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Publications and source records attributed to Jaydeb Sarkar.
The purpose of this paper is to study a class of quotient modules of the Hardy module $H^2(D^n)$. Along with the two variables quotient modules introduced by W. Rudin, we introduce and study a large class of quotient modules, namely Rudin's quotient modules of $H^2(D^n)$. By exploiting the structure of minimal representations we obtain an explicit co-rank formula for Rudin's quotient modules.
A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and geometric aspects of Hilbert module approach to operator theory and to survey several applications of the theory of Hilbert modules in multivariable operator theory. The topics which are studied include: generalized canonical models and Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert spaces, a class of simple submodules and quotient modules of the Hardy modules over polydisc, commutant lifting theorem, similarity and free Hilbert modules, left invertible multipliers, inner resolutions, essentially normal Hilbert modules, localizations of free resolutions and rigidity phenomenon. This article is a companion paper to "An Introduction to Hilbert Module Approach to Multivariable Operator Theory".
A commuting pair of operators (S, P) on a Hilbert space H is said to be a Gamma-contraction if the symmetrized bidisc is a spectral set of the tuple (S, P). In this paper we develop some operator theory inspired by Agler and Young's results on a model theory for Gamma-contractions. We prove a Beurling-Lax-Halmos type theorem for Gamma-isometries. Along the way we solve a problem in the classical one-variable operator theory. We use a "pull back" technique to prove that a completely non-unitary Gamma-contraction (S, P) can be dilated to a direct sum of a Gamma-isometry and a Gamma-unitary on the Sz.-Nagy and Foias functional model of P, and that (S, P) can be realized as a compression of the above pair in the functional model of P. Moreover, we show that the representation is unique. We prove that a commuting tuple (S, P) with |S| \leq 2 and |P \leq 1 is a Gamma-contraction if and only if there exists a compressed scalar operator X with the decompressed numerical radius not greater than one such that S = X + P X^*. In the commutant lifting set up, we obtain a unique and explicit solution to the lifting of S where (S, P) is a completely non-unitary Gamma-contraction. Our results concerning the Beurling-Lax-Halmos theorem of Gamma-isometries and the functional model of Gamma-contractions answers a pair of questions of J. Agler and N. J. Young.
Let $b(z) = \prod_{n=1}^\infty \frac{-\barα_n}{|α_n|} \frac{z - α_n}{1 - \barα_n z}$, where $\sum_{n=1}^\infty (1 - |α_n|) <\infty$, be the Blaschke product with zeros at $α_n \in \mathbb{D} \setminus \{0\}$. Then $\cls = \vee_{n=1}^\infty \big(z^n H^2(\mathbb{D})\big) \otimes \big(\prod_{k=n}^\infty \frac{-\barα_n}{|α_n|} \frac{z - α_n}{1 - \barα_n z} H^2(\mathbb{D})\big)$ is a joint $(M_{z_1}, M_{z_2})$ invariant subspace of the Hardy space $H^2(\mathbb{D}^2) \cong H^2(\mathbb{D}) \otimes H^2(\mathbb{D})$. This class of subspaces was originally introduced by Rudin in the context of infinite cardinality of generating sets of shift invariant subspaces of $H^2(\mathbb{D}^2)$. \noindent In this paper we prove that for a Rudin invariant subspace $\cls$ of $H^2(\mathbb{D}^2)$, the cross commutator $[(P_{\cls} M_{z_1}|_{\cls})^*, M_{z_2}|_{\cls}] = (P_{\cls} M_{z_1} |_{\cls})^* (M_{z_2}|_{\cls}) - (M_{z_2}|_{\cls}) (P_{\cls} M_{z_1}|_{\cls})^*$ is not compact. Consequently, Rudin's invariant subspaces are both infinitely generated and not essentially doubly commuting.
In this paper, we obtain a complete description of the class of n-tuples (n >= 2) of doubly commuting isometries. In particular, we present a several variables analogue of the Wold decomposition for isometries on Hilbert spaces. Our main result is a generalization of M. Slocinski's Wold-type decomposition of a pair of doubly commuting isometries.
We say that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ ($n >1$) is co-doubly commuting if the quotient module $H^2(\mathbb{D}^n)/\cls$ is doubly commuting. We show that a co-doubly commuting submodule of $H^2(\mathbb{D}^n)$ is essentially doubly commuting if and only if the corresponding one variable inner functions are finite Blaschke products or that $n = 2$. In particular, a co-doubly commuting submodule $\cls$ of $H^2(\mathbb{D}^n)$ is essentially doubly commuting if and only if $n = 2$ or that $\cls$ is of finite co-dimension. We obtain an explicit representation of the Beurling-Lax-Halmos inner functions for those submodules of $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ which are co-doubly commuting submodules of $H^2(\mathbb{D}^n)$. Finally, we prove that a pair of co-doubly commuting submodules of $H^2(\mathbb{D}^n)$ are unitarily equivalent if and only if they are equal.
