The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Henrik Winkler - One of the best experts on this subject based on the ideXlab platform.
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Sesquilinear forms corresponding to a non semibounded sturm liouville operator
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.
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Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.
Camillo Trapani - One of the best experts on this subject based on the ideXlab platform.
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Biorthogonal vectors, Sesquilinear forms, and some physical operators
Journal of Mathematical Physics, 2018Co-Authors: Fabio Bagarello, Hiroshi Inoue, Camillo TrapaniAbstract:Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and Sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular, we discuss what happens when they forms two D-quasi-bases.
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Representation Theorems for Solvable Sesquilinear Forms
Integral Equations and Operator Theory, 2017Co-Authors: Rosario Corso, Camillo TrapaniAbstract:New results are added to the paper (Di Bella and Trapani in J Math Anal Appl 451:64–83, 2017 ) about q-closed and solvable Sesquilinear forms. The structure of the Banach space $$\mathcal {D}[||\cdot ||_\Omega ]$$ D [ | | · | | Ω ] defined on the domain $$\mathcal {D}$$ D of a q-closed Sesquilinear form $$\Omega $$ Ω is unique up to isomorphism, and the adjoint of a Sesquilinear form has the same property of q-closure or of solvability. The operator associated to a solvable Sesquilinear form is the greatest which represents the form and it is self-adjoint if, and only if, the form is symmetric. We give more criteria of solvability for q-closed Sesquilinear forms. Some of these criteria are related to the numerical range, and we analyse in particular the forms which are solvable with respect to inner products. The theory of solvable Sesquilinear forms generalises those of many known Sesquilinear forms in literature.
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Representation Theorems for Solvable Sesquilinear Forms
arXiv: Functional Analysis, 2017Co-Authors: Rosario Corso, Camillo TrapaniAbstract:New results are added to the paper [4] about q-closed and solvable Sesquilinear forms. The structure of the Banach space $\mathcal{D}[||\cdot||_\Omega]$ defined on the domain $\mathcal{D}$ of a q-closed Sesquilinear form $\Omega$ is unique up to isomorphism, and the adjoint of a Sesquilinear form has the same property of q-closure or of solvability. The operator associated to a solvable Sesquilinear form is the greatest which represents the form and it is self-adjoint if, and only if, the form is symmetric. We give more criteria of solvability for q-closed Sesquilinear forms. Some of these criteria are related to the numerical range, and we analyse in particular the forms which are solvable with respect to inner products. The theory of solvable Sesquilinear forms generalises those of many known Sesquilinear forms in literature.
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Some representation theorems for Sesquilinear forms
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Salvatore Di Bella, Camillo TrapaniAbstract:Abstract The possibility of getting a Radon–Nikodym type theorem and a Lebesgue-like decomposition for a not necessarily positive Sesquilinear Ω form defined on a vector space D , with respect to a given positive form Θ defined on D , is explored. The main result consists in showing that a Sesquilinear form Ω is Θ-regular, in the sense that it has a Radon–Nikodym type representation, if and only if it satisfies a sort Cauchy–Schwarz inequality whose right hand side is implemented by a positive Sesquilinear form which is Θ-absolutely continuous. In the particular case where Θ is an inner product in D , this class of Sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace D of Hilbert space H we give a sufficient condition for the equality Ω ( ξ , η ) = 〈 T ξ | η 〉 , with T a closable operator, to hold on a dense subspace of H .
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Some Representation Theorems for Sesquilinear Forms
arXiv: Functional Analysis, 2016Co-Authors: Salvatore Di Bella, Camillo TrapaniAbstract:The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive Sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a Sesquilinear form $\Omega$ is $\Theta$-regular, in the sense that it has a Radon-Nikodym type representation, if and only if it satisfies a sort Cauchy-Schwarz inequality whose right hand side is implemented by a positive Sesquilinear form which is $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of Sesquilinear form covers all standard examples. In the case of a form defined on a dense subspace $\mathcal D$ of Hilbert space $\mathcal H$ we give a sufficient condition for the equality $\Omega(\xi,\eta)=\langle{T\xi}|{\eta}\rangle$, with $T$ a closable operator, to hold on a dense subspace of $\mathcal H$.
Andreas Fleige - One of the best experts on this subject based on the ideXlab platform.
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Sesquilinear forms corresponding to a non semibounded sturm liouville operator
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.
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Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.
Mohammad Sal Moslehian - One of the best experts on this subject based on the ideXlab platform.
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B-spline interpolation problem in Hilbert C*-modules
arXiv: Operator Algebras, 2020Co-Authors: Rasoul Eskandari, Michael Frank, Vladimir Manuilov, Mohammad Sal MoslehianAbstract:We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued Sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued Sesquilinear form on a Hilbert $C^*$-module to a Sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued Sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.
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Reverse Cauchy--Schwarz inequalities for positive C*-valued Sesquilinear forms
arXiv: Operator Algebras, 2009Co-Authors: Mohammad Sal Moslehian, Lars-erik PerssonAbstract:We prove two new reverse Cauchy--Schwarz inequalities of additive and multiplicative types in a space equipped with a positive Sesquilinear form with values in a C*-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy--Schwarz inequality due to G Polya and G. Szego.
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REVERSE CAUCHY-SCHWARZ INEQUALITIES FOR POSITIVE C*-VALUED Sesquilinear FORMS
Mathematical Inequalities & Applications, 2009Co-Authors: Mohammad Sal Moslehian, Lars-erik PerssonAbstract:We prove two new reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive Sesquilinear form with values in a C*-algebra. We apply our results to ...
Henk De Snoo - One of the best experts on this subject based on the ideXlab platform.
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Sesquilinear forms corresponding to a non semibounded sturm liouville operator
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.
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Sesquilinear forms corresponding to a non-semibounded Sturm–Liouville operator
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010Co-Authors: Andreas Fleige, Seppo Hassi, Henk De Snoo, Henrik WinklerAbstract:Let −DpD be a differential operator on the compact interval [−b, b] whose leading coefficient is positive on (0, b] and negative on [−b, 0), with fixed, separated, self-adjoint boundary conditions at b and −b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded Sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded Sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal Sesquilinear form. Moreover, among all closed forms associated with the self-adjoint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Kreĭn–Feller operators with a single concentrated mass.