Rozprawy doktorskie: „Hilbert space operators”

1

邱彩娜 and Choi-nai Charlies Tu. "Generalized spectral norms of Hilbert space operators." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31220010.

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Tu, Choi-nai Charlies. "Generalized spectral norms of Hilbert space operators /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19737452.

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Tian, Feng. "On commutativity of unbounded operators in Hilbert space." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1095.

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We study several unbounded operators with view to extending von Neumann's theory of deficiency indices for single Hermitian operators with dense domain in Hilbert space. If the operators are non-commuting, the problems are difficult, but special cases may be understood with the use representation theory. We will further study the partial derivative operators in the coordinate directions on the L2 space on various covering surfaces of the punctured plane. The operators are defined on the common dense domain of C∞ functions with compact support, and they separately are essentially selfadjoint, but the unique selfadjoint extensions will be non-commuting. This problem is of a geometric flavor, and we study an index formulation for its solution. The applications include the study of vector fields, the theory of Dirichlet problems for second order partial differential operators (PDOs), Sturm-Liouville problems, H.Weyl's limit-point/limit-circle theory, Schrödinger equations, and more.

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4

Kiteu, Marco M. "Orbits of operators on Hilbert space and some classes of Banach spaces." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1341850621.

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Hansen, A. C. "On the approximation of spectra of linear Hilbert space operators." Thesis, University of Cambridge, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603665.

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The main topic of this thesis is how to approximate and compute spectra of linear operators on separable Hilbert spaces. We consider several approaches including the finite section method, an infinite-dimensional version of the QR algorithm, as well as pseudospectral techniques. Several new theorems about convergence of the finite section method (and variants of it) for self-adjoint problems are obtained together with a rigorous analysis of the infinite-dimensional QR algorithm for normal operators. To attack the general spectral problem we look to the pseudospectral theory and introduce the complexity index. A generalization of the pseudospectrum is introduced, namely, the n-pseudospectrum. This set behaves very much like the original pseudospectrum, but has the advantage that it approximates the spectrum well for large n. The complexity index is a tool for indicating how complex or difficult it may be to approximate spectra of operators belonging to a certain class. We establish bounds on the complexity indeed and discuss some open problems regarding this new mathematical entity. As the approximation framework also gives rise to several computational methods, we analyze and discuss implementation techniques for algorithms that can be derived from the theoretical model. In particular, we develop algorithms that can compute spectra of arbitrary bounded operators on separable Hilbert spaces, and the exposition is followed by several numerical examples. The thesis also contains a thorough discussion on how to implement the QR algorithm in infinite dimensions. This is supported by numerical computations. These examples reveal several surprisingly nice features of the infinite-dimensional QR algorithm, and this leaves a number of open problems that we debate.

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6

Guven, Ayse. "Quantitative perturbation theory for compact operators on a Hilbert space." Thesis, Queen Mary, University of London, 2016. http://qmro.qmul.ac.uk/xmlui/handle/123456789/23197.

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This thesis makes novel contributions to a problem of practical and theoretical importance, namely how to determine explicitly computable upper bounds for the Hausdorff distance of the spectra of two compact operators on a Hilbert space in terms of the distance of the two operators in operator norm. It turns out that the answer depends crucially on the speed of decay of the sequence of singular values of the two operators. To this end, 'compactness classes', that is, collections of operators the singular values of which decay at a certain speed, are introduced and their functional analytic properties studied in some detail. The main result of the thesis is an explicit formula for the Hausdorff distance of the spectra of two operators belonging to the same compactness class. Along the way, upper bounds for the resolvents of operators belonging to a particular compactness class are established, as well as novel bounds for determinants of trace class operators.

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Alcántara, Bode Julio. "A criteria of completeness for compact operators in Hilbert space." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97122.

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A necessary and sufficient condition is given for completeness of the set of eigenfunctions and generalized eigenfunctions associated to the non zero eigenvalues of a compact operator on a Hilbert Space.

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Sutton, Daniel Joseph. "Structure of Invariant Subspaces for Left-Invertible Operators on Hilbert Space." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/28807.

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This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts. Ph. D.

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9

Raney, Michael W. "Abstract backward shifts of finite multiplicity /." view abstract or download file of text, 2002. http://wwwlib.umi.com/cr/uoregon/fullcit?p3061962.

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Thesis (Ph. D.)--University of Oregon, 2002. Typescript. Includes vita and abstract. Includes bibliographical references (leaf 55). Also available for download via the World Wide Web; free to University of Oregon users.

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Kazemi, Parimah. "Compact Operators and the Schrödinger Equation." Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5453/.

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In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.

