arXiv subjects

Jaydeb Sarkar

Publications and source records attributed to Jaydeb Sarkar.

At least 73 records · Page 4Linked to original sources

arXiv2018

Covariant representations of subproduct systems: Invariant subspaces and curvature

Let $X=(X(n))_{n \in \mathbb{Z_+}}$ be a standard subproduct system of $C^*$-correspondences over a $C^*$-algebra $\mathcal M.$ Assume $T=(T_n)_{n \in \mathbb{Z_+}}$ to be a pure completely contractive, covariant representation of $X$ on a Hilbert space $\mathcal H,$ and $\mathcal S$ to be a non-trivial closed subspace of $\mathcal H.$ Then $\mathcal{S}$ is invariant for $T$ if and only if there exist a Hilbert space $\mathcal{D},$ a representation $π$ of $\mathcal M$ on $\mathcal D,$ and a partial isometry $Π: \mathcal{F}_X\bigotimes_π\mathcal{D}\to \mathcal{H} $ such that $$Π(S_n(ζ)\otimes I_{\mathcal{D}})=T_n(ζ)Π~\mbox{whenever}~ζ\in X(n), ~n\in \mathbb{Z_+},~\mbox{and}$$ $\mathcal S$ is the range of $Π,$ or equivalently, $P_{\mathcal S}=ΠΠ^*.$ This result leads us to many important consequences including Beurling type theorem and other general observations on wandering subspaces. We extend the notion of curvature for completely contractive, covariant representations and analyze it in terms of the above results.

math.OA↗

arXiv2017

Characterization of Invariant subspaces in the polydisc

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.

math.FA↗

arXiv2017

Factorizations of Contractions

The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exists a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_Φ, M_Ψ)$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where \[ Φ(z)=(P+zP^{\perp})U^*\;\text{and}\; Ψ(z)=U(P^{\perp}+zP) \;;(z \in \mathbb{D}). \] Here we prove a similar factorization result for pure contractions. More particularly, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}} M_z|_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$. Then $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exists $\mathcal{B}(\mathcal{D})$-valued polynomials $φ$ and $ψ$ of degree $ \leq 1$ such that $\mathcal{Q}$ is a joint $(M_φ^*, M_ψ^*)$-invariant subspace, \[P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{φψ}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{ψφ}|_{\mathcal{Q}} \; \mbox{and} \;(T_1, T_2) \cong (P_{\mathcal{Q}} M_φ|_{\mathcal{Q}}, P_{\mathcal{Q}} M_ψ|_{\mathcal{Q}}).\]

math.FA↗

arXiv2017

Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$

We initiate a study of asymptotic Toeplitz operators on the Hardy space $H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$). We also study the Toeplitz operators in the polydisc setting. Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let $T_{z_i}$ denote the multiplication operator on $H^2(\mathbb{D}^n)$ by the $i^{th}$ coordinate function $z_i$, $i =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then the following hold: (i) $T$ is a Toeplitz operator (that is, $T = P_{H^2(\mathbb{D}^n)} M_φ|_{H^2(\mathbb{D}^n)}$, where $M_φ$ is the Laurent operator on $L^{2}(\mathbb{T}^n)$ for some $φ\in L^\infty(\mathbb{T}^n)$) if and only if $T_{z_i}^* T T_{z_i} = T$ for all $i = 1, \ldots, n$. (ii) $T$ is an asymptotic Toeplitz operator if and only if $T = \mbox{~Toeplitz} + \mbox{~compact}$. The case $n = 1$ is the well known results of Brown and Halmos, and Feintuch, respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.

math.FA↗

arXiv2017

Similarity of Quotient Hilbert modules in the Cowen-Douglas Class

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and sufficient conditions for a class of Cowen-Douglas Hilbert modules satisfying some positivity conditions to be similar to $\mathcal{M} \otimes \mathbb{C}^m$. We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen-Douglas Hilbert module is quasi-affinity to a submodule of the free module $\mathcal{M} \otimes \mathbb{C}^m$.

math.FA↗

arXiv2017

Rank of a co-doubly commuting submodule is 2

We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $φ, ψ\in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_φ = H^2(\mathbb{D})/ φH^2(\mathbb{D})$ and $\mathcal{Q}_ψ = H^2(\mathbb{D})/ ψH^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_φ \otimes \mathcal{Q}_ψ)^\perp = 2. \] An immediate consequence is the following: Let $\mathcal{S}$ be a co-doubly commuting submodule of $H^2(\mathbb{D}^2)$. Then $\mbox{rank~} \mathcal{S} = 1$ if and only if $\mathcal{S} = ΦH^2(\mathbb{D}^2)$ for some one variable inner function $Φ\in H^\infty(\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R. Yang.

