Orthonormal pairs of operators

TL;DR

This paper characterizes pairs of operators on Hilbert spaces that can be mapped to orthonormal pairs via isometries, providing necessary conditions, classifications for commuting normal operators, and metric characterizations within operator space structures.

Contribution

It introduces new criteria and classifications for operator pairs that can be orthonormalized through isometries, especially in the context of operator space structures.

Findings

Necessary conditions for orthonormal pairs of operators.

Complete classification of such pairs among commuting normal operators.

Metric characterization within C*-algebras.

Abstract

We consider pairs of operators $A,B\in B(H)$, where $H$ is a Hilbert space, such that there exist a linear isometry $f$ from the span of $\{A,B\}$ into $\mathbb{C}^2$ mapping $A,B$ into orthonormal vectors. We prove some necessary conditions for the existence of such an $f$ and determine all such pairs among commuting normal operators. Then we characterize all such pairs $A,B$ (in fact, we consider general sets instead of just pairs) under the additional requirement that $f$ is a complete isometry, when $H$ carries the column (or the row) operator space structure. We also metrically characterize elements in a C$^*$-algebra with orthogonal ranges.