Inequalities involving norm and numeri\nobreak cal radius of Hilbert space operators

The aim of this paper is to modify proximal operators in Hilbert spaces. We introduce an inter- mixed algorithm with viscosity technique to find the solution of fixed point problem of two proximal operators in a real Hilbert space, utilizing the modified proximal operators. Under some mild conditions, a strong conver- gence theorem is established for the proposed algorithm. We also apply our main result to the split feasibility problem. Finally we provide numerical examples for supporting the main result.

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 1 − g3, there exist xɛ ∈ H and a bounded linear operator S: H → H with ‖S‖ = 1 = ‖xɛ‖ such that $$\left\| {Sx_\varepsilon } \right\| = 1, \left\| {x_\varepsilon - x_0 } \right\| \leqslant \sqrt {2\varepsilon } + \sqrt[4]{{2\varepsilon }}, \left\| {S - T} \right\| \leqslant \sqrt {2\varepsilon } .$$

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

Approximating [formula omitted]-time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor type expansion

Previous article Next article Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville ExpressionsHans G. Kaper, Man Kam Kwong, and Anton ZettlHans G. Kaper, Man Kam Kwong, and Anton Zettlhttps://doi.org/10.1137/0517056PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented to characterize selfadjoint realizations of a singular Sturm–Liouville differential expression on a finite interval, where the singularities are of limit-circle type.[1] M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967xiii+144 35:6885 M. A. Naimark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968xv+352 41:7485 Google Scholar[2] N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1961xi+147 41:9015a N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. II, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1963v+218 41:9015b Google Scholar[3] H. Weyl, Üher gewöhnliche Differentialgleichungen mit Singulärituten and die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann, 68 (1910), 220–269 CrossrefGoogle Scholar[4] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 2, Oxford Univ. Press, Cambridge, 1962 CrossrefGoogle ScholarKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extension Previous article Next article FiguresRelatedReferencesCited ByDetails Friedrichs extensions of a class of singular Hamiltonian systemsJournal of Differential Equations, Vol. 293 | 1 Aug 2021 Cross Ref On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from belowJournal of Differential Equations, Vol. 269, No. 9 | 1 Oct 2020 Cross Ref Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indicesJournal of Mathematical Analysis and Applications, Vol. 461, No. 2 | 1 May 2018 Cross Ref On Properties of the Legendre Differential ExpressionResults in Mathematics, Vol. 42, No. 1-2 | 16 May 2013 Cross Ref The Friedrichs Extension of Singular Differential OperatorsJournal of Differential Equations, Vol. 160, No. 2 | 1 Jan 2000 Cross Ref Density, spectral theory and homoclinics for singular Sturm-Liouville systemsJournal of Computational and Applied Mathematics, Vol. 52, No. 1-3 | 1 Jul 1994 Cross Ref Singular Second-Order Operators: The Maximal and Minimal Operators, and Selfadjoint Operators in BetweenMojdeh Hajmirzaahmad and Allan M. KrallSIAM Review, Vol. 34, No. 4 | 2 August 2006AbstractPDF (1871 KB)Eigenvalue and eigenfunction computations for Sturm-Liouville problemsACM Transactions on Mathematical Software, Vol. 17, No. 4 | 1 Dec 1991 Cross Ref Volume 17, Issue 4| 1986SIAM Journal on Mathematical Analysis761-1035 History Submitted:17 August 1984Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extensionMSC codes34B2547E05PDF Download Article & Publication DataArticle DOI:10.1137/0517056Article page range:pp. 772-777ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth In specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space H has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator. In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space H on which the transfer operator T acts will then be an L 2 space on S, or a Hilbert space defined from an energy-quadratic form. This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in H; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.

By the use of inequalities for nonnegative Hermitian forms some new inequalities for numerical radius of bounded linear operators in complex Hilbert spaces are established.

Through our study of soft-liner operator in Hilbert space we discovered a new type, which is a Novel Type of Extended Soft quasinormal for Normal Operator in Hilbert Space In this paper we present some definitions and characteristics related to soft Ғ ̂- quasi Ⱪ ̂-normal operator Furthermore, we explain the different properties of this operator as well as direct addition and effective multiplication.

In this article, we suggest and study a new inertial algorithm combining inertial extrapolation, S‐iteration process and viscosity approximation method for computing a common solution of an equilibrium problem and a family of nonexpansive operators in real Hilbert space. We provide a strong convergence result of the suggested algorithm under some simple assumptions on control sequences. Further, we compare our suggested algorithm with some existing well‐known algorithms by an analytical example. Some prominent recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.

In this article, we consider a split common fixed point and null point problem which includes the split common fixed point problem, the split common null problem and other problems related to the fixed point problem and the null point problem. We introduce an algorithm for studying the split common fixed point and null problem for demicontractive operators and maximal monotone operators in real Hilbert spaces. We establish a strong convergence result under some suitable conditions and reduce our main result to above-mentioned problems. Moreover, we also apply our main results to the split equilibrium problem. Finally, we give numerical results to demonstrate the convergence of our algorithms.

A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given.

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

The concepts of bit and qubit, the fundamental units of information in classical and quantum computing respectively, are introduced. We discuss features of the binary number system, linear vector, and Hilbert spaces. We learn how to manipulate qubits and introduce Dirac’s bra-ket formalism to facilitate operations in Hilbert space. Scalar, direct and outer products of bra-kets in multi-qubit systems are introduced. We define Hermitian and unitary operators in a 2n-dimensional Hilbert space and use the bra-ket formalism to construct them. I summarize the foundational postulates of quantum mechanics, as espoused by the Copenhagen interpretation, for a finite set of qubits.

Remark on the decay for damped string and beam equations