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11

Turcu, George R. "Hypercyclic Extensions Of Bounded Linear Operators." Bowling Green State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1386189984.

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Höber, Marc [Verfasser]. "Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes / Marc Höber." Mainz : Universitätsbibliothek der Johannes Gutenberg-Universität Mainz, 2007. http://d-nb.info/1225402042/34.

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Atim, Alexandru Gabriel Kallman Robert R. "Uniqueness results for the infinite unitary, orthogonal and associated groups." [Denton, Tex.] : University of North Texas, 2008. http://digital.library.unt.edu/permalink/meta-dc-6136.

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Atim, Alexandru Gabriel. "Uniqueness Results for the Infinite Unitary, Orthogonal and Associated Groups." Thesis, University of North Texas, 2008. https://digital.library.unt.edu/ark:/67531/metadc6136/.

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Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *-automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space.

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15

Chakmak, Ryan. "Eigenvalues and Approximation Numbers." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/cmc_theses/2167.

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While the spectral theory of compact operators is known to many, knowledge regarding the relationship between eigenvalues and approximation numbers might be less known. By examining these numbers in tandem, one may develop a link between eigenvalues and l^p spaces. In this paper, we develop the background of this connection with in-depth examples.

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Merghni, Lobna. "Propriétés spectrales des opérateurs de composition et opérateurs de Hankel." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0029.

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Dans cette thèse nous nous intéressons aux opérateurs de composition sur les espaces de Hardy et Dirichlet et aux opérateurs de Hankel sur les espaces des fonction polyanalytiques. On s’'intéresse à l’'opérateur de composition sur les espaces de Dirichlet : $mathcal{D}_alpha=\left{f \in Hol(D): |f|_alpha^{2}=| f(0)| ^{2}+int_{D}| f'(z)| ^{2}dA_alpha(z) In this thesis we focus on the composition operators on Hardy and Dirichlet spaces and Hankel operators on spaces of polyanalytiques functions. We are interested in the composition operator on the Dirichlet spaces: $$ mathcal{D}_alpha=left{ f in Hol(D): |f|_alpha^{2}=| f(0)|^{2}+int_{D}| f'(z)| ^{2}dA_alpha(z)

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Khalil, Asma Mohammed. "Structure of scalar-type operators on Lp spaces and well-bounded operators on Hilbert spaces." Thesis, University of Edinburgh, 2002. http://hdl.handle.net/1842/10983.

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It is known that every scalar-type spectral operator on a Hilbert space H is similar to a multiplication operator on some L2 space. The purpose of the main theorem in Chapter 2 of this thesis is to show that every scalar-type spectral operator on an L1 space whose spectral measure has finite multiplicity is similar to a multiplication operator on the same L1 space. Then we prove a similar result for scalar-type spectral operators on Lp (Ω, SΩ, m), p ≠ 2, 1 < p < ∞, with spectral measure E(.) of finite uniform multiplicity provided an extra condition is satisfied. Also, we give conditions that make a scalar-type spectral operator on L2(Ω, SΩ, m) similar to a multiplication operator on the same L2(Ω, SΩ, m). In 1954, Dunford proved that a bounded operator T on a Banach space X is spectral if and only if it has the canonical decomposition T = S +Q, where S is a scalar-type operator and Q is a quasinilpotent operator which commutes with S. In Chapter 3, we prove that any well-bounded operator T on a Hilbert space H has the form T = A + Q, where A is a self-adjoint operator and Q is a quasinilpotent operator such that AQ - QA is quasinilpotent. Then we prove that a trigonometrically well-bounded operator T on H can be decomposed as T = U(Q + I) where U is a unitary operator and Q is quasinilpotent such that UQ = QU is also quasinilpotent. In Chapter 4 we prove that an AC-operator with discrete spectrum on H can be decomposed as a sum of a normal operator N and a quasinilpotent Q such that NQ - QN is quasinilpotent. However, the converse of each of the last three theorems is not true in general. In the final chapter we introduce a new class of operators on L2([a,b]) which is larger than the class of well-bounded operators on L2([a,b]) and we call them operators with an AC2-functional calculus. Then we give an example of an operator with an AC2-functional calculus on L2([0,1]) which can be decomposed as a sum of a self-adjoint operator and a quasinilpotent. We also discuss the possibility of decomposing every operator T with an AC2-functional calculus on L2([a,b]) into the sum of a self-adjoint operator A and a quasinilpotent operator Q such that AQ - QA is quasinilpotent.

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18

Boulton, Lyonell. "Topics in the spectral theory of non adjoint operators." Thesis, King's College London (University of London), 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.272412.

