A trace inequality for commuting tuple of operators

For a commuting $d$- tuple of operators $boldsymbol T$ defined on a complex separable Hilbert space $mathcal H$, let $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ be the $dtimes d$ block operator $big (!!big (big [ T_j^* , T_ibig ]big )!!big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, ldots, T_d,T_d^*)$ 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 $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ must be $0$. We show that if the $d$- tuple $boldsymbol T$ is cyclic, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ 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 $mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $big [!! big [ boldsymbol T^* , boldsymbol Tbig ] !!big ]$ 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.