Some results, concerning the variation of eigenvalues, when these depend on a real parameter

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Sequences of [formula omitted]-term deviations in Hilbert space

The purpose of this paper is to compute the filtering equations when the state process, given as the solution of a stochastic differential equation on an infinite dimensional Hilbert space, is observed through a counting observation.

The purpose of this paper is to compute the risk-sensitive filtering equations when the state process, given as the solution of a stochastic differential equation on an infinite dimensional Hilbert space, is observed through a counting observation.

In this paper we consider an augmented Lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a Hilbert space, an inequality constraint whose image space is finite dimensional, and an affine inequality constraint whose image space is in an infinite dimensional Hilbert space. We obtain local convergence of this method without imposing strict complementarity conditions when the equality, as well as the inequality constraint with finite dimensional image space are augmented. To the author's knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional.

In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable - and as a consequence they describe real degrees of freedom which display a form of "unsharpness" or "fuzzyness". For example, there are indications that this type of operators could arise with the description of space-time at the string or at the Planck scale, where some form of unsharpness or fuzzyness has long been conjectured. A priori, however, a potential problem with merely symmetric operators is the fact that, unlike self-adjoint operators, they do not generate unitaries - at least not straightforwardly. Here, we show for a large class of these operators that they do generate unitaries in a well defined way, and that these operators even generate the entire unitary group of the Hilbert space. This shows that merely symmetric operators, in addition to describing unsharp physical entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g. within a fundamental theory of quantum gravity.

For bounded linear operators, the study ofWeyl-type theorems and properties has been of significant interest for several non-normal classes of operators. In this paper, we extend this study to a class of unbounded posinormal operators. We define and study the spectral properties of unbounded posinormal and totally posinormal operators defined on an infinite dimensional complex Hilbert space H. For this class, under certain conditions several Weyl-type theorems and related properties are obtained.

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.

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results.

Iteration–discretization methods for variational inequalities over fixed point sets

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

This paper deals with solutions of the Yang–Baxter–like operator equation AXA = XAX on the infinite dimensional Hilbert space H , where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.