math.FA↗

arXiv2017

Contractions with Polynomial Characteristic Functions II. Analytic Approach

The simplest and most natural examples of completely nonunitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions are the nilpotent operators. The main purpose of this paper is to prove the following theorem: Let $T$ be a completely nonunitary contraction on a Hilbert space $\mathcal{H}$. If the characteristic function $Θ_T$ of $T$ is a polynomial of degree $m$, then there exist a Hilbert space $\mathcal{M}$, a nilpotent operator $N$ of order $m$, a coisometry $V_1 \in \mathcal{L}(\overline{ran} (I - N N^*) \oplus \mathcal{M}, \overline{ran} (I - T T^*))$, and an isometry $V_2 \in \mathcal{L}(\overline{ran} (I - T^* T), \overline{ran} (I - N^* N) \oplus \mathcal{M})$, such that \[ Θ_T = V_1 \begin{bmatrix} Θ_N & 0 0 & I_{\mathcal{M}} \end{bmatrix} V_2. \]

math.FA↗

arXiv2016

Dilations, Wandering Subspaces, and Inner Functions

The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb C^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1}, \ldots ,M_{z_n})$ can be described in terms of suitable $\mathcal{H}_K$-inner functions. We prove that $\mathcal{H}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way we prove a refinement of a result of Arveson on the uniqueness of minimal dilations of pure row contractions.

math.FA↗

arXiv2016

Analytic Model of Doubly Commuting Contractions

An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple $T$ can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H^2_{\mathcal{D}_{T^*}}(\mathbb{D}^n). As a consequence of the dilation theorem, we prove that there exists a closed subspace \mathcal{S}_T of the form \[\mathcal{H}_{T} := \sum_{i=1}^n Φ_{T_i} H^2_{\mathcal{E}_{T_i}}(\mathbb{D}^n),\] where \{\mathcal{E}_{T_i}\}_{i=1}^n are Hilbert spaces, Φ_{T_i} \in H^\infty_{\mathcal{B}(\mathcal{E}_{T_i}, \mathcal{D}_{T^*})}(\mathbb{D}^n) such that each Φ_{T_i} (1 \leq i \leq n) is either a one variable inner function in z_i, or the zero function. Moreover, \mathcal{H} \cong \mathcal{S}_T^\perp and \[(T_1, \ldots, T_n) \cong P_{\mathcal{S}_T^\perp} (M_{z_1}, \ldots, M_{z_n})|_{\mathcal{S}_T^\perp}.\]

math.FA↗

arXiv2016

Factorizations of Kernels and Reproducing Kernel Hilbert Spaces

The paper discusses a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, it is proved that for a large class of spaces isometric multipliers are trivial. One also gives for certain spaces conditions for obtaining a particular type of dilation, as well as a classification of Brehmer type submodules.

math.FA↗

arXiv2016

Factorizations of Characteristic Functions

Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and $X$ be a row operator from $\oplus_{i=1}^n \mathcal{H}_2$ to $\mathcal{H}_1$. Let $D_{A^*} = (I - A A^*)^{\frac{1}{2}}$ and $D_{B} = (I - B^* B)^{\frac{1}{2}}$ and $Θ_T$ be the characteristic function of $T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B \end{bmatrix}$. Then $Θ_T$ coincides with the product of the characteristic function $Θ_A$ of $A$, the Julia-Halmos matrix corresponding to $L$ and the characteristic function $Θ_B$ of $B$. More precisely, $Θ_T$ coincides with \[ \begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix}, \] where $Γ$ is the full Fock space. Similar results hold for constrained row contractions.

math.FA↗

arXiv2016

Operator Positivity and Analytic Models of Commuting Tuples of Operators

We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an $m$-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space $\mathcal{H}$, there exists a Hilbert space $\mathcal{E}$ and a partially isometric multiplier $θ\in \mathcal{M}(H^2(\mathcal{E}), A^2_m(\mathcal{H}))$ such that \[\mathcal{H} \cong \mathcal{Q}_θ = A^2_m(\mathcal{H}) \ominus θH^2(\mathcal{E}), \quad \quad \mbox{and} \quad \quad T \cong P_{\mathcal{Q}_θ} M_z|_{\mathcal{Q}_θ},\]where $A^2_m$ is the weighted Bergman space and $H^2$ is the Hardy space over the unit disc $\mathbb{D}$. We then proceed to study and develop analytic models for doubly commuting $n$-tuples of operators and investigate their applications to joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over polydisc. In particular, we completely analyze doubly commuting quotient modules of a large class of reproducing kernel Hilbert modules, in the sense of Arazy and Englis, over the unit polydisc $\mathbb{D}^n$.