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Hansmann, Marcel. "On the discrete spectrum of linear operators in Hilbert spaces." Clausthal-Zellerfeld Universitätsbibliothek Clausthal, 2010. http://d-nb.info/1001898664/34.

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Zandler, Andersson Nils. "Boundedness of a Class of Hilbert Operators on Modulation Spaces." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-84932.

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In this work we take interest in frames and modulation spaces. On the basis of their properties, we show how frame expansions can be used to prove the boundedness of a particular class of Hilbert operators on modulation spaces taking advantage of the special category of piece-wise polynomial functions known as B-splines.

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21

Khadivi, Mohammad Reza. "Operator theory and infinite networks." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/30019.

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Hansmann, Marcel [Verfasser]. "On the discrete spectrum of linear operators in Hilbert spaces / Marcel Hansmann." Clausthal-Zellerfeld : Universitätsbibliothek Clausthal, 2010. http://d-nb.info/1001898664/34.

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Srithharan, T. "Theory and applications of Hilbert's and Thompson's metrics to positive operators in ordered spaces." Thesis, University of Sussex, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262302.

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Hofmann, B., and O. Scherzer. "Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800957.

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The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.

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Formisano, Teresa. "Minimax in the theory of operators on Hilbert spaces and Clarkson-McCarthy estimates for lq (Sp) spaces of operators in the Schatten ideals." Thesis, London Metropolitan University, 2014. http://repository.londonmet.ac.uk/1099/.

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The main results in this thesis are the minimax theorems for operators in Schatten ideals of compact operators acting on separable Hilbert spaces, generalized Clarkson-McCarthy inequalities for vector lq-spaces lq (Sp) of operators from Schatten ideals Sp, inequalities for partitioned operators and for Cartesian decomposition of operators. All Clarkson-McCarthy type inequalities are in fact some estimates on the norms of operators acting on the spaces lq (Sp) or from one such space into another.

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26

Cassier, Gilles. "Algebres duales d'operateurs sur l'espace de hilbert." Paris 6, 1988. http://www.theses.fr/1988PA066122.

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Etude des algebres duales d'operateurs sur l'espace de hilbert subdivisee en quatre parties. La 1ere traite des questions relatives aux topologies definies sur l'algebre duale engendree par un operateur et a la classification proposee par h. Bercovici, c. Foias et c. Pearcy de ces algebres. On montre au cours de la 2eme partie des proprietes de convexite de l'image numerique simultanee lorsque les operateurs se trouvent dans une algebre duale uniforme. Elles sont en defaut si l'algere duale n'est pas unifrome. La 3eme contient quelques remarques sur les algebres duales engendrees par un seul operateur, une decomposition de celles-ci lorsqu'elles sont uniformes et ses corollaires. La derniere est une application des algebres duales d'operateurs aux problemes de caracterisation des suites d'interpolation

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Hofmann, B. "On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801185.

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In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed.

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Abdillah, Said Amana. "Extensions au cadre Banachique de la notion d'opérateur de Hilbert-Schmidt." Thesis, Bordeaux 1, 2012. http://www.theses.fr/2012BOR14622/document.

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Cette thèse est consacrée à l’extension au cadre Banachique de la notion d’opérateur de Hilbert-Schmidt. Dans un premier temps, on étudie d’une part les opérateurs p-sommants dans un espace de Banach X vers un autre espace de Banach Y et d’autre part, les opérateurs gamma-radonifiants dans un espace de Hilbert vers un autre espace de Banach.Dans un second temps, on s'intéresse aux opérateurs gamma-sommants dans des espaces de Banach, qui coïncident avec les opérateurs de Rademacher-bornés, ce qui nous amène aux opérateurs presque sommants. Enfin, on en déduit plusieurs généralisations naturelles de la notion d’opérateur de Hilbert-Schmidt aux espaces de Banach.-Les classes des opérateurs p-sommants de X dans Y .-La classe des opérateurs presque sommants de X dans Y qui coïncide avec la classe des opérateurs gamma-radonifiants de X dans Y.-La classe des opérateurs faible* 1-nucléaires de X dans Y This thesis is devoted to extending the notion of Banach Hilbert-Schmidt operator to the framework of Banach spaces. In a first step, we study p-summing operators from a Banach space X into a Banach space Y and gamma-radoniyfing operators from a Hilbert space into a Banach space. In a second step, we discuss gamma-summing operators between Banach spaces, which coincide with Rademacher-bounded operators, which leads to the notion of almost summing operators. Finally, we present serval natural generalizations of the notion of Hilbert-Schmidt operator to Banach spaces.- Classes of p-summing operators from X into Y. - The class of almost summing operators from X into Y, which coincides with the class of gamma-radoniyfing operators from X into Y.- The class of weak*1-nuclear operators from X into Y

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Giulini, Ilaria. "Generalization bounds for random samples in Hilbert spaces." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0026/document.

