Analyzing specific cases of an operator inequality

Abstract

Let H and K be Hilbert spaces, and suppose A ε B ( H ) and B ε B ( K ) are self-adjoint operators with dist(σ( A), σ( B))≥ δ > 0. In 1983 Bhatia, Davis, and McIntosh showed that for any Q ε B ( K , H ) we must have ( π ;2 )‖AQ−QB‖≥δ‖Q‖ . In this paper we specialize their inequality to the case where A, Q, and B are 2 × 2 or 3 × 3 matrices, and give sharp estimates. Doing so, we illustrate one way that bounds on the norm of the Schur product of two matrices have applications to perturbation theory. By specializing the Fourier transform used in the proof of the theorem above, we also obtain sharp estimates in two Fourier interpolation problems.