Browsing by Author "Trivedi, Shailesh"

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Analytic m-isometries without the wandering subspace property
(AMS, 2020) Trivedi, Shailesh
The wandering subspace problem for an analytic norm-increasing -isometry on a Hilbert space asks whether every -invariant subspace of can be generated by a wandering subspace. An affirmative solution to this problem for is ascribed to Beurling-Lax-Halmos, while that for is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic -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 . We also show that if the wandering subspace property fails for an analytic norm-increasing -isometry, then it fails miserably in the sense that the smallest -invariant subspace generated by the wandering subspace is of infinite codimension.
An analytic model for left-invertible weighted shifts on directed trees
(Wiley, 2016-06) Trivedi, Shailesh
Let be a rooted directed tree with finite branching index , and let be a left-invertible weighted shift on . We show that can be modelled as a multiplication operator on a reproducing kernel Hilbert space of -valued holomorphic functions on a disc centred at the origin, where . The reproducing kernel associated with is multi-diagonal and of bandwidth Moreover, admits an orthonormal basis consisting of polynomials in with at most non-zero coefficients. As one of the applications of this model, we give a spectral picture of Unlike the case , the approximate point spectrum of could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed tree with finite branching index.
Bounded point evaluation for a finitely multicyclic commuting tuple of operators
(Elsevier, 2020-09) Trivedi, Shailesh
We generalize the notion of bounded point evaluation introduced by Williams for a cyclic operator to a finitely multicyclic commuting d-tuple of bounded linear operators on a complex separable Hilbert space. We show that the set of all bounded point evaluations for T is a unitary invariant and we characterize it in terms of the dimension of the joint cokernel of T. Using this, we show that if has non-empty interior, then T can be realized as the d-tuple of multiplication operators on a reproducing kernel Hilbert space of functions on . We further characterize the largest open subset of on which all the elements of are analytic, which we refer to as the set of all analytic bounded point evaluations. As an application, we describe the set of all analytic bounded point evaluations for toral and spherical isometries, and also, derive an analytic model of a commuting d-tuple of composition operators.
Bounded point evaluation for operators with the wandering subspace property
(ARXIV, 2021) Trivedi, Shailesh
We extend and study the notion of bounded point evaluation introduced by Williams for a cyclic operator to the class of operators with the wandering subspace property. We characterize the set bpe(T) of all bounded point evaluations for an operator T with the wandering subspace property in terms of the invertibility of certain projections. This result generalizes the earlier established characterization of bpe(T) for a finitely cyclic operator T. Further, if T is a left-invertible operator with the wandering subspace property, then we determine the bpe(T) and the set abpe(T) of all analytic bounded point evaluations for T. We also give examples of left-invertible operator T with the wandering subspace property for which D(0,r(T′)−1)⫋abpe(T)⊆bpe(T), where r(T′) is the spectral radius of the Cauchy dual T′ of T.
Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs
(ARXIV, 2017-09) Trivedi, Shailesh
Given a directed Cartesian product T of locally finite, leafless, rooted directed trees T1,…,Td of finite joint branching index, one may associate with T the Drury-Arveson-type C[z1,…,zd]-Hilbert module Hca(T) of vector-valued holomorphic functions on the open unit ball Bd in Cd, where a>0. In case all directed trees under consideration are without branching vertices, Hca(T) turns out to be the classical Drury-Arveson-type Hilbert module Ha associated with the reproducing kernel 1(1−⟨z,w⟩)a defined on Bd. Unlike the case of d=1, the above association does not yield a reproducing kernel Hilbert module if we relax the assumption that T has finite joint branching index. The main result of this paper classifies all directed Cartesian product T for which the Hilbert modules Hca(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 Hca(T). Further, a careful analysis of these Hilbert modules allows us to prove that the cardinality of the kth generation (k=0,1,…) of T1,…,Td are complete invariants for Hca(⋅) provided ad≠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 Mz on Hca(T) with orthogonal direct sum of tensor products of certain hyperplanes.
Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces
(Elsevier, 2018-10) Trivedi, Shailesh
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 on a reproducing kernel Hilbert space of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for , under some natural conditions on the -valued kernel associated with , the commutant of is shown to be the algebra of bounded holomorphic -valued functions on Ω, provided satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of and ). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree of finite branching index. As the second main result of this paper, we show that a multiplication d-tuple on 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 -valued kernels for which the associated multiplication operators are non-hyponormal reflexive operators with commutants equal to .
Decomposability of Weighted Composition Operators on $L^p$ of Atomic Measure Space
(PMF, 2015) Trivedi, Shailesh
In this paper, we discuss the decomposability of weighted composition operator $uC_\phi$ on $L^p(X)(1\leq p<\infty)$ of a $\sigma$-finite atomic measure space $(X,\mathcal{S},\mu)$ with the assumption that $u\in L^\infty(X)$ and $|u|$ has positive ess inf. We prove that if the analytic core of $uC_\phi$ is zero and $uC_\phi$ is not quasinilpotent, then it is not decomposable. We also show that if $\phi$ is either injective almost everywhere or surjective almost everywhere but not both, then $uC_\phi$ is not decomposable. Finally, we give a necessary condition for decomposability of $uC_\phi$.
