arXiv subjects

Sameer Chavan

Publications and source records attributed to Sameer Chavan.

32 records · Page 2Linked to original sources

arXiv2021

Weighted Join Operators on Directed Trees

A rooted directed tree $\mathscr T=(V, E)$ with can be extended to a directed graph $\mathscr T_\infty=(V_\infty, E_\infty)$ by adding a vertex $\infty$ to $V$ and declaring each vertex in $V$ as a parent of $\infty.$ One may associate with the extended directed tree a family of semigroup structures $\sqcup_{b}$ with extreme ends being induced by the join operation $\sqcup$ and the meet operation $\sqcap$. Each semigroup structure among these leads to a family of densely defined linear operators $W^{b}_{λ_u}$ acting on $\ell^2(V),$ which we refer to as weighted join operators at a given base point $b \in V_{\infty}$ with prescribed vertex $u \in V$. The extreme ends of this family are weighted join operators $W^{\mathsf{root}}_{λ_u}$ and weighted meet operators $W^{\infty}_{λ_u}$. In this paper, we systematically study these operators. We also present a more involved counter-part of weighted join operators on rootless directed trees. In both cases, the class of weighted join operators overlaps with the well-studied classes of complex Jordan operators and $n$-symmetric operators. An important half of this paper is devoted to the study of rank one extensions $W_{f, g}$ of weighted join operators, where $f \in \ell^2(V)$ and $g : V \to \mathbb C$ is unspecified. Unlike weighted join operators, these operators are not necessarily closed. We provide a couple of compatibility conditions involving the weight system $λ_u$ and $g$ to ensure closedness of $W_{f, g}$. We discuss the role of the Gelfand-triplet in the realization of the Hilbert space adjoint of $W_{f, g}$. Further, we describe various spectral parts of $W_{f, g}$ in terms of the weight system and the tree data. We also provide sufficient conditions for $W_{f, g}$ to be a sectorial operator. In case $\mathscr T$ is leafless, we characterize rank one extensions $W_{f, g}$, which admit compact resolvent.

math.FA↗

arXiv2020

The Cauchy dual subnormality problem for cyclic $2$-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a $2$-isometry is subnormal. Recently this problem has been solved in the negative. Here we show that it has a negative solution even in the class of cyclic $2$-isometries.

math.FA↗

arXiv2020

Dirichlet-type spaces on the unit ball and joint 2-isometries

We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint $2$-isometries.

math.FA↗

arXiv2019

Taylor spectrum approach to Brownian-type operators with quasinormal entry

In this paper, we introduce operators that are represented by upper triangular $2\times 2$ block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class $\mathcal Q,$ briefly operators of class $\mathcal Q.$ These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. It turns out that the class $\mathcal Q$ is closely related to the Cauchy dual subnormality problem which asks whether the Cauchy dual of a completely hyperexpansive operator is subnormal. Since the class $\mathcal Q$ is closed under the operation of taking the Cauchy dual, the problem itself becomes a part of a more general question of investigating subnormality in this class. This issue, along with the analysis of nonstandard moment problems, covers a large part of the paper. Using the Taylor spectrum technique culminates in a full characterization of subnormal operators of class $\mathcal Q.$ As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class $\mathcal Q$ in the affirmative, showing that the original problem can surprisingly be extended to a class of operators that are far from being completely hyperexpansive. The Taylor spectrum approach turns out to be fruitful enough to allow us to characterize other classes of operators including $m$-isometries. We also study linear operator pencils associated with operators of class $\mathcal Q$ proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints.

math.FA↗

arXiv2019

Analytic $m$-isometries without the wandering subspace property

The wandering subspace problem for an analytic norm-increasing $m$-isometry $T$ on a Hilbert space $\mathcal H$ asks whether every $T$-invariant subspace of $\mathcal H$ can be generated by a wandering subspace. An affirmative solution to this problem for $m=1$ is ascribed to Beurling-Lax-Halmos, while that for $m=2$ is due to Richter. In this paper, we capitalize on the idea of weighted shift on one-circuit directed graph to construct a family of analytic cyclic $3$-isometries, which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one dimensional space, their norms can be made arbitrarily close to $1$. We also show that if the wandering subspace property fails for an analytic norm-increasing $m$-isometry, then it fails miserably in the sense that the smallest $T$-invariant subspace generated by the wandering subspace is of infinite codimension.

