relationships between normal operators with equal exponentials, providing conditions under which their absolute values are equal or related, and offers formulas and generalizations for spectral projections and operator differences.

Contribution

It introduces new conditions and formulas for normal operators with equal exponentials, extending previous results by Schmoeger and covering unbounded operators.

Findings

If $e^X=e^Y$ and spectra are in a specific strip, then $|X|=|Y|$.

For bounded $Y$, $|X|Y=Y|X|$ when $e^X=e^Y$.

Provides a spectral projection formula for $X-Y$ when $X,Y$ are normal with equal exponentials.

Abstract

Let $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip $\s$ of the complex plane defined by $|\Im(z)|\leq \pi$, we show that $|X|=|Y|$. If $Y$ is only assumed to be bounded, then $|X|Y=Y|X|$. We give a formula for $X-Y$ in terms of spectral projections of $X$ and $Y$ provided that $X,Y$ are normal and $e^X=e^Y$. If $X$ is an unbounded self-adjoint operator, which does not have $(2k+1) \pi$, $k \in \ZZ$, as eigenvalues, and $Y$ is normal with spectrum in $\s$ satisfying $e^{iX}=e^Y$, then $Y \in \{\, e^{iX} \, \}"$. We give alternative proofs and generalizations of results on normal operator exponentials proved by Ch. Schmoeger.