Polar decompositions of C-0(N) contractions

Abstract

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C-0(N) contraction if and only if A = U(I - Sigma(d)(j=1) r(j)(x(j) circle times x(j))), where U is a singular unitary operator with multiplicity d <= N, 0 < r(1),...,r(d) < 1 and x(1),...,x(d) are orthonormal vectors satisfying V{U-k x(j) : k >= 0, 1 <= j <= d} = H. For a C-0(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.