The Stieltjes Moment Problem

1981 ◽  
Vol 24 (3) ◽  
pp. 279-282
Author(s):  
G. Klambauer
Keyword(s):  
Hilbert Space ◽  
Moment Problem ◽  
Symmetric Operator ◽  
Extension Theorem ◽  
Spectral Theorem ◽  
Friedrichs Extension ◽  
Stieltjes Moment Problem ◽  
Operator Version ◽  
Adjoint Extension ◽  
Adjoint Operators

We shall apply the spectral theorem for self adjoint operators in Hilbert space to study an operator version of the Stieltjes moment problem [1]. In the course of the work we shall make use of the Friedrichs extension theorem which states that any non-negative symmetric operator in Hilbert space has a non-negative self adjoint extension.

On the Right Hamiltonian for Singular Perturbations: General Theory

1997 ◽  
Vol 09 (05) ◽  
pp. 609-633 ◽  
Author(s):  
Hagen Neidhardt ◽  
Valentin Zagrebnov
Keyword(s):  
General Theory ◽  
Symmetric Operator ◽  
Stability Domain ◽  
Bounded Operators ◽  
Friedrichs Extension ◽  
Abstract Problem ◽  
Dense Domain ◽  
Adjoint Extension ◽  
The Right ◽  
Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

scholarly journals Application of Localization to the Multivariate Moment Problem

2014 ◽  
Vol 115 (2) ◽  
pp. 269 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Hilbert Space ◽  
Moment Problem ◽  
Existence And Uniqueness ◽  
Positive Semidefinite ◽  
Linear Functionals ◽  
Localization Technique ◽  
Adjoint Operators

It is explained how the localization technique introduced by the author in [19] leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of $\mathsf{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$. It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure $\mu$ and density of $\mathsf{C}[\underline{x}]$ in $\mathscr{L}^s(\mu)$ and, at the same time, to give new proofs of old results of Fuglede [11], Nussbaum [21], Petersen [22] and Schmüdgen [27], results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.

scholarly journals Self-Adjoint Extensions with Friedrichs Lower Bound

2020 ◽  
Vol 14 (7) ◽  
Author(s):  
Matteo Gallone ◽  
Alessandro Michelangeli
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Symmetric Operator ◽  
Simple Criterion ◽  
Abstract Result ◽  
Friedrichs Extension ◽  
Lower Semi

Abstract We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

A Spectral Theorem for Hermitian Operators of Meromorphic Type on Banach Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 703-708
Author(s):  
T. Owusu-Ansah
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Adjoint Operator ◽  
Spectral Projection ◽  
Spectral Theorem ◽  
Adjoint Operators

It is well known that if T is a compact self-adjoint operator on a Hilbert space whose distinct non-zero eigenvalues {λn} are arranged so that |λn|≥|λn+1| for n = 1, 2…. and if En in the spectral projection corresponding to λn, then with convergence in the uniform operator topology. With the generalisation of self-adjoint operators on Hilbert spaces to Hermitian operators on Banach spaces by Vidav and Lumer, Bonsall gave a partial analogue of this result for Banach spaces when he proved the following theorem.

Operator m-convex functions

2018 ◽  
Vol 25 (1) ◽  
pp. 93-107
Author(s):  
Jamal Rooin ◽  
Akram Alikhani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Convex Functions ◽  
Weight Functions ◽  
Operator Inequality ◽  
Jensen Inequality ◽  
Hilbert Space Operators ◽  
Operator Version ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Comprehensive Study

AbstractThe aim of this paper is to present a comprehensive study of operatorm-convex functions. Let{m\in[0,1]}, and{J=[0,b]}for some{b\in\mathbb{R}}or{J=[0,\infty)}. A continuous function{\varphi\colon J\to\mathbb{R}}is called operatorm-convex if for any{t\in[0,1]}and any self-adjoint operators{A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained inJ, we have{\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuousm-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operatorm-convexity, we extend the Choi–Davis–Jensen inequality for operatorm-convex functions. We also present an operator version of the Jensen–Mercer inequality form-convex functions and generalize this inequality for operatorm-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operatorm-convex functions, we construct them-Jensen operator functional and obtain an upper bound for it.

scholarly journals Symmetric multiparameter problems and deficiency index theory

1988 ◽  
Vol 31 (3) ◽  
pp. 481-488 ◽  
Author(s):  
Patrick J. Browne ◽  
Hamlet Isaev
Keyword(s):  
Hilbert Space ◽  
Half Plane ◽  
Classical Result ◽  
Symmetric Operator ◽  
Null Space ◽  
Index Theory ◽  
Lower Half ◽  
Deficiency Index ◽  
Lower Half Plane ◽  
Adjoint Extension

In this article we study the multiparameter generalization of standard deficiency index theory. A classical result in this area states that if T is a symmetric operator in a Hilbert space then the dimension of the null space of T*−λI, λ∈ℂ, is constant for λ belonging to the upper (or lower) half-plane and further, when these two constants are equal, T admits a self-adjoint extension.

scholarly journals A polynomial iteration for the spectral family of an operator

1963 ◽  
Vol 6 (2) ◽  
pp. 65-69
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Finite Dimension ◽  
Symmetric Matrix ◽  
Complete Solution ◽  
Spectral Theorem ◽  
Natural Question ◽  
Eigenvalues And Eigenvectors ◽  
Strong Limit ◽  
Spectral Family

Let T be a bounded symmetric operator in a Hilbert space H. According to the spectral theorem, T can be expressed as an integral in terms of its spectral family {Eλ}, each Eλ being a certain projection which is known to be the strong limit of some sequence of polynomials in T. It is a natural question to ask for an explicit sequence of polynomials in T that converges strongly to Eλ. So far as the author knows, no complete solution of this problem has been given even when H has finite dimension, i.e. when T is a finite symmetric matrix. Since a knowledge of the spectral family {Eλ} of a finite symmetric matrix carries with it a knowledge of the eigenvalues and eigenvectors, a solution of the problem may have some practical value.

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