On \mu-scale invariant operators

TL;DR

This paper introduces f-scale invariant operators in Hilbert spaces and demonstrates that certain symmetric operators' extensions preserve this invariance, advancing the understanding of operator invariance properties.

Contribution

The paper defines f-scale invariance for operators and proves that Friedrichs and Krein-von Neumann extensions maintain this invariance.

Findings

Nonnegative symmetric operators can be f-scale invariant.

Friedrichs and Krein-von Neumann extensions preserve f-scale invariance.

Abstract

We introduce the concept of a \mu-scale invariant operator with respect to unitary transformation in a separable complex Hilbert space. We show that if a nonnegative densely defined symmetric operator is \mu-scale invariant for some \mu >0, then both the Friedrichs and the Krein-von Neumann extensions are also \mu-scale invariant.