We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results Koldobsky and Wójcik. Moreover, in an arbitrary dimension, we construct examples non-isometric pairs non-smooth admit norm preserving homogeneous bicontinuous prese...

Dedicated to Professor Hans Schneider. Abstract The interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated. It is shown that one can characterize Hermitian matrices, positive semi-definite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Also, characterizations are given to ma...