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model subspaces techniques to study Fourier expansions in L² spaces associated to singular measures

JOURNAL OF FUNCTIONAL ANALYSIS 2020年第10期279卷 108725-108725页

作者： Antezana, Jorge Guadalupe Garcia, Maria IAM CONICET Inst Argentino Matemat Alberto P Calderon Buenos Aires DF Argentina Univ Nacl La Plata Dept Matemat Fac Ciencias Exactas La Plata Argentina

Let mu be a probability measure on T that is singular with respect to the Haar measure. In this paper we study Fourier expansions in L-2(T, mu) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials {z(n)}(n is an element of N) is effective in L-2(T, mu), it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame P-phi(z(n)) for the model subspace H(phi) = H-2 circle minus phi H-2, where P-phi, is the orthogonal projection from the Hardy space H-2 onto H(phi). The study of Fourier expansions in L-2(T, mu) also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures it which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization. (C) 2020 Elsevier Inc. All rights reserved.

关键词： model subspaces Kaczmarz algorithm Fourier expansions Parseval frames

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Interpolating by Functions from model subspaces in H¹

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2018年第4期90卷 42-42页

作者： Dyakonov, Konstantin M. Univ Barcelona IMUB Dept Matemat & Informat Gran Via 585 E-08007 Barcelona Spain Univ Barcelona BGSMath Gran Via 585 E-08007 Barcelona Spain ICREA Pg Lluis Co 23 Barcelona 08010 Spain

Given an interpolating Blaschke product B with zeros , we seek to characterize the sequences of values for which the interpolation problem f( aj) = wj ( j = 1, 2,...) can be solved with a function f from the model sub... 详细信息

关键词： Inner function Interpolating Blaschke product Hardy spaces model subspaces

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TRUNCATED TOEPLITZ OPERATORS OF FINITE RANK

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2014年第4期142卷 1301-1313页

作者： Bessonov, R. V. Russian Acad Sci Math Inst St Petersburg Dept Steklov St Petersburg 191023 Russia

We give a complete description of the finite-rank truncated Toeplitz operators.

关键词： Toeplitz operators model subspaces finite-rank

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Unitary Perturbations of Compressed n-Dimensional Shifts

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COMPLEX ANALYSIS AND OPERATOR THEORY 2013年第4期7卷 765-799页

作者： Martin, R. T. W. Univ Cape Town Dept Math & Appl Math ZA-7925 Cape Town South Africa

Given any contractive matrix-valued analytic function on the unit disc , we construct a -parameter family of unitary operators which correspond to in a natural way. These operators are unitarily equivalent to higher dimensional analogues of Clark's unitary perturbations, a family of unitary operators associated to any self-map of the unit disc. Clark's unitary perturbations were introduced in a seminal paper of Clark which has inspired the study of what are now called Aleksandrov-Clark measures. Our higher dimensional analogues of Clark's unitary perturbations are applied to obtain matrix-generalizations of several classical results on the Aleksandrov-Clark measures associated to any holomorphic self-map of the unit disc. In particular we establish a matrix-valued Aleksandrov disintegration theorem for the Aleksandrov-Clark measures associated with matrix-valued contractive analytic functions , and, by following results of Clark and Fricain in the scalar case, we provide a necessary and sufficient condition for the de Branges-Rovnyak space associated with to contain a total orthogonal set of point evaluation vectors.

关键词： Hardy space model subspaces Aleksandrov disintegration theorem Clark's unitary perturbations Aleksandrov-Clark measures Matrix-analytic functions Symmetric/isometric linear transformations

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Semigroups of Partial Isometries and Symmetric Operators

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2011年第2期70卷 205-226页

作者： Martin, R. T. W. Univ Cape Town Dept Math & Appl Math ZA-7925 Cape Town South Africa

Let {V(t)vertical bar t is an element of [0, infinity)} be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let V(-t) : = V*(t) for l is an element of [0, infinity). It is shown that if V(t) is a partial isometry for all t is an element of [-t(0) , t(0)], t(0) > 0, then the pointwise two-sided derivative of V(t) exists on a dense domain of vectors. This derivative B is necessarily a densely defined symmetric operator. This result can be viewed as a generalization of Stone's theorem for one-parameter strongly continuous unitary groups, and is used to establish sufficient conditions for a self-adjoint operator on a Hilbert space K to have a symmetric restriction to a dense linear manifold of a closed subspace H subset of K. A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the independent variable in K := circle plus(n)(i=1) L-2 (R) to certain model subspaces of the Hardy space of n-compenent vector valued functions which are analytic in the upper half plane is presented.

关键词： Symmetric operators Hardy space model subspaces strongly continuous one parameter operator semigroups partial isometries

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Aleksandrov-Clark Theory for Drury-Arveson Space

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2018年第4期90卷 1-42页

作者： Jury, M. T. Martin, R. T. W. Univ Florida Gainesville FL 32611 USA Univ Cape Town Cape Town South Africa

Recent work has demonstrated that Clark's theory of unitary perturbations of the backward shift on a deBranges-Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury-Arveson multiplier algebra and the Aleksandrov-Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system , where is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury-Arveson space by establishing the existence of a canonical 'tight' extension of any Aleksandrov-Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov-Clark maps.

关键词： Hardy space Drury-Arveson space model subspaces deBranges-Rovnyak spaces Multiplier algebra Non-commutative disk algebra Aleksandrov-Clark measures

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Composition of Inner Functions

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CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 2014年第2期66卷 387-399页

作者： Mashreghi, J. Shabankhah, M. Univ Laval Dept Math & Stat Quebec City PQ G1K 7P4 Canada Univ Tehran Coll Engn Dept Engn Sci Tehran 111554563 Iran

We study the image of the model subspace K theta under the composition operator C-phi, where phi and theta are inner functions, and find the smallest model subspace which contains the linear manifold C phi K theta. Then we characterize the case when Cc, maps K-theta into itself. This case leads to the study of the inner functions cc and theta such that the composition psi circle phi is a divisor of 0 in the family of inner functions.

关键词： composition operators inner functions Blaschke products model subspaces

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