A trace inequality for commuting d-tuples of operators

Abstract

For a commuting d-tuple of operators TT defined on a complex separable Hilbert space H, let [[TT∗,TT]] be the d×d block operator (([T∗j,Ti])) of the commutators [T∗j,Ti]:=T∗jTi-TiT∗j. We define the determinant of [[TT∗,TT]] by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of [[TT∗,TT]] equals the generalized commutator of the 2d - tuple of operators, (T1,T∗1,…,Td,T∗d) introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting d-tuple of d-normal operators, the determinant of [[TT∗,TT]] must be 0. We show that if the d-tuple TT is cyclic, the determinant of [[TT∗,TT]] is non-negative and the compression of a fixed set of words in T∗j and Ti-to a nested sequence of finite dimensional subspaces increasing to H-does not grow very rapidly, then the trace of the determinant of the operator [[TT∗,TT]] is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting d-tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.