We characterize the nuclearity of Toeplitz operators $T_\mu : F_\alpha ^p \to F_\alpha ^q$ with Borel measure symbols for $1\leq p,q\leq \infty $. For positive measures $\mu $ and $q\leq p$, we provide necessary and sufficient conditions in terms of the Berezin transform and establish a rigidity property for nuclearity across this range. In the case $p<q$, we obtain separate necessary and sufficient conditions, indicating that the Berezin transform alone is insufficient for a complete characterization. Our results extend to Fock spaces on $\mathbb {C}^n$.

We provide a characterization of equivariant Fock covariant injective representations for product systems. We show that this characterization coincides with Nica covariance for compactly aligned product systems over right least common multiple semigroups of Kwaśniewski and Larsen and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao–Ng isomorphism problem for generalized gauge actions by discrete groups.

A categorical axiomatic theory of creation/annihilation operators on symmetric Fock space is introduced, and the combinatorial model that motivated it is presented. Commutation relations and coherent states are considered in both frameworks.

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

The subject of this article is operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from [Ans07, BL09, BL11, LS08] and extend a number of results from these articles to our more general setting. The results include the quadratic relation satisfied by the kernel of the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$. The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$, where $x \in \mathbb{R}^{n+1}$. The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

Inspired by Xiao’s work on Hankel measures for Hardy and Bergman spaces [‘Pseudo-Carleson measures for weighted Bergman spaces’. Michigan Math. J. 47 (2000), 447–452], we introduce Hankel measures for Fock space $F^p_\alpha $. Given $p\ge 1$, we obtain several equivalent descriptions for such measures on $F^p_\alpha $.

We investigate unbounded, linear operators arising from a finite sum of composition operators on Fock space. Real symmetry and complex symmetry of these operators are characterised.

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

Given a positive Borel measure ${\it\mu}$ on the $n$-dimensional Euclidean space $\mathbb{C}^{n}$, we characterise the boundedness (and compactness) of Toeplitz operators $T_{{\it\mu}}$ between Fock spaces $F^{\infty }({\it\varphi})$ and $F^{p}({\it\varphi})$ with $0<p\leq \infty$ in terms of $t$-Berezin transforms and averaging functions of ${\it\mu}$. Our result extends recent work of Mengestie [‘On Toeplitz operators between Fock spaces’, Integral Equations Operator Theory78 (2014), 213–224] and others.

The level l Fock space admits canonical bases and . They correspond to and -module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

For a general vector field we exhibit two Hilbert spaces, namely the space of so called closed functions and the space of exact functions and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg–Landau field and for the case of the fourth-order Ginzburg–Landau field.