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Ce travail de thèse porte sur l'obtention de bornes de généralisation pour des échantillons statistiques à valeur dans des espaces de Hilbert définis par des noyaux reproduisants. L'approche consiste à obtenir des bornes non asymptotiques indépendantes de la dimension dans des espaces de dimension finie, en utilisant des inégalités PAC-Bayesiennes liées à une perturbation Gaussienne du paramètre et à les étendre ensuite aux espaces de Hilbert séparables. On se pose dans un premier temps la question de l'estimation de l'opérateur de Gram à partir d'un échantillon i. i. d. par un estimateur robuste et on propose des bornes uniformes, sous des hypothèses faibles de moments. Ces résultats permettent de caractériser l'analyse en composantes principales indépendamment de la dimension et d'en proposer des variantes robustes. On propose ensuite un nouvel algorithme de clustering spectral. Au lieu de ne garder que la projection sur les premiers vecteurs propres, on calcule une itérée du Laplacian normalisé. Cette itération, justifiée par l'analyse du clustering en termes de chaînes de Markov, opère comme une version régularisée de la projection sur les premiers vecteurs propres et permet d'obtenir un algorithme dans lequel le nombre de clusters est déterminé automatiquement. On présente des bornes non asymptotiques concernant la convergence de cet algorithme, lorsque les points à classer forment un échantillon i. i. d. d'une loi à support compact dans un espace de Hilbert. Ces bornes sont déduites des bornes obtenues pour l'estimation d'un opérateur de Gram dans un espace de Hilbert. On termine par un aperçu de l'intérêt du clustering spectral dans le cadre de l'analyse d'images This thesis focuses on obtaining generalization bounds for random samples in reproducing kernel Hilbert spaces. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations and then in generalizing the results in a separable Hilbert space. We first investigate the question of estimating the Gram operator by a robust estimator from an i. i. d. sample and we present uniform bounds that hold under weak moment assumptions. These results allow us to qualify principal component analysis independently of the dimension of the ambient space and to propose stable versions of it. In the last part of the thesis we present a new algorithm for spectral clustering. It consists in replacing the projection on the eigenvectors associated with the largest eigenvalues of the Laplacian matrix by a power of the normalized Laplacian. This iteration, justified by the analysis of clustering in terms of Markov chains, performs a smooth truncation. We prove nonasymptotic bounds for the convergence of our spectral clustering algorithm applied to a random sample of points in a Hilbert space that are deduced from the bounds for the Gram operator in a Hilbert space. Experiments are done in the context of image analysis

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Agora, Elona. "Boundedness of the Hilbert Transform on Weighted Lorentz Spaces." Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/108930.

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The main goal of this thesis is to characterize the weak-type (resp. strong-type) boundedness of the Hilbert transform H on weighted Lorentz spaces Λpu(w). The characterization is given in terms of some geometric conditions on the weights u and w and the weak-type (resp. strong-type) boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results extend and unify simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Ariño-Muckenhoupt weights Bp. Títol: Acotaciò de l'operador de Hilbert sobre espais de Lorentz amb pesos Resum: L'objectiu principal d'aquesta tesi es caracteritzar l'acotació de l'operador de Hilbert sobre els espais de Lorentz amb pesos Λpu(w). També estudiem la versió dèbil. La caracterització es dona en terminis de condicions geomètriques sobre els pesos u i w, i l'acotació de l'operador maximal de Hardy-Littlewood sobre els mateixos espais. Els nostres resultats unifiquen dues teories conegudes i aparentment no relacionades entre elles, que tracten l'acotació de l'operador de Hilbert sobre els espais de Lebegue amb pesos Lp(u) per una banda i els espais de Lorentz clàssics Λp(w) per altre banda.

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Seddik, Ameur. "Intersection de l'adhérence de l'image d'une dérivation "delta"A avec le commutant de A* dans des cas particuliers." Montpellier 2, 1988. http://www.theses.fr/1988MON20107.

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A un element a de l(h) algebre des operateurs lineaires bornes definis sur l'espace de hilbert complexe h, on associe l'operateur de derivation delta ::(a) defini sur l(h) par: delta ::(a)(x)=ax-xa, x appartient a l(h). L'objet de cette etude est de trouver des operateurs de l(h) appartenant a la classe m=(a appartient a l(h):r(delta ::(a)) inter (a*)'=(o)). On a montre que cette classe contient les operateurs de la forme a cercle+b, ou p(a)=o pour un certain polynome p du second degre et b appartient a m, les operateurs de jordan d'ordre quelconque, et on donne quelques exemples d'operateurs nilpotents d'ordre 3 appartenant a m

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32

Niedzialomski, Robert. "Extension of positive definite functions." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/2595.

