A trace inequality for commuting 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,Ti])) of commutators: [T∗j,Ti]:=T∗jTi−TiT∗j. We define an operator on the Hilbert space H, to be designated the determinant operator, corresponding to the block operator [[T∗,T]]. We show that if the d- tuple is cyclic, the determinant operator is positive 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 (([T∗j,Ti])) is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for certain small 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.