math.FA↗

arXiv2015

Ando dilations, von Neumann inequality, and distinguished varieties

Let $\mathbb{D}$ denote the unit disc in the complex plane $\mathbb{C}$ and let $\mathbb{D}^2 = \mathbb{D} \times \mathbb{D}$ be the unit bidisc in $\mathbb{C}^2$. Let $(T_1, T_2)$ be a pair of commuting contractions on a Hilbert space $\mathcal{H}$. Let $\mbox{dim } \mbox{ran}(I_{\mathcal{H}} - T_j T_j^*) < \infty$, $j = 1, 2$, and let $T_1$ be a pure contraction. Then there exists a variety $V \subseteq \overline{\mathbb{D}}^2$ such that for any polynomial $p \in \mathbb{C}[z_1, z_2]$, the inequality \[ \|p(T_1,T_2)\|_{\mathcal{B}(\mathcal{H})} \leq \|p\|_V \] holds. If, in addition, $T_2$ is pure, then \[V = \{(z_1, z_2) \in \mathbb{D}^2: \det (Ψ(z_1) - z_2 I_{\mathbb{C}^n}) = 0\}\]is a distinguished variety, where $Ψ$ is a matrix-valued analytic function on $\mathbb{D}$ that is unitary on $\partial \mathbb{D}$. Our results comprise a new proof, as well as a generalization, of Agler and McCarthy's sharper von Neumann inequality for pairs of commuting and strictly contractive matrices.

math.FA↗

arXiv2015

Characterizations of Symmetrized Polydisc

Let $Γ_n$, $n \geq 2$, denote the symmetrized polydisc in $\mathbb{C}^n$, and $Γ_1$ be the closed unit disc in $\mathbb{C}$. We provide some characterizations of elements in $Γ_n$. In particular, an element $(s_1, \ldots, s_{n-1}, p) \in \mathbb{C}^n$ is in $Γ_n$ if and only if $s_j = β_j + \overline{β_{n-j}} p$, $j = 1, \ldots, n-1$, for some $(β_1, \ldots, β_{n-1}) \in Γ_{n-1}$, and $|p| \leq 1$.

math.CV↗

arXiv2015

Inner multipliers and Rudin type invariant subspaces

Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ be the $\mathcal{E}$-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well known Beurling-Lax-Halmos theorem states that every shift invariant subspace of $H^2_{\mathcal{E}}(\mathbb{D})$ other than $\{0\}$ has the form $ΘH^2_{\mathcal{E}_*}(\mathbb{D})$, where $Θ$ is an operator-valued inner multiplier in $H^\infty_{B(\mathcal{E}_*, \mathcal{E})}(\mathbb{D})$ for some Hilbert space $\mathcal{E}_*$. In this paper we identify $H^2(\mathbb{D}^n)$ with $H^2(\mathbb{D}^{n-1})$-valued Hardy space $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ and classify all such inner multiplier $Θ\in H^\infty_{\mathcal{B}(H^2(\mathbb{D}^{n-1}))}(\mathbb{D})$ for which $ΘH^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ is a Rudin type invariant subspace of $H^2(\mathbb{D}^n)$.

math.FA↗

arXiv2015

An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II

This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T_1, \ldots, T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \mathcal{H} and \mathcal{S} be a non-trivial closed subspace of \mathcal{H}. One of our main results states that: \mathcal{S} is a joint T-invariant subspace if and only if there exists a partially isometric operator Π\in \mathcal{B}(H^2_n(\mathcal{E}), \mathcal{H})$ such that $\mathcal{S} = ΠH^2_n(\mathcal{E})$, where H^2_n is the Drury-Arveson space and \mathcal{E} is a coefficient Hilbert space and T_i Π= ΠM_{z_i}, i = 1, \ldots, n. In particular, our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in \mathbb{C}^n.

math.FA↗

arXiv2014

Rudin's Submodules of $H^2(\mathbb{D}^2)$

Let $\{α_n\}_{n\geq 0}$ be a sequence of scalars in the open unit disc of $\mathbb{C}$, and let $\{l_n\}_{n\geq 0}$ be a sequence of natural numbers satisfying $\sum_{n=0}^\infty (1 - l_n|α_n|) <\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace \[\mathcal{S}_Φ = \vee_{n=0}^\infty \Big( z_1^n \prod_{k=n}^\infty \left(\frac{-\barα_k}{|α_k|} \frac{z_2 - α_k}{1 - \barα_k z_2}\right)^{l_k} H^2(\mathbb{D}^2)\Big),\] is called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that \[ \text{dim} (\mathcal{S}_Φ\ominus (z_1 \mathcal{S}_Φ+ z_2\mathcal{S}_Φ))= 1+\#\{n\ge 0: α_n=0\}<\infty. \]In particular, this answer a question earlier raised by Douglas and Yang (2000).

math.FA↗

arXiv2014

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. The above module is usually called the Hilbert module over $\mathbb{C}[z_1, \ldots, z_n]$. Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ (or natural function algebras) were first introduced by R. G. Douglas and C. Foias in 1976. The two main driving forces were the algebraic and complex geometric views to multivariable operator theory. This article gives an introduction of Hilbert modules over function algebras and surveys some recent developments. Here the theory of Hilbert modules is presented as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces and its applications to the theory of $n$-tuples ($n \geq 1$) of commuting operators. The topics which are studied include: model theory from Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, curvature and Fredholm Hilbert modules. More developments in the study of Hilbert module approach to operator theory can be found in a companion paper, "Applications of Hilbert Module Approach to Multivariable Operator Theory".

math.FA↗