The sandwich rule for sequences of self-adjoint operators and some applications

TL;DR

This paper establishes a sandwich rule for convergent sequences of self-adjoint operators on Hilbert spaces, extending its validity across all main topologies on bounded operators, with applications to operator theory.

Contribution

It introduces a generalized squeeze theorem for self-adjoint operator sequences applicable to all key topologies on bounded operators.

Findings

The sandwich rule holds for all three main topologies on $B(H)$.

The theorem applies to sequences of self-adjoint operators.

Extensions to various topologies enhance its applicability.

Abstract

In this paper, we mainly deal with sequences of bounded linear operators on Hilbert space. The main result is the so-called squeeze theorem (or sandwich rule) for convergent sequences of self-adjoint operators. We show that this theorem remains valid for all three main topologies on $B(H)$.