A Gaussian Method for the Square Root of Accretive Operators

Convergence of composite iterative methods for finding zeros of accretive operators

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.

Remark on the decay for damped string and beam equations

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.

We investigate a class of new <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with <svg style="vertical-align:-2.3205pt;width:41.25px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 41.25 15.0875" width="41.25" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,24.728,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,31.425,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,35.301,12.138)"><use xlink:href="#x29"/></g> </svg>-cocoercive operators to the <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M4" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators and prove that the resolvent operator of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M5" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M6" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M7" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

We obtained sharp "multiplicative" and sharp "additive" inequalities of Hardy-Littlewood-Polya type for operators in Hilbert space.

Some new inequalities for the Cebysev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are given. 2010 Mathematics Subject Classifications. 47A63; 47A99. .

We consider a two-point boundary value problem for the equation $u_{tt} - A(t)u = f(t,u)$ where, for each t, $A(t)$ is an unbounded operator in Hilbert space whose domain varies with t. We establish the existence (but not the uniqueness) of “weak” solutions of this problem by using finite difference methods. When $A(t)$ represents a differential operator in the space variables, the method of proof can be converted into a convergent numerical procedure.

Some new inequalities of Grüss' type for functions of selfadjoint operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Several examples for particular functions of interest are provided as well. 2000 Mathematics Subject Classification. 47A63, 47A99

\( \newcommand\norm[1]{\left\lVert#1\right\rVert}\newcommand\normx[1]{\left\Vert#1\right\Vert} \)In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping $f$$$\norm{(1\otimes B-A\otimes 1)^{-1}[\operatorname{exp}(1\otimes B)-\operatorname{exp}(A\otimes 1)]- \operatorname{exp}\left(\frac{A\otimes 1+1\otimes B}{2}\right)}$$$$\leqslant \norm{1\otimes B-A\otimes 1}^{2}\frac{\norm{f''}_{I,+\infty}}{24}.$$

Stability theory for absolutely continuous parts of commutative m-tuples of self-adjoint operators in complex Hilbert spaces

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.