ON A THEOREM OF WIELANDT AND THE COMPOUNDS OF UNITARY MATRICES

Let A and B be normal endomorphisms with prescribed eigenvalues define d on a finite dimensional unitary space. A celebrated theorem of Wiela ndt states that the eigenvalues of A - B are forced to lie in a certai n subset of the complex plane. The primary objective of this article i s to extend Wielandt's result to give information about the joint dist ribution of the eigenvalues of A - B. The main tool in establishing th is extension is a result on the compounds of unitary matrices. If U is an n X n unitary matrix, then Birkhoff's famous result on doubly stoc hastic matrices is often applied to write the matrix (\u(jk)\2)jk as a convex combination of permutation matrices. The natural generalizatio n of this process to the compound of a unitary matrix is known to fail . Here we show that it succeeds if one considers only a certain restri cted subset of entries.