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Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

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Acevedo, Jeovanny de Jesus Muentes. "O fluxo espectral de caminhos de operadores de Fredholm auto-adjuntos em espaços de Hilbert." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-01122017-214259/.

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O objetivo principal desta dissertação é apresentar o fluxo espectral de um caminho de operadores de Fredholm auto-adjuntos em um espaço de Hilbert e suas propriedades. Pelos resultados clássicos de teoria espectral, sabemos que se H é um espaço de Hilbert e L : H &#8594 H é um operador linear, limitado e auto-adjunto, H pode ser escrito como soma direta ortogonal H+(L)&#8853 H-(L)&#8853 Ker L, onde H+(L) e H-(L) são os subespaços espectrais positivo e negativo de L, respectivamente. No trabalho damos uma definição de fluxo espectral baseada na decomposição acima, aprofundando as conexões deste conceito com a teoria espectral dos operadores de Fredholm em espaços de Hilbert. Entre as propriedades do fluxo espectral, será analisada a invariância homotópica que se apresenta em várias formas. Veremos o conceito de índice de Morse relativo, que estende o clássico índice de Morse, e sua relação com o fluxo espectral. A construção do fluxo espectral dada neste trabalho segue a abordagem de P. M. Fitzpatrick, J. Pejsachowicz e L. Recht em [9]. The main purpose of this dissertation is to present the spectral flow of a path of selfadjoint Fredholm operators in a Hilbert space and its properties. By classical results in spectral theory, we know that, if H is a Hilbert space and L : H &#8594 H is a bounded self-adjoint linear operator, H may be written as the following orthogonal direct sum H = H+(L)&#8853 H-(L)&#8853 Ker L, where H+(L) and H-(L) are the positive and negative spectral subspaces of L, respectively. In this work we give a definition of spectral flow which is based on the above splitting, examining in depth the connection between this concept and the spectral theory of Fredholm operators in Hilbert spaces. Among the properties of the spectral flow we will analyze the homotopic invariance, which appears on different ways. We will see the concept of relative Morse index, which generalize the classical Morse index, and its relation with the spectral flow. The construction of the spectral flow given in this work follows the approach of P. M. Fitzpatrick, J. Pejsachowicz and L. Recht in [9].

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34

Hofmann, B., and G. Fleischer. "Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800987.

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In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

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35

Amaya, Austin J. "Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/27636.

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Given a full-range simply-invariant shift-invariant subspace M of the vector-valued L2 space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function W so that M may be represented as the image of of the Hardy space H2 on the disc under multiplication by W. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces (M,MÃ ) which together form a direct-sum decomposition of L2. In the case where the pair (M,MÃ ) are finite-dimensional perturbations of the Hardy space H2 and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function W; this realization was parameterized in terms of zero-pole data computed from the pair (M,MÃ ). Later work by Ball-Raney extended this analysis to the case of nonrational functions W where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs (M,MÃ ) of the L2 spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans. Ph. D.

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36

Dora, Seleši. "Uopšteni stohastički procesi u beskonačno-dimenzionalnim prostorima sa primenama na singularne stohastičke parcijalne diferencijalne jednačine." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2007. https://www.cris.uns.ac.rs/record.jsf?recordId=6018&source=NDLTD&language=en.

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Doktorska disertacija je posvećena raznim klasama uop&scaron;tenih stohastičkih procesa i njihovim primenama na re&scaron;avanje singularnih stohastičkih parcijalnih diferencijalnih jednačina. U osnovi, disertacija se može podeliti na dva dela. Prvi deo disertacije (Glava 2) je posvećen strukturnoj karakterizaciji uop&scaron;tenih stohastičkih procesa u vidu haos ekspanzije i integralne reprezentacije. Drugi deo disertacije (Glava 3) čini primena dobijenih rezultata na re·savanje stohastičkog Dirihleovog problema u kojem se množenje modelira Vikovim proizvodom, a koefcijenti eliptičnog diferencijalnog operatora su Kolomboovi uop&scaron;teni stohastički procesi. Subject of the dissertation are various classes of generalizedstochastic processes and their applications to solving singular stochasticpartial di®erential equations. Basically, the dissertation can be divided intotwo parts. The ¯rst part (Chapter 2) is devoted to structural characteri-zations of generalized random processes in terms of chaos expansions andintegral representations. The second part of the dissertation (Chapter 3)involves applications of the obtained results to solving a stochastic Dirichletproblem, where multiplication is modeled by the Wick product, and thecoe±cients of the elliptic di®erential operator are Colombeau generalizedrandom processes.

