Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space

2009 ◽  
Vol 61 (10) ◽  
pp. 1533-1540
Author(s):  
V. F. Babenko ◽  
R. O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Type Inequality ◽  
Adjoint Operators

scholarly journals Some inequalities for P-class functions

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4555-4566
Author(s):  
Ismail Nikoufar ◽  
Davuod Saeedi
Keyword(s):  
Hilbert Space ◽  
Upper Bound ◽  
Type Inequality ◽  
Operator Version ◽  
Adjoint Operators

In this paper, we provide some inequalities for P-class functions and self-adjoint operators on a Hilbert space including an operator version of the Jensen?s inequality and the Hermite-Hadamard?s type inequality. We improve the H?lder-MacCarthy inequality by providing an upper bound. Some refinements of the Jensen type inequality for P-class functions will be of interest.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

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.

The Spectra of Semi-Normal Singular Integral Operators

1970 ◽  
Vol 22 (1) ◽  
pp. 134-150 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Principal Value ◽  
Image Position ◽  
Adjoint Operators

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

scholarly journals Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 20 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Difference Equation ◽  
Difference Equations ◽  
Lyapunov Equation ◽  
Stability Test ◽  
Norm Estimates ◽  
Point Estimates ◽  
Adjoint Operators ◽  
Solution Estimates

The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

Improved Young and Heinz operator inequalities for unitarily invariant norms

2020 ◽  
Vol 70 (2) ◽  
pp. 453-466
Author(s):  
A. Beiranvand ◽  
Amir Ghasem Ghazanfari
Keyword(s):  
Hilbert Space ◽  
Type Inequality ◽  
Young Inequality ◽  
Unitarily Invariant Norm ◽  
Operator Inequalities ◽  
Schmidt Norm ◽  
Unitarily Invariant Norms ◽  
Invariant Norm ◽  
Kantorovich Constant ◽  
New Inequalities

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn(ℂ): $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r0 = min{ν, 1 – ν}.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

Entropy of dynamical systems and perturbations of operators

1991 ◽  
Vol 11 (4) ◽  
pp. 779-786 ◽  
Author(s):  
Dan Voiculescu
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Hausdorff Measure ◽  
Spectral Measure ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
Hilbert Space Operators ◽  
Neumann Class ◽  
Adjoint Operators

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

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