A note on equicontinuity of families of operators and automorphisms

TL;DR

This paper investigates conditions under which families of operators and automorphisms on a Hilbert space are uniformly equicontinuous, providing a geometric criterion and linking automorphisms to their implementing unitaries.

Contribution

It establishes a necessary and sufficient geometric condition for uniform equicontinuity of operator families and characterizes automorphisms via their implementing unitaries.

Findings

Automorphisms are uniformly equicontinuous iff their implementing unitaries are.

A geometric condition characterizes uniform equicontinuity of operator families.

One-parameter automorphism groups are uniformly equicontinuous under specific conditions.

Abstract

This note concerns uniform equicontinuity of families of operators on a separable Hilbert space H, and of families of maps on B(H). It is shown that a one parameter group of automorphisms is uniformly equicontinuous if and only if the group of unitaries which implements it is so. A "geometrical" necessary and sufficient condition is given for a family of operators to be uniformly equicontinuous.