Nonnegative unitary operators

Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.