A trace inequality for commuting tuple of operators

TL;DR

This paper introduces a new trace inequality for the determinant of a commutator block operator associated with commuting tuples of operators, linking it to generalized commutators and exploring conditions for finiteness and bounds of the trace.

Contribution

It defines a determinant for the commutator block operator, relates it to generalized commutators, and establishes conditions under which the trace of this determinant is finite and bounded.

Findings

Determinant of the commutator block operator equals the generalized commutator.

For commuting d-normal operators, the determinant must be zero.

Under certain conditions, the trace of the determinant is finite and bounded, with bounds shown to be sharp.

Abstract

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 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 $\big [\!\! \big [ \boldsymbol T^* , \boldsymbol T\big ] \!\!\big ]$ 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.