Orthonormal pairs of operators

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.