Unbounded Linear Operators in Hilbert Spaces

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. .

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.

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.

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)

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.

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.

Classes of Linear Operators in Hilbert Space

Remark on the decay for damped string and beam equations

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

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).

The monogenic functional calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator \(\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})\) is associated with polynomial \(z\mapsto z_{1}z_{2}\) in two variables by the monogenic functional calculus. The same formula applies just when the spectrum \(\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})\) of a finite linear combination \(\xi _{1}A_{1} +\xi _{2}A_{2}\) of A 1 and A 2 is a subset of the real numbers for any \(\xi _{1},\xi _{2} \in \mathbb{R}\). The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system \((A_{1},\ldots ,A_{n})\) of closed unbounded operators such that the spectrum \(\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )\) of the closed operator \(\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\) is contained in a two-sided sector in \(\mathbb{C}\) for almost all \(\xi \in \mathbb{R}^{n}\). The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system \(D_{\Sigma } = i(D_{1},\ldots ,D_{n})\) of differentiating operators on a Lipschitz surface \(\Sigma \) so that an H ∞-functional calculus \(f\mapsto f(D_{\Sigma })\) is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to \(\Sigma \).KeywordsBounded Linear OperatorClifford AlgebraFunctional CalculusSelfadjoint OperatorMonogenic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

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.