Author

# Sneh Lata

Other affiliations: University of Houston

Bio: Sneh Lata is an academic researcher from Shiv Nadar University. The author has contributed to research in topics: Hilbert space & Hardy space. The author has an hindex of 5, co-authored 15 publications receiving 49 citations. Previous affiliations of Sneh Lata include University of Houston.

##### Papers

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TL;DR: In this paper, the fundamental operators of a tetrablock contraction were investigated and the results obtained for Γ-contractions were then applied to tetrabelablock contractions.

15 citations

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01 Nov 2009TL;DR: In this paper, a short direct proof of Agler's factorization theorem using the Blecher-Ruan-Sinclair characterization of operator algebras is given. But this proof is restricted to polynomials.

Abstract: We give a short direct proof of Agler's factorization theorem that uses the Blecher-Ruan-Sinclair characterization of operator algebras. The key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.

14 citations

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TL;DR: In this article, it was shown that for reproducing kernel Hilbert spaces, the Feichtinger Conjecture (FC) is equivalent to the Kadison-Singer Problem in the sense that every norm bounded below Bessel sequence in a Hilbert space can be partitioned into finitely many Riesz basic sequences.

Abstract: In this dissertation, we study the Feichtinger Conjecture(FC), which has been shown to be equivalent to the celebrated Kadison-Singer Problem. The FC states that every norm-bounded below Bessel sequence in a Hilbert space can be partitioned into finitely many Riesz basic sequences. This study is divided into two parts. In the first part, we explore the FC in the setting of reproducing kernel Hilbert spaces. The second part of this study introduces two new directions to explore the FC further, which are based on a factorization of positive operators in B(`). The results presented in the later part have a mixed flavor in the sense that some of them point in the direction of finding a negative answer to the FC, whereas others prove the FC for some special cases. In the first part of the thesis, we show that in order to prove the FC it is enough to prove that in every Hilbert space, contractively contained in the Hardy space H, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and show that the above holds in them. We also look at products and tensor products of kernels, where using Schur products we obtain some interesting results. These results allows us to prove that in the Bargmann-Fock spaces on the n-dimensional complex plane and the weighted Bergman spaces on the unit ball, the Bessel sequences of normalized kernel functions split into finitely many Riesz basic sequences. We also prove that the same result holds in the H α,β spaces as well.

10 citations

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TL;DR: A short direct proof of Agler's factorization theorem that uses the Blecher-Ruan-Sinclair characterization of operator algebras and provides some additional information about these factorizations in the case of polynomials.

Abstract: We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.

9 citations

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TL;DR: In this article, it was shown that for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences.

Abstract: We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.

5 citations

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Abstract: The Schur–Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur–Agler-class theory to the free noncommutative function setting. This includes the positive-kerneldecomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur–Agler class coincides with the contractivemultiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.

50 citations

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TL;DR: In this article, the authors present general theorems about operator algebras, i.e., functions on sets that can be represented as scalar multipliers of a reproducing kernel Hilbert space.

27 citations

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TL;DR: In this paper, reproducing kernels and associated reproducing kernel Hilbert spaces (RKHSs) are studied over infinite, discrete and countable sets V. In this setting, the distributions of the corresponding Dirac point-masses of V are analyzed in detail.

Abstract: We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) H over infinite, discrete and countable sets V. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of V. Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural network-configuration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs H of functions defined on a prescribed countable infinite discrete set V, we characterize those which contain the Dirac masses δx for all points x in V. Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set V of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields.

25 citations

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TL;DR: In this article, the authors look at many of the known equivalent forms of the Kadison-Singer Problem and see what are the best new theorems available in each area of research as a consequence of the work of Marcus, Spielman and Srivastava.

Abstract: It is known that the famous, intractable 1959 Kadison-Singer problem in C∗-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. The recent surprising solution to this problem by Marcus, Spielman and Srivastava was a significant achievement and a significant advance for all these areas of research. We will look at many of the known equivalent forms of the Kadison-Singer Problem and see what are the best new theorems available in each area of research as a consequence of the work of Marcus, Spielman and Srivastava. In the cases where constants are important for the theorem, we will give the best constants available in terms of a generic constant taken from (A. Marcus, D. Spielman and N. Srivastava, Interlacing families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem, arXiv 1306.3969v4). Thus, if better constants eventually become available, it will be simple to adapt these new constants to the theorems.

19 citations

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TL;DR: In this article, it was shown that every distinguished variety in the tetrablock is one-dimensional and can be represented as Ω(x 1,x 2,x 3 )-approximation matrices.

Abstract: We show an interplay between the complex geometry of the tetrablock $\mathbb E$ and the commuting triples of operators having $\overline{\mathbb E}$ as a spectral set. We prove that every distinguished variety in the tetrablock is one-dimensional and can be represented as \begin{equation}\label{eqn:1} \Omega=\{ (x_1,x_2,x_3)\in \mathbb E \,:\, (x_1,x_2) \in \sigma_T(A_1^*+x_3A_2\,,\, A_2^*+x_3A_1) \}, \end{equation} where $A_1,A_2$ are commuting square matrices of the same order satisfying $[A_1^*,A_1]=[A_2^*,A_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb E$. We show that for a triple of commuting operators $\Upsilon = (T_1,T_2,T_3)$ having $\overline{\mathbb E}$ as a spectral set, there is a one-dimensional subvariety $\Omega_{\Upsilon}$ of $\overline{\mathbb E}$ depending on $\Upsilon$ such that von-Neumann's inequality holds, i.e, \[ f(T_1,T_2,T_3)\leq \sup_{(x_1,x_2,x_3)\in\Omega_{\Upsilon}}\, |f(x_1,x_2,x_3)|, \] for any holomorphic polynomial $f$ in three variables, provided that $T_3^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $\Omega_\Upsilon$ has been shown to have representation like (\ref{eqn:1}), where $A_1,A_2$ are the unique solutions of the operator equations \begin{gather*} T_1-T_2^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_1(I-T_3^*T_3)^{\frac{1}{2}} \text{ and } \\ T_2-T_1^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_2(I-T_3^*T_3)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $\Omega_{\Upsilon}$ is a distinguished variety in $\mathbb E$. We produce an explicit dilation and a concrete functional model for such a triple $(T_1,T_2,T_3)$ in which the unique operators $A_1,A_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.

13 citations