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

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.

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. .

Let B(H) be the space of all bounded linear operators on a complex Hilbert space H, and let A1,...,An ∈ B(H). In this article, we obtain new upper bounds for n P j=1 A j . Moreover, we establish and generalize inequalities for the operator norm of sums of bounded linear operators in Hilbert spaces.

Comparing the ill-posedness for linear operators in Hilbert spaces

Analytic problems and results in the theory of linear operators in Hilbert space (abstract)

Summation formulae are used in the theory of elastic stability so that approximate estimates of the critical load factors of a complex problem are obtained by combining the load factors of subproblems in different ways. If the critical load factors are directly added, then the formula is called a Southwell type formula. If the reciprocals of the critical load factors are added, then the formula is called a Dunkerley type formula. The practical advantage of the summation formulae is that, for the subproblems, there are usually solutions available, or it is easy to determine them while, for the original problem the solution would be difficult to obtain. In this paper we will present the main theorems and formulae (Southwell theorem, Dunkerley theorem, Foppl-Papkovich theorem, Kollar conjecture, Melan theorem), and show, where possible, the conditions under which the results are on the safe side. The proofs are given in mathematical way, based on the relationships for eigenvalues of linear operators in Hilbert space. The theoretical results are illustrated by examples.

This work presents some basic principles concerned with two general settings for stable approximating orthogonal generalized inverses in Hilbert spaces. In these two settings, perfect convergence and perfect uniform convergence are two important concepts for making theoretical or numerical analysis. Our basic principles characterize the two important concepts by some necessary conditions: stability with other compensated conditions. These basic principles include as simple corollaries some well-known results.

By N. I. Akhiezer and I. M. Glazman; translated by E. R. Dawson; edited by W. N. Everitt: pp. 312 and 218. £38.50 and £29.50. (Pitman Books Ltd‥ 1981.) (New translation of the third Russian edition)

Classes of Linear Operators in Hilbert Space

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

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

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional case is also considered. In this setting, the convergence is shown to be of exponential type. The method is also tested for the computation of a generic fractional power.

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

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.