math.FA↗

arXiv2018

Complete systems of unitary invariants for some classes of $2$-isometries

The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.

math.FA↗

arXiv2018

Commutants and Reflexivity of Multiplication tuples on Vector-valued Reproducing Kernel Hilbert Spaces

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the commutant of $\mathscr M_z$ is shown to be the algebra $H^{\infty}_{_{B(E)}}(Ω)$ of bounded holomorphic $B(E)$-valued functions on $Ω$, provided $\mathscr M_z$ satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of $\dim E=1$ and $d=1$). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree $\mathscr T$ of finite branching index. As the second main result of this paper, we show that a multiplication $d$-tuple $\mathscr M_z$ on $\mathscr H$ satisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonal $B(\mathbb C^2)$-valued kernels for which the associated multiplication operators $\mathscr M_z$ are non-hyponormal reflexive operators with commutants equal to $H^{\infty}_{_{B(\mathbb C^2)}}(\mathbb D)$.

math.FA↗

arXiv2018

A solution to the Cauchy dual subnormality problem for 2-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{\prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{\prime}$ is subnormal.

math.FA↗

arXiv2017

Weyl's Theorem for pairs of commuting hyponormal operators

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbolλ) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbolλ})^*), $$ for every $\boldsymbolλ$ in the Taylor spectrum $σ(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $ω(\mathbf{T})$, satisfies the identity $$ ω(\mathbf{T})=σ(\mathbf{T}) \setminus π_{00}(\mathbf{T}), $$ where $π_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity. Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.

math.FA↗

arXiv2017

Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

Given a directed Cartesian product $\mathscr T$ of locally finite, leafless, rooted directed trees $\mathscr T_1, \ldots, \mathscr T_d$ of finite joint branching index, one may associate with $\mathscr T$ the Drury-Arveson-type $\mathbb C[z_1, \ldots, z_d]$-Hilbert module $\mathscr H_{\mathfrak c_a}(\mathscr T)$ of vector-valued holomorphic functions on the open unit ball $\mathbb B^d$ in $\mathbb C^d$, where $a >0.$ In case all directed trees under consideration are without branching vertices, $\mathscr H_{\mathfrak c_a}(\mathscr T)$ turns out to be the classical Drury-Arveson-type Hilbert module $\mathscr H_{a}$ associated with the reproducing kernel $\frac{1}{(1 - \langle{z}, {w}\rangle)^a}$ defined on $\mathbb B^d$. Unlike the case of $d=1$, the above association does not yield a reproducing kernel Hilbert module if we relax the assumption that $\mathscr T$ has finite joint branching index. The main result of this paper classifies all directed Cartesian product $\mathscr T$ for which the Hilbert modules $\mathscr H_{\mathfrak c_a}(\mathscr T)$ are isomorphic in case $a$ is a positive integer. One of the essential tools used to establish this isomorphism is an operator-valued representing measure arising from $\mathscr H_{\mathfrak c_a}(\mathscr T).$ Further, a careful analysis of these Hilbert modules allows us to prove that the cardinality of the $k^{\tiny \mbox{th}}$ generation $(k =0, 1, \ldots)$ of $\mathscr T_1, \ldots, \mathscr T_d$ are complete invariants for $\mathscr H_{\mathfrak c_a}(\cdot)$ provided $ad \neq 1$. Failure of this result in case $ad =1$ may be attributed to the von Neumann-Wold decomposition for isometries. Along the way, we identify the joint cokernel $E$ of the multiplication $d$-tuple $\mathscr M_{z}$ on $\mathscr H_{\mathfrak c_a}(\mathscr T)$ with orthogonal direct sum of tensor products of certain hyperplanes.