In this paper, we present a unified approach to problems of tensor product of quotient modules of Hilbert modules over $\mathbb{C}[z]$ and corresponding submodules of reproducing kernel Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ and the doubly commutativity property of module multiplication operators by the coordinate functions. More precisely, for a reproducing kernel Hilbert module $\clh$ over $\mathbb{C}[z_1, \ldots, z_n]$ of analytic functions on the polydisc in $\mathbb{C}^n$ which satisfies certain conditions, we characterize the quotient modules $\q$ of $\clh$ such that $\q$ is of the form $\q_1 \otimes \cdots \otimes \q_n$, for some one variable quotient modules $\{\q_1, \ldots, \q_n\}$. For $\clh$ the Hardy module over polydisc $H^2(\mathbb{D}^n)$, this reduces to some recent results by Izuchi, Nakazi and Seto and the third author. This is used to obtain a classification of co-doubly commuting submodules for a class of reproducing kernel Hilbert modules over the unit polydisc. These results are applied to compute the cross commutators of co-doubly commuting submodules. This is used to give further insight into the wandering subspaces and ranks of submodules of the Hardy module case. Our results includes the case of weighted Bergman modules over the unit polydisc in $\mathbb{C}^n$.
Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Π: H^2_D(\mathbb{D}) \raro H such that ΠM_z = T Πand that S = ran Π, or equivalently, P_S = ΠΠ^*. As an application we completely classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.
In [11] the authors investigated a family of quotient Hilbert modules in the Cowen-Douglas class over the unit disk constructed from classical Hilbert modules such as the Hardy and Bergman modules. In this paper we extend the results to the multivariable case of higher multiplicity. Moreover, similarity as well as isomorphism results are obtained.
In this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in $H^\infty(\mathbb{D}^n)$.
Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.
Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $α=\{α_1,\dots,α_k\}$ of $\{1,\dots,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_α|_\s:=(M_{z_{α_1}}|_\s,\dots, M_{z_{α_k}}|_\s)$ always possesses wandering subspace of the form \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]
We develop a several variables analog of the Jordan blocks of the Hardy space $H^2(\mathbb{D})$. In this consideration, we obtain a complete characterization of the doubly commuting quotient modules of the Hardy module $H^2(\mathbb{D}^n)$. We prove that a quotient module $\clq$ of $H^2(\mathbb{D}^n)$ ($n \geq 2$) is doubly commuting if and only if \[\clq = \clq_{Θ_1} \otimes \cdots \otimes \clq_{Θ_n},\]where each $\clq_{Θ_i}$ is either a one variable Jordan block $H^2(\mathbb{D})/Θ_i H^2(\mathbb{D})$ for some inner function $Θ_i$ or the Hardy module $H^2(\mathbb{D})$ on the unit disk for all $i = 1, \ldots, n$. We say that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ is a co-doubly commuting if the quotient module $H^2(\mathbb{D}^n)/\cls$ is doubly commuting. We obtain a Beurling like theorem for the class of co-doubly commuting submodules of $H^2(\mathbb{D}^n)$. We prove that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ is co-doubly commuting if and only if \[\cls = \mathop{\sum}_{i=1}^m Θ_i H^2(\mathbb{D}^n),\]for some integer $m \leq n$ and one variable inner functions $\{Θ_i\}_{i=1}^m$.
One can view contraction operators given by a canonical model of Sz.-Nagy and Foias as being defined by a quotient module where the basic building blocks are Hardy spaces. In this note we generalize this framework to allow the Bergman and weighted Bergman spaces as building blocks, but restricting attention to the case in which the operator obtained is in the Cowen-Douglas class and requiring the multiplicity to be one. We view the classification of such operators in the context of complex geometry and obtain a complete classification up to unitary equivalence of them in terms of their associated vector bundles and their curvatures.
Let H^2_m be the Drury-Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function (z, w) \in B^m \times B^m \raro (1 - )^{-1}. We investigate for which multipliers θ: \mathbb{B}^m \raro \cll(\cle, \cle_*) the quotient module \clh_θ is similar to H^2_m \otimes \clf for some Hilbert space \clf, where M_θ is the corresponding multiplication operator in \cll(H^2_m \otimes \cle, H^2_m \otimes \cle_*) for Hilbert spaces \cle and \cle_* and \clh_θ is the quotient module (H^2_m \otimes \cle_*)/ {clos} [M_θ(H^2_m \otimes \cle)]. We show that a necessary condition is the existence of a multiplier $ψ$ in \clm(\cle_*, \cle) such that θψθ= θ. Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of H^2_m \otimes \cle for a Hilbert space \cle, which is valid for the case of m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator tuple or algebra is not necessarily commuting. In this case the converse to the similarity result is always valid.
In this note we study the completely non unitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form T = S & * & * 0 & N & * 0& 0& C, where $S$ and C^* are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that dimension of ker S^* and dimension of ker C are unitary invariants of $T$ and that N, up to a quasi-similarity is uniquely determined by T. Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.
In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z, w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S^{-1}(z, w) k(z, w) is a positive kernel function, where S(z, w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale \cite{Ath} and Ambrozie, Englis and Müller. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in {C}^m. Some consequences of this more general result are then explored in the case of several natural function algebras.
In a recent paper, Trent and Wick establish a strong relation between the corona problem and the Toeplitz corona problem for a family of spaces over the ball and the polydisk. Their work is based on earlier work of Amar. In this note, several of their lemmas are reinterpreted in the language of Hilbert modules, revealing some interesting facts and raising some questions about quasi-free Hilbert modules. Moreover, a modest generalization of their result is obtained.