Hilbert Space (hilbert + space)

Distribution by Scientific Domains

Selected Abstracts

Time asymmetric quantum theory , I. Modifying an axiom of quantum physics

A.R. Bohm
A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in-states of scattering theory) and the space of observables (out-"states" of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law. [source]

Landweber scheme for compact operator equation in Hilbert space and its applications

Gangrong Qu
Abstract We study the Landweber scheme for linear compact operator equation in infinite Hilbert spaces. Using the singular value decomposition for compact operators, we obtain a formula for the Landweber scheme after n iterations and iterative truncated error and consequently establish its convergence conditions. Our results extend known results on convergence conditions. As applications, we apply the Landweber scheme to the X-ray tomography and extrapolation of band-limited functions, and establish accelerated strategies for each application. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Solutions of linear and semilinear distributed parameter equations with a fractional Brownian motion

T. E. Duncan
Abstract In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parameter H,,,(0, 1). For H,=,½ the process is Brownian motion. Solutions of these linear and semilinear equations are given for each H,,,(0, 1) with the assumptions differing for the cases H,,,(0, ½) and H,,,(½, 1). For the linear equations, the solutions are mild solutions and limiting Gaussian measures are characterized. For the semilinear equations, the solutions are either mild or weak. The weak solutions are obtained by transforming the measure of the associated linear equation by a Radon,Nikodym derivative (likelihood function). An application to identification is given by obtaining a strongly consistent family of estimators for an unknown parameter in a linear equation with distributed noise or boundary noise. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Cleanthes A. Nicolaides
Abstract Stationary-state quantum mechanics presents no difficulties in defining and computing discrete excited states because they obey the rules established in the properties of Hilbert space. However, when this idealization has to be abandoned to formulate a theory of excited states dissipating into a continuous spectrum, the problem acquires additional interest in many fields of physics. In this article, the theory of resonances in the continuous spectrum is formulated as a problem of decaying states, whose treatment can entail time-dependent as well as energy-dependent theories. The author focuses on certain formal and computational issues and discusses their application to polyelectronic atomic states. It is argued that crucial to the theory is the understanding and computation of a multiparticle localized wavepacket, ,0, at t = 0, having a real energy E0. Assuming this as the origin, without memory of the excitation process, the author discusses aspects of time-dependent dynamics, for t , 0 as well as for t , ,, and the possible significance of nonexponential decay in the understanding of timeasymmetry. Also discussed is the origin of the complex eigenvalue Schrödinger equation (CESE) satisfied by resonance states and the state-specific methodology for its solution. The complex eigenvalue drives the decay exponentially, with a rate ,, to a good approximation. It is connected to E0 via analytic continuation of the complex self-energy function, A(z), (z is complex), into the second Riemann sheet, or, via the imposition of outgoing wave boundary conditions on the stationary state Schrödinger equation satisfied by the Fano standing wave superposition in the vicinity of E0. If the nondecay amplitude, G(t), is evaluated by inserting the unit operator I = ,dE|E>, then the resulting spectral function is real, g(E) =|<,0|E>|2, and does not differentiate between positive and negative times. The introduction of time asymmetry, which is associated with irreversibility, is achieved by starting from < ,0|,(t)e,iHt|,0 >, where ,(t) is the step function at the discontinuity point t = 0. In this case, the spectral function is complex. Within the range of validity of exponential decay, the complex spectral function is the same as the coefficient of ,0 in the theory of the CESE. A calculation of G(t) using the simple pole approximation and the constraints that t > 0 and E > 0 results in a nonexponential decay (NED) correction for t , 1/, that is different than when a real g(E) is used, representing the contribution of both "in" and "out" states. Earlier formal and computational work has shown that resonance states close to threshold are good candidates for NED to acquire nonnegligible magnitude. In this context, a pump-probe laser experiment in atomic physics is proposed, using as a paradigm the He, 1s2p24P shape resonance. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source]

Predictive control of parabolic PDEs with state and control constraints

Stevan Dubljevic
Abstract This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with state and control constraints. Initially, the PDE is written as an infinite-dimensional system in an appropriate Hilbert space. Next, modal decomposition techniques are used to derive a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional system, and express the infinite-dimensional state constraints in terms of the finite-dimensional system state constraints. A number of model predictive control (MPC) formulations, designed on the basis of different finite-dimensional approximations, are then presented and compared. The closed-loop stability properties of the infinite-dimensional system under the low order MPC controller designs are analysed, and sufficient conditions that guarantee stabilization and state constraint satisfaction for the infinite-dimensional system under the reduced order MPC formulations are derived. Other formulations are also presented which differ in the way the evolution of the fast eigenmodes is accounted for in the performance objective and state constraints. The impact of these differences on the ability of the predictive controller to enforce closed-loop stability and state constraints satisfaction in the infinite-dimensional system is analysed. Finally, the MPC formulations are applied through simulations to the problem of stabilizing the spatially-uniform unstable steady-state of a linear parabolic PDE subject to state and control constraints. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Characterizing arbitrarily slow convergence in the method of alternating projections

