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 $d\times d$ block operator $\big (\!\!\big (\big [T_j^* , T_i\big ]\big )\!\!\big )$ of commutators: $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define an operator on the Hilbert space $\mathcal H$, to be designated the determinant operator, corresponding to the block operator $\big [\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\big ]$. 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 $T_i$ -- 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 $\big (\!\!\big (\big [ T_j^* , T_i\big ]\big )\!\!\big )$ 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.