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Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having $$\overline{\boldsymbol\Theta}_n:=\{\boldsymbol\theta(z)=(\theta_1(z),\ldots,\theta_n(z)):z\in\overline{\mathbb D}^n\}$$ as a spectral set, where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $\boldsymbol\Theta_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $\boldsymbol\Theta_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $\boldsymbol\Theta_n$-isometries is obtained., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1301.2837

Let $\Omega$ be a bounded symmetric domain in $\mathbb C^n$ and $f :\Omega \to \Omega^\prime$ be a proper holomorphic mapping factored by (automorphisms) a finite complex reflection group $G.$ We define an appropriate notion of the Hardy space $H^2(\Omega^\prime)$ on $\Omega^\prime$ which can be realized as a closed subspace of an $L^2$-space on the \v{S}ilov boundary of $\Omega^\prime$. We study various algebraic properties of Toeplitz operators (such as the finite zero product property, commutative and semi-commutative property etc.) on $H^2(\Omega^\prime)$. We prove a Brown-Halmos type characterization for Toeplitz operators on $H^2(\Omega^\prime),$ where $\Omega^\prime$ is an image of the open unit polydisc in $\mathbb C^n$ under a proper holomorphic mapping factored by an irreducible finite complex reflection group., Comment: 29 pages

High resolution simulations of incompressible flows have become routine across a range of engineering applications. Despite their routine use, due to the high dimensional parameter space present for most practical applications, a comprehensive exploration of the available parameter space is often impractical. In this work, we demonstrate the ability of physics-constrained deep learning methods to provide an efficient means of exploring high-dimensional parameter spaces with minimal amounts of data from high resolution computational fluid dynamic simulations. As a specific application, we choose the well established problem of a 2D lid driven cavity flow. While giving an extensive treatment of the classic case of a square cavity, we extend the analysis to treat an isosceles trapezoid. In so doing, the number of parameters determining the solution includes not just the Reynolds number, but also two additional parameters characterizing the geometry of the cavity. Thus, together with the $\left( x, y \right)$ variation of the flow and pressure in configuration space, the presence of these three parameters results in the solution varying in a 5D space. It is shown that in the absence of data, physics-constrained methods are able to provide an accurate description of the cavity flow in this 5D space up to intermediate values of the Reynolds number, but fails to train for sufficiently high Reynolds numbers. In contrast, using a small quantity of flow data, a single neural network is able to provide an accurate description for a broad range of Reynolds numbers and cavity geometries. Once trained, such a model provides a rapid surrogate that can be used to efficiently explore the 5D space. This 5D surrogate model is subsequently used to identify critical parameter values for the merger and splitting of vortices as the Reynolds number and cavity geometry are varied.

Let $ \Omega \subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on $\Omega$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subset\Omega$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(\Omega)$ of $\Omega$ depending upon a subgroup $G$ of Aut$(\Omega)$ whenever the tuple of multiplication operators on $\mathcal{M}$ is homogeneous with respect to $G$ and both $\mathcal{M}$ as well as $\mathcal{M}_q$ are in the Cowen-Douglas class. We show that these compression of multiplication operators might be reducible even if the tuple of multiplication operators on $\mathcal{M}$ is irreducible by exhibiting a concrete example. Moreover, the irreducible components of these reducible operators are identified as Generalized Wilkins' operators., Comment: Primary version

Let $G$ be a finite pseudoreflection group, $\Omega\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(\Omega)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on $\mathcal H,$ where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbol\theta = (\theta_1, \ldots, \theta_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(\Omega)$ of holomorphic functions on $\Omega$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbol\theta(\Omega)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $\Omega$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(\Omega).$, Comment: Extensively revised with new results and applications that includes version 1 as one of the sections

This note introduces a special class of tuples of bounded operators on a Hilbert space. It is called the Agler Young class. Major results about this class include a Wold decomposition and a dilation theorem. The structure of the dilation is completely spelt out. A characterization of this class using the hereditary functional calculus of Agler is obtained and examples are discussed. Toeplitz operators play a major role in this note. An Agler-Young pair arising from a truncated Toeplitz operator is characterized. Thus, we extend results obtained in the case of commuting operators by several authors over many decades to the non-commutative situation. The results for the commuting case can be recovered as special cases., Comment: A co-author added

The weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbb D^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1) \ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent., Comment: 22 pages