A Trace Inequality for Commuting d-Tuples of Operators

For a commuting d-tuple of operators T defined on a complex separable Hilbert space H, let [T*, T] be the d x d block operator (([T-j*, T-i])) n of the commutators [T-j*, T-i] := T-j*T-i - TiTj*. We define the determinant of [T*, T] by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of [T*, T] equals the generalized commutator of the 2d - tuple of operators, (T-1, T-1*, ..., T-d, 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 [T*, T] must be 0. We show that if the d-tuple T is cyclic, the determinant of [T*, T] is non-negative and the compression of a fixed set of words in T-j* and T-i -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 [T*, T] 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.