Orthonormal pairs of operators

Abstract

We consider pairs of operators A,B∈B(H), where H is a Hilbert space, such that there exist a linear isometry f from the span of {A,B} into C2 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 row) operator space structure. We also metrically characterize elements in a C⁎-algebra with orthogonal ranges.