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× d block operator ( ( [ T_j^* , T_i ] ) ) of the commutators [T^*_j , T_i] := T^*_j T_i - T_iT_j^* . 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 2 d - 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.