Trace Estimate For The Determinant Operator And K Homogeneous Operators

Abstract

Let $\boldsymbol T=(T_1, \ldots , T_d)$ be a $d$- tuple of commuting operators on a Hilbert space $\mathcal H$. Assume that $\boldsymbol T$ is hyponormal, that is, $\big [\!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\! \big ]:=\big (\!\!\big ( \big [ T_j^*,T_i] \big )\!\!\big )$ acting on the $d$ - fold direct sum of the Hilbert space $\mathcal H$ is non-negative definite. The commutator $[T_j^*,T_i]$, $1\leq i,j \leq d$, of a finitely ctyclic and hyponormal $d$ - tuple is not necessarily compact and therefore the question of finding trace inequalities for such a $d$- tuple does not arise. A generalization of the Berger-Shaw theorem for a commuting tuple $\boldsymbol T$ of hyponormal operators was obtained by Douglas and Yan decades ago. We discuss several examples of this generalization in an attempt to understand if the crucial hypothesis in their theorem requiring the Krull dimension of the Hilbert module over the polynomial ring defined by the map $p\to p(\boldsymbol T)$, $p\in \mathbb C[\boldsymbol z]$, is optimal or not. Indeed, we find examples $\boldsymbol T$ to show that there is a large class of operators for which $\text{trace}\,[T_j^*,T_i]$, $1\leq j,i \leq d$, is finite but the $d$ - tuple is not finitely polynomially cyclic, which is one of the hypotheses of the Douglas-Yan theorem. We also introduce the weaker notion of ``projectively hyponormal operatorsquot and show that the Douglas-Yan thorem remains valid even under this weaker hypothesis. We introduce the determinant operator $\text{dEt}\,(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big) $, which coincides with the generalized commutator introduced by Helton and Howe earlier... newline