Linear maps preserving G-unitary 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

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.

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. .

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

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

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

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.

K-POTENT PRESERVING LINEAR MAPS

The following theorem is proved: Suppose H is a complex Hilbert space, and T : H → H T:H \to H is a monotonic, nonexpansive operator on H , and f ∈ H f \in H . Define S : H → H S:H \to H by S u = − T u + f Su = - Tu + f for all u ∈ H u \in H . Suppose 0 ⩽ t n ⩽ 1 0 \leqslant {t_n} \leqslant 1 for all n = 1 , 2 , 3 , … , n = 1,2,3, \ldots , and Σ n = 1 ∞ t n ( 1 − t n ) \Sigma _{n = 1}^\infty \;{t_n}(1 - {t_n}) diverges. Then the iterative process V n + 1 = ( 1 − t n ) V n + t n S V n {V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n} converges to the unique solution u = p u = p of the equation u + T u = f u + Tu = f .

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).