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 operators" 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. We identify a class $BS_{m, \vartheta}(\Omega)$ consisting of commuting $d$- tuples of hyponormal operators $\boldsymbol T$, $\sigma(\boldsymbol T) = \overbar{\Omega}$, satisfying a growth condition for which the dEt is a non-negative definite operator. We then obtain the trace estimate given in the Theorem below. \begin{thmAbs} Let $\boldsymbol{T}=(T_1,\ldots, T_d)$ be a commuting tuple of operators on a Hilbert space $\mathcal{H}$ such that $\boldsymbol{T}$ is in the class $BS_{m, \vartheta}(\Omega)$. Then the determinant operator $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big)$ is in trace-class and \[\text{trace}\,\big (\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big]\big)\big )\leq m\, \vartheta \,d!\prod_{i=1}^{d}\|T_i\|^2.\] \end{thmAbs} In the case of a commuting $d$ - tuple $\boldsymbol T$ of operators, where $\sigma(\boldsymbol T)$ is of the form $\overbar{\Omega}_1 \times \cdots \times \overbar{\Omega}_d$, we obtain a slightly different but a related estimate for the trace of $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big]\big)\big )$. Explicit computation of $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big ]\big)$ in several examples and based on some numerical evidence, we make the following conjecture refining the estimate from the Theorem: \begin{conjAbs} Let $\boldsymbol{T}=(T_1,\ldots, T_d)$ be a commuting tuple of operators on a Hilbert space $\mathcal{H}$ such that $\boldsymbol{T}$ is in the class $BS_{m, \vartheta}(\Omega)$. Then the determinant operator $\text{dEt}\,\big(\big[\!\!\big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big)$ is in trace-class, and \[\text{\rm trace}\,\big (\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big ]\big) \big )\leq \frac{m d!}{\pi^d} \nu(\overline{\Omega}), \] where $\nu$ is the Lebesgue measure. \end{conjAbs}

Let $\Omega$ be an irreducible classical bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T$ of commuting bounded linear operators by the rule: \[k\cdot\boldsymbol{T}:=\big(k_1(T_1, \ldots, T_d), \ldots, k_d(T_1, \ldots, T_d)\big),\,\,k\in \mathbb K, \] where $k_1(\boldsymbol z), \ldots, k_d(\boldsymbol z)$ are linear polynomials. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. We realize a certain class of $\mathbb{K}$-homogeneous $d$-tuples $\boldsymbol{T}$ as a $d$ -tuple of multiplication by the coordinate functions $z_1,\ldots ,z_d$ on a reproducing kernel Hilbert space $\mathcal H_K$. (The Hilbert space $\mathcal H_K$ consisting of holomorphic functions defined on $\Omega$ and $K$ is the reproducing kernel.) Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\Omega)$. For an irreducible bounded symmetric domain $\Omega$ of rank $2$, an explicit description of the operator $\sum_{i=1}^d T_i^*T_i$ is given. Based on this formula, a conjecture giving the form of this operator in any rank $r \geq 1$ was made. This conjecture was recently verified by H. Upmeier.