math.FA↗

arXiv2017

Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees

Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $\mathbb D$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel \begin{eqnarray*} κ_{\mathscr H_q}(z, w) = \sum_{n=0}^{\infty}\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf{root}}\rangle} + \sum_{v \in V_{\prec}} \sum_{n=0}^{\infty} \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{eqnarray*} where $V_{\prec}$ denotes the set of branching vertices of $\mathscr T$, $n_v$ denotes the depth of $v \in V$ in $\mathscr T,$ and $P_{\langle e_{\mathsf{root}}\rangle}$, $~P_{v}~(v \in V_{\prec})$ are certain orthogonal projections. We also discuss some structural properties of the operator $\mathscr M_{z, q}$ of multiplication by $z$ on $\mathscr H_q.$ Further, we discuss the question of unitary equivalence of operators $\mathscr M^{(1)}_z$ and $\mathscr M^{(2)}_z$ of multiplication by $z$ on Dirichlet spaces $\mathscr H_q$ associated with directed trees $\mathscr T_1$ and $\mathscr T_2$ respectively.

math.CV↗

arXiv2016

Multishifts on Directed Cartesian Product of Rooted Directed Trees

We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of product of directed graphs, we introduce and study the notion of multishifts on directed Cartesian product of rooted directed trees. This framework unifies the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings into picture some new phenomena such as the appearance of system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the first half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including finer analysis of various joint spectra and wandering subspace property for multishifts. In the second half, we separate out two special classes of multishifts, which we refer to as torally balanced and spherically balanced multishifts. The classification of these two classes is closely related to toral and spherical polar decompositions of multishifts. Furthermore, we exhibit a family of spherically balanced multishifts on $d$-fold directed Cartesian product $\mathscr T$ of rooted directed trees. These multishifts turn out be multiplication $d$-tuples $\mathscr M_{z, a}$ on certain reproducing kernel Hilbert spaces $\mathscr H_a$ of vector-valued holomorphic functions defined on the unit ball $\mathbb B^d$ in $\mathbb C^d$, which can be thought of as tree analogs of the multiplication $d$-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels $\frac{1}{(1-\langle{z},{{w}\rangle})^a}~(z, w \in \mathbb B^d, a \in \mathbb N).$

math.FA↗

arXiv2016

Module tensor product of subnormal modules need not be subnormal

Let $κ: \mathbb D \times \mathbb D \to \mathbb C$ be a diagonal positive definite kernel and let $\mathscr H_κ$ denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc $\mathbb D$. Assume that $zf \in \mathscr H$ whenever $f \in \mathscr H.$ Then $\mathscr H$ is a Hilbert module over the polynomial ring $\mathbb C[z]$ with module action $p \cdot f \mapsto pf$. We say that $\mathscr H_κ$ is a subnormal Hilbert module if the operator $\mathscr M_{z}$ of multiplication by the coordinate function $z$ on $\mathscr H_κ$ is subnormal. %If $κ_1$ and $κ_2$ are two diagonal positive definite kernels then so is their pointwise (tensor) product $κ:=κ_1κ_2$. In [Oper. Theory Adv. Appl, 32: 219-241, 1988], N. Salinas asked whether the module tensor product $\mathscr H_{κ_1} \otimes_{\mathbb C[z]} \mathscr H_{κ_2}$ of subnormal Hilbert modules $\mathscr H_{κ_1}$ and $\mathscr H_{κ_2}$ is again subnormal. In this regard, we describe all subnormal module tensor products $L^2_a(\mathbb D, w_{s_1}) \otimes_{\mathbb C[z]} L^2_a(\mathbb D, w_{s_2})$, where $L^2_a(\mathbb D, w_s)$ denotes the weighted Bergman Hilbert module with radial weight $$w_s(z)=\frac{1}{s π}|z|^{\frac{2(1-s)}{s}}~(z \in \mathbb D, ~s > 0).$$ In particular, the module tensor product $L^2_a(\mathbb D, w_{s}) \otimes_{\mathbb C[z]} L^2_a(\mathbb D, w_{s})$ is never subnormal for any $s \geq 6$. Thus the answer to this question is no.

math.FA↗

arXiv2015

An Analytic Model for Left-Invertible Weighted Shifts on Directed Trees

Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_λ \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_λ$ can be modelled as a multiplication operator $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S^*_λ$. The reproducing kernel associated with $\mathscr H$ is multi-diagonal and of bandwidth $k_{\mathscr T}.$ Moreover, $\mathscr H$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{\mathscr T}+1$ non-zero coefficients. As one of the applications of this model, we give a complete spectral picture of $S_λ.$ Unlike the case $\dim E = 1,$ the approximate point spectrum of $S_λ$ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed trees with finite branching index.

math.FA↗