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37

Smith, Lidia. "On Orbits of Operators on Hilbert Space." 2009. http://hdl.handle.net/1969.1/ETD-TAMU-2009-08-842.

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In this dissertation we treat some problems about possible density of orbits for non-hypercyclic operators and we enlarge the class of known non-orbit-transitive operators. One of the questions related to hypercyclic operators that we answer is whether the density (in the set of positive real numbers) of the norms of the elements in the orbit for each nonzero vector in the Hilbert space is sufficient to imply that at least one vector has orbit dense in the Hilbert space. We show that the density of the norms is not a sufficient condition to imply hypercyclicity by constructing a weighted bilateral shift that, on one hand, satisfies the orbit-density property (in the sense defined above), but, on the other hand, fails to be hypercyclic. The second major topic that we study refers to classes of operators that are not hypertransitive (or orbit-transitive) and is related to the invariant subspace problem on Hilbert space. It was shown by Jung, Ko and Pearcy in 2005 that every compact perturbation of a normal operator is not hypertransitive. We extend this result, after introducing the related notion of weak hypertransitivity, by giving a sufficient condition for an operator to belong to the class of non-weakly-hypertransitive operators. Next, we study certain 2-normal operators and their compact perturbations. In particular, we consider operators with a slow growth rate for the essential norms of their powers. Using a new idea, of accumulation of growth for each given power on a set of different orthonormal vectors, we establish that the studied operators are not hypertransitive.

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38

Stephen, Matthew A. "Spectral Theory for Bounded Operators on Hilbert Space." 2013. http://hdl.handle.net/10222/35345.

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This thesis is an exposition of spectral theory for bounded operators on Hilbert space. Detailed proofs are given for the functional calculus, the multiplication operator, and the projection-valued measure versions of the spectral theorem for self-adjoint bounded operators. These theorems are then generalized to finite sequences of self-adjoint and commuting bounded operators. Finally, normal bounded operators are discussed, as a particular case of the generalization.

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39

Hung, Ching-Nam. "The numerical range and the core of Hilbert-space operators." 2004. http://link.library.utoronto.ca/eir/EIRdetail.cfm?Resources__ID=94714&T=F.

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40

Singh, Pravin. "Polynomial approximations to functions of operators." Thesis, 1994. http://hdl.handle.net/10413/5140.

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To solve the linear equation Ax = f, where f is an element of Hilbert space H and A is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate -1 the inverse operator A by an operator V which is a polynomial in A. Using the spectral theory of bounded normal operators the problem is reduced to that of approximating a function of the real variable by polynomials of best uniform approximation. We apply two different techniques of evaluating A-1 so that the operator V is chosen either as a polynomial P (A) when P (A) approximates the n n function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn (A) approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A) satisfy three point recurrence relations, thus the approximate solution vectors P (A)f n and Q (A)f can be evaluated iteratively. We compare the procedures involving n Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite. We also show that the technique can be applied to an operator which is not selfadjoint, but close, in the sense of operator norm, to a selfadjoint operator. The iterative techniques we develop are used to solve linear systems arising from the discretization of Freedholm integral equations of the second kind. Both smooth and weakly singular kernels are considered. We show that earlier work done on the approximation of linear functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse operator and are thus special cases of a general result involving an approximation of arbitrary degree to A -1 . Thesis (Ph.D.)-University of Natal, 1994.

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41

Carter, James Michael. "Commutants of composition operators on the Hardy space of the disk." 2013. http://hdl.handle.net/1805/3659.

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Indiana University-Purdue University Indianapolis (IUPUI) The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.

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42

Ghara, Soumitra. "Decomposition of the tensor product of Hilbert modules via the jet construction and weakly homogeneous operators." Thesis, 2018. https://etd.iisc.ac.in/handle/2005/4909.

