Aspects of multivariable operator theory on weighted symmetric Fock spaces (Q2923448)

In the paper under review, the author systematically develops the function theory and operator theory on weighted symmetric Fock spaces. To have an idea about the work presented in this paper, let us reproduce some notations and notions. Let \(E_0:=\mathbb C\) and let \(E^n\) denote the symmetric tensor product of \(n\) copies of \(E=\mathbb{C}^d\) for \(n \geq 1 \). Recall that NEWLINE\[NEWLINE\{e_ {i_1}e_ {i_2}\cdots e_ {i_n} : 1 \leq i_1 \leq \cdots \leq i_n \leq d\}NEWLINE\]NEWLINE is an orthogonal basis for \(E^n\), where \(\xi \eta\) denotes the symmetric tensor product of \(\xi\) and \(\eta\). Set NEWLINE\[NEWLINE\mathcal{F}(E):=\bigoplus_{n=0}^{\infty}{E^n}.NEWLINE\]NEWLINE A \textit{weighted symmetric Fock space} \(\mathcal{F}_a(E)\) associated with a sequence of non-negative real numbers \(\{{a_n}\}_{n \geq 0}\) is defined by NEWLINE\[NEWLINE\{\oplus \xi_n \in \mathcal{F}(E) : \sum_{n=0}^{\infty}{a_n} \langle{\xi_n},{\xi_n}\rangle_{E^n} < \infty\};NEWLINE\]NEWLINE \(\mathcal{F}_a(E)\) is endowed with the inner product NEWLINE\[NEWLINE \langle{\oplus \xi_n},{\oplus \eta_n}\rangle : =\sum_{n=0}^{\infty} {a_n} \langle{\xi_n},{\eta_n}\rangle_{E^n}.NEWLINE\]NEWLINE The choice \(a_k=1\) for all positive integers \(k\) yields the symmetric (unweighted) Fock space that appeared in the work of \textit{W. B. Arveson} [Acta Math. 123, 141--224 (1969; Zbl 0194.15701)]. It turns out that \(\mathcal{F}_a(E)\) is a reproducing kernel Hilbert space with \(\mathcal U\)-invariant kernel. Moreover, under some mild regularity conditions on the sequence \(\{a_n\}_{n \geq 0}\), the elements of \(\mathcal{F}_a(E)\) can be realized as holomorphic functions on the unit ball in \(\mathbb C^d\) (Theorems 3.2 and 3.3). In Section 4, several examples of weighted Fock spaces are discussed which include, in particular, a one parameter family Dirichlet-type spaces \(\mathcal D_q\) on the unit ball. A complete analysis of the effect of the real parameter \(q\) in various topics from function theory is presented. For instance, Theorem 5.2 from Section 5 says that \(\mathcal F_b\) is an analytic Hilbert module if and only if \(b \notin l^1\). Specializing this result to Dirichlet-type spaces, we obtain that \(\mathcal D_q\) is an analytic Hilbert module if and only if \(q \geq -(1+d)\). Sections 6--11 are devoted to the operator theory on symmetric Fock spaces. In particular, the author obtains von Neumann inequalities for row contractions on a Hilbert space with respect to each \(\mathcal D_q\). The author also studies the Toeplitz \(C^*\)-algebra generated by the multiplication tuples \(M_z\) on \(\mathcal D_q\).NEWLINENEWLINE Some of the results in this paper are new even in the one-dimensional case. This interesting work not only unifies many existing results in the literature, but also provides a framework for posing and scrutinizing a myriad range of problems.