ABSTRACT

The space Mm,n(C) of all m × n complex matrices can be made into a Banach space in a number of ways and so provides a fertile ground for the study of isometries. The earliest norm to be considered on this matrix space was the familiar operator norm given by

‖A‖ = sup ‖Ax‖2‖x‖2 ,

where x ∈ Cn and ‖ · ‖2 denotes the Euclidean norm. As early as 1925, the isometries on Mm,n were described by Schur, and in the next section we will give Morita’s proof of that result. It may be that this theorem of Schur is the very first one that characterizes the isometries of a specific Banach space.