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Let ½ Cm be a bounded domain and K :£!C be a sesqui-analytic function. We show that if ®,¯ È 0 be such that the functions K® and K¯, defined on £, are non-negative definite kernels, then theMm(C) valued function K(®,¯)(z,w) :Æ K®Å¯(z,w) ³ ¡ @i¯@ j logK ¢ (z,w) ´m i , jÆ1 , z,w 2, is also a non-negative definite kernel on £. Then a realization of the Hilbert space (H,K(®,¯)) determined by the kernel K(®,¯) in terms of the tensor product (H,K®)(H,K¯) is obtained. For two reproducing kernel Hilbert modules (H,K1) and (H,K2), let An, n ¸ 0, be the submodule of the Hilbert module (H,K1)(H,K2) consisting of functions vanishing to order n on the diagonal set ¢ :Æ {(z, z) : z 2}. Setting S0 ÆA? 0 , Sn ÆAn¡1ªAn, n ¸ 1, leads to a natural decomposition of (H,K1)(H,K2) into infinite direct sum L1 nÆ0Sn. A theorem of Aronszajn shows that the module S0 is isomorphic to the push-forward of the module (H,K1K2) under the map ¶ : !£, where ¶(z) Æ (z, z), z 2 . We prove that if K1 Æ K® and K2 Æ K¯, then the module S1 is isomorphic to the push-forward of the module (H,K(®,¯)) under the map ¶. Let Möb denote the group of all biholomorphic automorphisms of the unit disc D. An operator T in B(H) is said to be weakly homogeneous if ¾(T ) µ ¯D and '(T ) is similar to T for each ' inMöb. For a sharp non-negative definite kernel K : D£D!Mk(C), we show that the multiplication operator Mz on (H,K) is weakly homogeneous if and only if for each ' in Möb, there exists a g' 2Hol(D,GLk(C)) such that the weighted composition operator Mg'C'¡1 is bounded and invertible on (H,K). We also obtain various examples and nonexamples of weakly homogeneous operators in the class FB2(D). Finally, it is shown that there exists a Möbius bounded weakly homogeneous operator which is not similar to any homogeneous operator. We also show that if K1 and K2 are two positive definite kernels on D£D such that the multiplication operators Mz on the corresponding reproducing kernel Hilbert spaces are subnormal, then the multiplication operator Mz on the Hilbert space determined by the sum K1ÅK2 need not be subnormal. This settles a recent conjecture of Gregory T. Adams, Nathan S. Feldman and Paul J.McGuire in the negative.

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43

Bhattacharjee, Monojit. "Analytic Models, Dilations, Wandering Subspaces and Inner Functions." Thesis, 2017. http://etd.iisc.ac.in/handle/2005/4241.

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This thesis concerns dilation theory, analytic models, joint invariant subspaces, reproducing kernelHilbert spaces and multipliers associated to commuting tuples of bounded linear operators on Hilbert spaces. The main contribution of this thesis is twofold: dilation and analytic model theory for n-tuples of (1) commuting contractions (in the setting of the unit polydisc), and (2) commuting row contractions (in the setting of the unit ball). On n-tuples of commuting contractions: We study analytic models of operators with some positivity assumptions and quotient modules of function Hilbert spaces over polydisc. We prove that for an m-hypercontraction T 2 C¢0 on a Hilbert space H, there exist Hilbert spaces E and E¤, and a partially isometric multiplier µ 2M ¡H2 E (D), A2 m(E¤) ¢ such that H » Æ Qµ Æ A2 m(E¤)ªµH2 E (D), and T » Æ PQµMz jQµ , where A2 m(E¤) is the E¤-valued weighted Bergman space and H2 E (D) is the E -valued Hardy space over the unit disc 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 Dn. On commuting row contractions: We study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. We prove that for a large class of analytic functional Hilbert spaces HK on the unit ball in Cn, wandering subspaces for restrictions of the multiplication tupleMz Æ (Mz1 , . . . ,Mzn ) can be described in terms of suitable HK -inner functions. We prove that, HK -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.

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44

Martin, Robert. "Bandlimited functions, curved manifolds, and self-adjoint extensions of symmetric operators." Thesis, 2008. http://hdl.handle.net/10012/3698.

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Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

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45

Thompson, Derek Allen. "Restrictions to Invariant Subspaces of Composition Operators on the Hardy Space of the Disk." 2014. http://hdl.handle.net/1805/3881.

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Indiana University-Purdue University Indianapolis (IUPUI) Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.

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46

Pal, Sourav. "Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction." Thesis, 2011. https://etd.iisc.ac.in/handle/2005/2182.

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A pair of commuting bounded operators (S, P) for which the set r = {(z 1 +z 2,z 1z 2) : |z 1| ≤1, |z 2| ≤1} C2 is a spectral set, is called a r-contraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation S –S*P = (I –P*P)½ X(I –P*P)½ where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a r-contraction. We call the unique solution, the fundamental operator of the r-contraction (S,P). As the title indicates, there are three parts of this thesis and the main role in all three parts is played by the fundamental operator. The existence of the fundamental operator allows us to explicitly construct a r-isometric dilation of a r-contraction (S,P), whereas its uniqueness guarantees the uniqueness of the minimal r-isometric dilation. The fundamental operator helps us to produce a genuine functional model for pure r-contractions. Also it leads us to a complete unitary invariant for pure r-contractions. We decipher the structures of r-isometries and r-unitaries by characterizing them in several different ways. We establish the fact that for every pure r-contraction (S,P), there is a bounded operator C with numerical radius being not greater than 1 such that S = C + C* P. When (S,P) is a r-isometry, S has the same form where P is an isometry commuting with C and C*. Also when (S,P) is a r-unitary, S has the same form too with P and C being commuting unitaries. Examples of r-contractions on reproducing kernel Hilbert spaces and their dilations are discussed.

