An extension of the Toeplitz-Hausdorff theorem

Abstract

The Toeplitz-Hausdorff Theorem asserts that for any operator A acting on a complex Hilbert space H, the set of numbers of the form <Az, z>, where z varies over the unit sphere of H, is always a convex subset of C. In this paper we obtain the same result for non-homogeneous quadratic functions of the form <Az, z> + <a, z> + <z, beta> + c. This implies, in particular, that the set of numbers of the form <Az, z>, where z varies over any sphere in H, centered or not at the origin, is always convex. We also show by an example that the corresponding result is not true for pairs of operators on a real Hilbert space.