Dirichlet Spaces Associated with Locally Finite Rooted Directed Trees
(Springer, 2017-09) Trivedi, Shailesh
Let T = (V, E) be a leafless, locally finite rooted directed tree. We associate with T a one parameter family of Dirichlet spaces Hq (q 1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel κH q (z,w) = ∞ n=0 (1)n (q)n znwn P eroot + v∈V≺ ∞ n=0 (nv + 2)n (nv + q + 1)n znwn Pv (z,w ∈ D), where V≺ denotes the set of branching vertices of T , nv denotes the depth of v ∈ V in T , and P eroot , Pv (v ∈ V≺) are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators M(1) z and M(2) z of multiplication by z on Dirichlet spaces Hq associated with directed trees T1 and T2 respectively
Failure of the wandering subspace property for analytic norm-increasing 3-isometries
(ARXIV, 2022) Trivedi, Shailesh
We construct an analytic norm-increasing 3-isometric weighted shift on a rootless directed tree, which does not have the wandering subspace property. This answers a question of Shimorin [S2001, p. 185] in the negative. The counterexample in question is built over the rootless quasi-Brownian directed tree of valency 2.
Local Spectral Properties of a Composition Operator on LP Spaces
(Informatics Journal, 2015-12) Trivedi, Shailesh
In this paper, we discuss the decomposability and single valued extension property of composition operators CÏ† on Lp(X)(1 â‰¤ p < âˆž) spaces. We give a sufficient condition for non-decomposability of CÏ† in terms of Radon-Nikodym derivative. Further, we prove that if Ï† is conservative or it is invertible with non-singular inverse, then CÏ† has single valued extension property.
Local spectral theory of endomorphisms of the disk algebra
(De Gruyter, 2016) Trivedi, Shailesh
LetApDqdenote the disk algebra. Every endomorphism ofApDqis inducedby someφPApDqwith}φ}≤1. In this paper, it is shown that ifφis not an automorphismofDandφhas a fixed point in the open unit disk then the endomorphism induced byφis decomposable if and only if the fixed set ofφis singleton. Further, we determine thelocal spectra of the endomorphism induced byφin the cases when the fixed set ofφeitherincludes unit circle or is a singleton.
Multishifts on Directed Cartesian Product of Rooted Directed Trees
(ARXIV, 2017) Trivedi, Shailesh
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 uni es 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 rst half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including ner 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 classi cation 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 T of rooted directed trees. These multishifts turn out be multiplication d-tuples Mz;a on certain reproducing kernel Hilbert spaces Ha of vector-valued holomorphic functions de ned on the unit ball Bd in Cd, which can be thought of as tree analogs of the multiplication d-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels 1 (1􀀀hz;wi)a (z;w 2 Bd; a 2 N): Indeed, the reproducing kernels associated with Ha are certain operator linear combinations of 1 (1􀀀hz;wi)a and multivariable hypergeometric functions 2F1( v + a + 1; 1; v + 2; ) de ned on Bd Bd, where v denotes the depth of a branching vertex v in T . We also classify joint subnormal and joint hyponormal multishifts within the class of spherically balanced multishifts.
A short note on the similarity of operator-valued multishifts
(Springer, 2024-07) Trivedi, Shailesh
A complete characterization of the similarity between two operator-valued multishifts with invertible operator weights is obtained purely in terms of operator weights. This generalizes several existing results of the unitary equivalence of two (multi)shifts. Further, we utilize the aforementioned similarity criteria to determine the similarity between two tuples of operators of multiplication by the coordinate functions on certain reproducing kernel Hilbert spaces determined by diagonal kernels.
Some results on local spectral theory of Composition operators on lp spaces
(EMIS, 2014-09) Trivedi, Shailesh
In this paper, we give a condition under which a bounded linear operator on a complex Banach space has Single Valued Extension Property (SVEP) but does not have decomposition property (±). We also discuss the analytic core, decomposability and SVEP of composition operators CÁ on lp (1 · p < 1) spaces. In particular, we prove that if Á is onto but not one-one then CÁ is not decomposable but has SVEP. Further, it is shown that if Á is one-one but not onto then CÁ does not have SVEP.
Von Neumann’s inequality for commuting operator-valued multishifts
(AMS, 2019) Trivedi, Shailesh
Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann’s inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In fact, we show that if and are commuting contractive -tuples of operators such that satisfies the matrix-version of von Neumann’s inequality and is in the algebraic spectrum of , then the tensor product satisfies von Neumann’s inequality if and only if satisfies von Neumann’s inequality. We also exhibit several families of operator-valued multishifts for which von Neumann’s inequality always holds.
Wold-type decomposition for left-invertible weighted shifts on a rootless directed tree
(2025-01) Trivedi, Shailesh
Let $S_{\lambdab}$ be a bounded left-invertible weighted shift on a rootless directed tree T=(V,E). We address the question of when $S_{\lambdab}$ has Wold-type decomposition. We relate this problem to the convergence of the series $\displaystyle {\tiny \sum_{n = 1}^{\infty} \sum_{u \in G_{v, n}\backslash G_{v, n-1}} \Big(\frac{\lambdab^{(n)}(u)}{\lambdab^{(n)}(v)}\Big)^2},$ v∈V, involving the moments $\lambdab^{(n)}$ of $S^*_{\lambdab}$, where $G_{v, n}=\childn{n}{\parentn{n}{v}}.$ The main result of this paper characterizes all bounded left-invertible weighted shifts $S_{\lambdab}$ on T, which have Wold-type decomposition.