Linear Operators In Hilbert Spaces Research Articles

Ill-posed problems, [formula omitted]-semigroups and the Showalter regularization

We consider linear operator equations A x = y in Banach space, where − A generates a bounded C 0 -semigroup. If the range of A is not closed, this equation is ill-posed. It can be solved approximately by Showalter's method, which is known for bounded linear operators in Hilbert space. We show convergence and estimate convergence rates in Banach space.

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION

Let A denote the class of bounded linear Hilbert space operators with the property that jA 2 j ‚ jAj 2 . In this paper we show that A-operators are flnitely ascensive and that, for non-zero operators A and B, A › B is in A if and only if A and B are in A. Also, it is shown that if A is an operator such that p(A) is in A for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.

Discretization strategy for linear ill-posed problems in variable Hilbert scales

The authors study the regularization of projection methods for solving linear ill-posedproblems with compact and injective linear operators in Hilbert spaces. The smoothness ofthe unknown solution is given in terms of general source conditions, such that theframework of variable Hilbert scales is suitable.The structure of the error is analysed in terms of the noise level, the regularizationparameter and as a function of other parameters, driving the discretization. As aresult, a strategy is proposed which automatically adapts to the unknown sourcecondition, uniformly for certain classes, and provides the optimal order of accuracy.

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space†

The conjugate gradient method for nonsymmetric linear operators in Hilbert space is investigated. Conditions on the coincidence of the full and truncated versions, known from the finite-dimensional case, are extended to the Hilbert space setting. The focus is on preconditioning by the symmetric part of the operator, in which case estimates are given for the resulting condition number. An important motivation for this study is given by differential operators, for which the obtained estimates yield mesh independent conditioning properties of the full CG method, and are in fact achieved by the simpler truncated version. †Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied Mathematics.

The representation and approximation of the W-weighted Drazin inverse of linear operators in Hilbert space

We present a unified representation theorem for the W-weighted Drazin inverse of linear operators in Hilbert space and a general error bound. Five specific expressions, computational procedures, and their error bounds for the W-weighted Drazin inverse are uniformly derived from the unified representation theorem.

Invertibility of the Poiseuille Linearization for Stationary Two-Dimensional Channel Flows: Symmetric Case

It was shown in the first part that in Sobolev spaces that incorporate symmetry requirements, the linearization at the Poiseuille solution of the Navier-Stokes system is an isomorphism regardless of the flux of the solution. This paper is devoted to the same question when the symmetry assumptions are removed. While the linearization remains one-to-one with dense range for all the fluxes, we are only able to prove its invertibility up to an upper bound for the flux. Although results of this type are known, the upper bound found here is more than 50 times larger than the currently known one. Some of the tools developed to derive this new bound have independent interest. This should be especially true of a simple criterion for the surjectivity of some linear operators in Hilbert space derived from Bessel‚s inequality. The applications to the Navier-Stokes system are similar to those in the symmetric case.

The representation and approximation of the Drazin inverse of a linear operator in Hilbert space

We present a unified representation theorem for the Drazin inverse of linear operators in Hilbert space and a general error bound. Five specific expressions, computational procedures, and their error bounds for the Drazin inverse are uniformly derived from the unified representation theorem.

Stabilisation par viscosité de sous-maille pour l'approximation de Galerkin des opérateurs linéaires monotones

This paper presents a stabilized Galerkin technique for approximating monotone linear operators in Hilbert spaces. The key idea consists in introducing an approximation space that is broken up into resolved and subgrid scales so that the bilinear form associated with the problem satisfies a uniform inf-sup condition with respect to this decomposition. An optimal Galerkin approximation is obtained by introducing an artificial diffusion on the subgrid scales.

On the Continuity of Generalized Inverses of Linear Operators in Hilbert Spaces

On the Continuity of Generalized Inverses of Linear Operators in Hilbert Spaces

On the continuity of generalized inverses of linear operators in Hilbert spaces

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. We study the continuity problem of the generalized inverse of T and related least squares solutions to the operator equation Tx = y.

Perturbation of Generalized Inverses of Linear Operators in Hilbert Spaces

LetXandYbe Hilbert spaces, and letT:X→Ybe a bounded linear operator with closed range. In this paper we present some new results of the perturbation analysis on the generalized inverse ofTand the related least squares solution to the operator equationTx=y.

The Wigner transform and the Wigner-poisson system

Abstract The Wigner method in (quantum) statistical mechanics was introduced in 19321 as an alternative to the standard formulation2 which involves computation of eigenvalues and eigenvectors (and traces) of linear operators in Hilbert spaces. The Wigner approach uses only integrations over phase space –R6N in the case of an N-particle system–of ordinary functions f:R6n → C.

Optimal linear randomized methods for linear operators in Hilbert spaces

Basic questions of information-based complexity are strongly related to n-widths and s-numbers. In this paper we study Monte Carlo methods or randomized methods for linear operators. Similar as in the worst case, Mathé defined linear stochastic n-widths. Our main result is the characterization of these widths in the case of linear operators in Hilbert spaces.

Singular perturbation of a nonlinear time-dependent problem in hilbert space

The convergence of the solution of the nonlinear singular perturbation problem Ud'J + aU´ { + e2AnUe £ + g(AUe e) + h(t, A)]f(A)Ue e = 0, C/e e(0) = x0, f/´ €'(0) = zi for ϵ > 0 to the solution of the degenerate problem (ϵ - 0) is studied. Here a [math001] 0 is a constant and A is a (generally unbounded) self- adjoint linear operator in Hilbert space. The problem is nonlocal, nonlinear, and time dependent because of g(‖AU‖)f(A)U and h(t,A). The existence of solutions is proven simultaneously with asymptotic behaviour.

Spectral condition numbers for defective elements of linear operators in hilbert spaces

In this paper we generalise matrix eigenelements results on condition numbers to linear bounded operators in complex Hilbert infinite dimensional spaces. Suitable nornls are introduced in order to interpret these results geometrically.

The strong Hamburger moment problem and self-adjoint operators in Hilbert space

The strong Hamburger moment problem is solved using spectral theory of (unbounded) self-adjoint linear operators in Hilbert space. By means of orthogonal projections of finite dimensional range certain two-point Padé approximants to the moment generating function are obtained.

The strong Hamburger moment problem and self-adjoint operators in Hilbert space

The strong Hamburger moment problem is solved using spectral theory of (unbounded) self-adjoint linear operators in Hilbert space. By means of orthogonal projections of finite dimensional range certain two-point Padé approximants to the moment generating function are obtained.

Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville Expressions

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

Partial *-Algebras of Closed Linear Operators In Hilbert Space (Addendum/Erratum to Vol.21, No. 1, 1985, 205-236)

Partial *-Algebras of Closed Linear Operators In Hilbert Space (Addendum/Erratum to Vol.21, No. 1, 1985, 205-236)

Partial *-Algebras of Closed Linear Operators In Hilbert Space

Given a dense domain \mathcal D of a Hilbert space, we consider the class of all closed operators which, together with their adjoint, have \mathcal D in their domain. A partial * -algebra of operators on \mathcal D is a subset of that class, stable under suitable operations of involution, addition and multiplication, the latter when it is defined. We present two types of such objects and study their properties, both algebraic and topological.