arXiv subjects

B. Krishna Das

Publications and source records attributed to B. Krishna Das.

31 records · Page 2Linked to original sources

arXiv2019

Toeplitz operators and pseudo-extensions

There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by studying the unique canonical unitary pseudo-extension of a tuple of commuting contractions. The third one is about the $C^*$-algebra generated by the Toeplitz operators determined by a commuting tuple of contractions. With the help of a special completely positive map, a different proof of the existence of the unique canonical unitary pseudo-extension is given.

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arXiv2019

Hypercontractions and factorizations of multipliers in one and several variables

We introduce the notion of characteristic functions for commuting tuples of hypercontractions on Hilbert spaces, as a generalization of the notion of Sz.-Nagy and Foias characteristic functions of contractions. We present an explicit method to compute characteristic functions of hypercontractions and relate characteristic functions by means of the factors of Schur-Agler class of functions and universal multipliers on the unit ball in $\mathbb{C}^n$. We also offer some factorization properties of multipliers. Characteristic functions of hypercontrctions are complete unitary invariant. The Drury-Arveson space and the weighted Bergman spaces on the unit ball continues to play a significant role in our consideration. Our results are new even in the special case of single hypercontractions.

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arXiv2018

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von Neumann inequality for a large class of $n$-tuples, $n \geq 3$, of commuting contractions.

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arXiv2018

On Quotient modules of $H^2(\mathbb{D}^n)$: Essential Normality and Boundary Representations

Let $\mathbb{D}^n$ be the open unit polydisc in $\mathbb{C}^n$, $n \geq 1$, and let $H^2(\mathbb{D}^n)$ be the Hardy space over $\mathbb{D}^n$. For $n\ge 3$, we show that if $θ\in H^\infty(\mathbb{D}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1}, \ldots, C_{z_n})$ on the Beurling type quotient module $\mathcal{Q}_θ$ is not essentially normal, where \[\mathcal{Q}_θ = H^2(\mathbb{D}^n)/ θH^2(\mathbb{D}^n) \quad \mbox{and} \quad C_{z_j} = P_{\mathcal{Q}_θ} M_{z_j}|_{\mathcal{Q}_θ}\quad (j = 1, \ldots, n).\] Rudin's quotient modules of $H^2(\mathbb{D}^2)$ are also shown to be not essentially normal. We prove several results concerning boundary representations of $C^*$-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.

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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, \[P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{φψ}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{ψφ}|_{\mathcal{Q}} \; \mbox{and} \;(T_1, T_2) \cong (P_{\mathcal{Q}} M_φ|_{\mathcal{Q}}, P_{\mathcal{Q}} M_ψ|_{\mathcal{Q}}).\]

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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.

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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.

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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 $H^2(\mathbb{D}^n)$.

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arXiv2014

Star-Generating Vectors of Rudin's Quotient Modules

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.

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arXiv2014

Cross commutators of Rudin's submodules

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.

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arXiv2013

Tensor product of quotient Hilbert modules

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 $\mathbb{C}^n$.

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arXiv2013

Maximal contractive tuples

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.

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arXiv2013

Wandering subspaces of the Bergman space and the Dirichlet space over polydisc

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 \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]

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