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47

Pal, Sourav. "Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction." Thesis, 2011. http://etd.iisc.ernet.in/handle/2005/2182.

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A pair of commuting bounded operators (S, P) for which the set r = {(z 1 +z 2,z 1z 2) : |z 1| ≤1, |z 2| ≤1} C2 is a spectral set, is called a r-contraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation S –S*P = (I –P*P)½ X(I –P*P)½ where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a r-contraction. We call the unique solution, the fundamental operator of the r-contraction (S,P). As the title indicates, there are three parts of this thesis and the main role in all three parts is played by the fundamental operator. The existence of the fundamental operator allows us to explicitly construct a r-isometric dilation of a r-contraction (S,P), whereas its uniqueness guarantees the uniqueness of the minimal r-isometric dilation. The fundamental operator helps us to produce a genuine functional model for pure r-contractions. Also it leads us to a complete unitary invariant for pure r-contractions. We decipher the structures of r-isometries and r-unitaries by characterizing them in several different ways. We establish the fact that for every pure r-contraction (S,P), there is a bounded operator C with numerical radius being not greater than 1 such that S = C + C* P. When (S,P) is a r-isometry, S has the same form where P is an isometry commuting with C and C*. Also when (S,P) is a r-unitary, S has the same form too with P and C being commuting unitaries. Examples of r-contractions on reproducing kernel Hilbert spaces and their dilations are discussed.

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48

Hota, Tapan Kumar. "Subnormality and Moment Sequences." Thesis, 2012. http://etd.iisc.ac.in/handle/2005/3242.

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In this report we survey some recent developments of relationship between Hausdorﬀ moment sequences and subnormality of an unilateral weighted shift operator. Although discrete convolution of two Haudorﬀ moment sequences may not be a Hausdorﬀ moment sequence, but Hausdorﬀ convolution of two moment sequences is always a moment sequence. Observing from the Berg and Dur´an result that the multiplication operator on Is subnormal, we discuss further work on the subnormality of the multiplication operator on a reproducing kernel Hilbert space, whose kernel is a point-wise product of two diagonal positive kernels. The relationship between infinitely divisible matrices and moment sequence is discussed and some open problems are listed.

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49

Hota, Tapan Kumar. "Subnormality and Moment Sequences." Thesis, 2012. http://hdl.handle.net/2005/3242.

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In this report we survey some recent developments of relationship between Hausdorﬀ moment sequences and subnormality of an unilateral weighted shift operator. Although discrete convolution of two Haudorﬀ moment sequences may not be a Hausdorﬀ moment sequence, but Hausdorﬀ convolution of two moment sequences is always a moment sequence. Observing from the Berg and Dur´an result that the multiplication operator on Is subnormal, we discuss further work on the subnormality of the multiplication operator on a reproducing kernel Hilbert space, whose kernel is a point-wise product of two diagonal positive kernels. The relationship between infinitely divisible matrices and moment sequence is discussed and some open problems are listed.

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50

Kumar, Poornendu. "Interaction of distinguished varieties and the Nevanlinna-Pick interpolation problem in some domains." Thesis, 2023. https://etd.iisc.ac.in/handle/2005/6148.

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This thesis explores the interplay between complex geometry and operator theory, focusing on characterizing certain objects from algebraic geometry. Two concepts that have been of prime importance in recent times in the analysis of Hilbert space operators are distinguished varieties, which are a priori geometric in nature, and joint spectra, which are a priori algebraic in nature. This thesis brings them together to characterize all distinguished varieties with respect to the bidisc, more generally the polydisc and the symmetrized bidisc in terms of the joint spectrum of certain linear pencils. Some of the results are shown to refine earlier work in these directions. The binding force is provided by an operator-theoretic result, the Berger-Coburn-Lebow characterization of a tuple of commuting isometries. The thesis then turns to studying the uniqueness of solutions of the solvable NevanlinnaPick interpolation problems on the symmetrized bidisc and its connection with distinguished varieties. Several sucient conditions have been identified for a given data to have a unique solution. Moreover, for a class of solvable data on the symmetrized bidisc, there exists a distinguished variety where all solutions agree. Additionally, the thesis explores the more general concept of the determining sets.

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