Hilbert Space Case Research Articles

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.

Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper, we focus on the case of Hilbert spaces and introduce, in particular, a version of the Hardy space. Then we define Blaschke factors and Blaschke products and consider an interpolation problem. In the second part of the paper, we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carathéodory functions.

The holomorphic functional calculus approach to operator semigroups

In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille-Yosida and Gomilko-Shi-Feng generation theorems, Trotter-Kato apprximation theorem, Euler approximation formula, Gearhart-Prüss theorem) can be elegantly obtained in this framework. Then we discuss the notions of bounded H∞-calculus and m-bounded calculus on half-planes and their relation to weak bounded variation conditions over vertical lines for powers of the resolvent. Finally we discuss the Hilbert space case, where semigroup generation is characterised by the operator having a strong m-bounded calculus on a half-plane.

Stability estimates for semigroups on Banach spaces

For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjostrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.

Greedy Algorithms for Reduced Bases in Banach Spaces

Given a Banach space X and one of its compact sets $\mathcal{F}$ , we consider the problem of finding a good n-dimensional space X n ⊂X which can be used to approximate the elements of $\mathcal{F}$ . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$ . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886---911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.

Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

Scattering Theory for Klein–Gordon Equations with Non-Positive Energy

We study the scattering theory for charged Klein-Gordon equations: \[\{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x, D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, {array}. \] where: \[\epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x),\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \[ h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. \] We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case.

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

The Lagrange interpolation problem in Banach spaces is approached by cardinal basis interpolation. Some error estimates are given and the results of several numerical tests are reported in order to show the approximation performances of the proposed interpolants. A comparison between some examples of interpolants is presented in the noteworthy case of Hilbert spaces, with some considerations about the possible localization of the formulas. Finally, some remarks about the cardinal basis interpolation framework are made from the application point of view.

Fractional relaxation equations on Banach spaces

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation (0.1) u ′ ( t ) − A D t α u ( t ) + u ( t ) = f ( t ) , 0 < α < 1 , t ≥ 0 , u ( 0 ) = 0 , on a complex Banach space X , where A is a closed linear operator, D t α is the Caputo derivative of fractional order α ∈ ( 0 , 1 ) , and f is an X -valued function. We also study conditions under which the solution operator has the properties of maximal regularity and L p integrability. We characterize these properties in the Hilbert space case.

Two-Isometries on Pontryagin Spaces

A model for a class of cyclic analytic 2-isometries acting on Pontryagin spaces is given, generalizing a Hilbert space version given by S. Richter. Furthermore, an example is constructed showing that, unlike in the Hilbert space case, a cyclic analytic 2-isometry need not have a cyclic vector in the orthogonal complement of its range.

$\epsilon$-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems

By introducing the concept of $\epsilon$-convertibility, we extend Nielsen's and Vidal's theorems to the entanglement transformation of infinite-dimensional systems. Using an infinite-dimensional version of Vidal's theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong to a higher rank of entanglement class in terms of SLOCC convertibility. For the case of Hilbert spaces of countable, but infinite dimensionality, we show that there are actually an uncountable number of classes of pure non-interconvertible bipartite entangled states.

Monoidal Categories, Symmetries, and Compound Physical Systems

The conceptually non-trivial problem of relating the notion of a compound physical system and the mathematical descriptions of its constituent parts is dramatically illustrated in standard quantum physics by the use of the Hilbert tensor product of the spaces representing the subsystems, instead of the more familiar cartesian product, as it is the case for classical physical systems. Aspects of the general structure of this relationship can be explained by endowing suitable categories that arise in the mathematical descriptions of classical systems and of quantum systems with their natural monoidal structures, and constructing a monoidal functor, relating the monoidal structures of the domain and codomain categories in a coherent way. To highlight some of the structural aspects involved, I will confine myself in this paper to the simple case of finite sets or finite-dimensional Hilbert spaces, on which finite groups act.

Surjectivity of linear operators from a Banach space into itself

We show that linear operators from a Banach space into itself which satisfy some relaxed strong accretivity conditions are invertible. Moreover, we characterise a particular class of such operators in the Hilbert space case. By doing so we manage to answer a problem posed by B. Ricceri, concerning a linear second order partial differential operator.

Computability of compact operators on computable Banach spaces with bases

AbstractWe develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non‐constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable Schauder bases that are well‐behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non‐uniform result on composition with bounded linear operators from right. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an Lp,q space and M∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.

On Computable Compact Operators on Banach Spaces

We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable Schauder bases that are well-behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non-uniform result on composition with bounded linear operators from right.

Radius problems for holomorphic mappings on the unit ball in Cn

AbstractIn this paper we obtain certain results related to radius of starlikeness, convexity, parametric representation and Bloch radius for some classes of holomorphic mappings on the unit ball B n in ℂn . In particular, we consider the class ℳ︁ of mappings of “positive real part” and some related classes. We also consider two classes of mappings defined by coefficient inequalities. In the last section we consider the complex Hilbert space case of certain results which are treated in the previous sections. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Asymptotic behaviour of a family of gradient algorithms in ℝ d and Hilbert spaces

The asymptotic behaviour of a family of gradient algorithms (including the methods of steepest descent and minimum residues) for the optimisation of bounded quadratic operators in ℝd and Hilbert spaces is analyzed. The results obtained generalize those of Akaike (1959) in several directions. First, all algorithms in the family are shown to have the same asymptotic behaviour (convergence to a two-point attractor), which implies in particular that they have similar asymptotic convergence rates. Second, the analysis also covers the Hilbert space case. A detailed analysis of the stability property of the attractor is provided.

Characterization of elliptic distributions

We characterize elliptic distributions in ℝ d by an extremal property for a class of probability distances. A generalization to the case of Hilbert spaces is given. Bibliography: 2 titles.