Variance of operators and derivations

TL;DR

This paper characterizes when two operators have identical variance at all vectors, explores inequalities involving operator variances, and examines their implications for derivations and functional calculus in Hilbert space operators.

Contribution

It provides a complete characterization of operators with equal variances, establishes conditions for variance inequalities via Lipschitz functions, and links these to derivation range inclusions in C*-algebras.

Findings

Operators with the same variance differ by a scalar or are related via conjugation and scalar addition.

Variance inequalities hold iff one operator is a Lipschitz function of the other on the spectrum.

Connections between derivation range inclusion and operator inequalities are established, especially for subnormal operators.

Abstract

The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a unit vector $h$ is defined by $D_h(a)=\|ah\|^2-|<ah,h>|^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $h\in H$ if and only if there exist scalars $\sigma,\lambda$ with $|\sigma|=1$ such that $b=\sigma a+\lambda1$ or $a$ is normal and $b=\sigma a^*+\lambda1$. Further, if $a$ is normal, then the inequality $D_h(b)\leq\kappa D_h(a)$ holds for some constant $\kappa$ and all unit vectors $h$ if and only if $b=f(a)$ for a Lipschitz function $f$ on the spectrum of $a$. Variants of these results for C$^*$-algebras are also proved. We also study the related, but more restrictive inequalities $\|bx-xb\|\leq \|ax-xa\|$ supposed to hold for all $x\in B(H)$ or for all $x\in B(H^n)$ and all positive integers $n$. We consider the connection between such inequalities and the range inclusion $d_b(B(H))\subseteq d_a(B(H))$, where $d_a$ and $d_b$ are the derivations on $B(H)$ induced by $a$ and $b$. If $a$ is subnormal, we study these conditions in particular in the case when $b$ is of the form $b=f(a)$ for a function $f$.