Heinz H. Bauschke
Abstract Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem itself is correct. We give a different proof that uses the multiplicative form of the spectral theorem, and the theorem holds in any real or complex Hilbert space, not just in a real Hilbert space. [source]

The play operator on the rectifiable curves in a Hilbert space

Vincenzo Recupero
Abstract The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data, the existence of solutions is easily obtained from general results on maximal monotone operators. If the datum is a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Traces of Sobolev functions with one square integrable directional derivative

M. Gregoratti
Abstract We consider the Sobolev spaces of square integrable functions v, from ,n or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ,,,,v. In these spaces we define closed trace operators on the boundaries ,Q and on the hyperplanes {r,, = z}, z , ,\{0}, which turn out to be possibly unbounded with respect to the usual L2 -norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L2, we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ,n and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Diffusion in poro-plastic media

R. E. Showalter
Abstract A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial-boundary-value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto-visco-plastic type. The variational form of this problem in Hilbert space is a non-linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi-static momentum equation. The essential sufficient conditions for the well-posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Sufficient conditions of non-uniqueness for the Coulomb friction problem

Riad Hassani
Abstract We consider the Signorini problem with Coulomb friction in elasticity. Sufficient conditions of non-uniqueness are obtained for the continuous model. These conditions are linked to the existence of real eigenvalues of an operator in a Hilbert space. We prove that, under appropriate conditions, real eigenvalues exist for a non-local Coulomb friction model. Finite element approximation of the eigenvalue problem is considered and numerical experiments are performed. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On a formula for the spectral flow and its applications

Pierluigi Benevieri
Abstract We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Two singular point linear Hamiltonian systems with an interface condition

Horst Behncke
Abstract We consider the problem of a linear Hamiltionian system on , with an interface condition which we take to be at x = 0. Assuming limit point conditions at ±,, we prove the problem is uniquely solvable, and a resolvent is constructed. Our method of solution is to map the problem onto a half line problem of double size and apply the theory of half line problems. A Titchmarsh-Weyl function is associated with the problem, and a unitary transform is constructed which maps the differential operator onto the multiplication operator in the Hilbert space determined by the spectral function , (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Variational principles for symmetric bilinear forms

Jeffrey Danciger
Abstract Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Quasi-splitting subspaces in a pre-Hilbert space

David Buhagiar
Abstract Let S be a pre-Hilbert space. Two classes of closed subspaces of S that can naturally replace the lattice of projections in a Hilbert space are E (S) and F (S), the classes of splitting subspaces and orthogonally closed subspaces of S respectively. It is well-known that in general the algebraic structure of E (S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert space. In the present note we introduce the class Eq(S) of quasi-splitting subspaces of S. First it is shown that Eq(S) falls between E (S) and F (S). It is also shown that, in contrast to the other two classes, Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. At the end, the algebraic structure of Eq(S) is used to characterize Hilbert spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On the point spectrum of ,,2 -singular perturbations

Sergio Albeverio
Abstract We prove that for any self-adjoint operator A in a separable Hilbert space , and a given countable set , = {,i}i ,, of real numbers, there exist ,,2 -singular perturbations à of A such that , , ,p(Ã). In particular, if , = {,1,,, ,n} is finite, then the operator à solving the eigenvalues problem, Ã,k = ,k,k, k = 1,,, n, is uniquely defined by a given set of orthonormal vectors {,k}nk =1 satisfying the condition span {,k}nk =1 , dom (|A |1/2) = {0}. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Deficiency indices and spectral theory of third order differential operators on the half line

Horst Behncke
Abstract We investigate the spectral theory of a general third order formally symmetric differential expression of the form acting in the Hilbert space ,2w(a ,,). A Kummer,Liouville transformation is introduced which produces a differential operator unitarily equivalent to L . By means of the Kummer,Liouville transformation and asymptotic integration, the asymptotic solutions of L [y ] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L . For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On stable implicit difference scheme for hyperbolic,parabolic equations in a Hilbert space

Allaberen Ashyralyev
Abstract The first-order of accuracy difference scheme for approximately solving the multipoint nonlocal boundary value problem for the differential equation in a Hilbert space H, with self-adjoint positive definite operator A is presented. The stability estimates for the solution of this difference scheme are established. In applications, the stability estimates for the solution of difference schemes of the mixed type boundary value problems for hyperbolic,parabolic equations are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

Symmetries and anisotropies of the electronic states within full spin,orbit coupling

G. E. Marques
Abstract We have analyzed how the symmetries and the anisotropies of energy dispersions and of the spinor states, within full spin,orbit interaction, form the two independent circular spin polarizations. We also compare how the effects produced by Rashba and Dresselhauss interaction terms act on the structure of the Hilbert space. These aspects are used to envisage a voltage controlled multichannel spin-filtering regime in nonmagnetic diode structures. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Is polaron effect important for resonant Raman scattering in self-assembled quantum dots?

M. I. Vasilevskiy
Abstract While the diagonal (or intra-level) interaction of a confined exciton with optical phonons in self-assembled quantum dots (SAQD's) is rather weak, the non-diagonal one can lead to a considerable change of the exciton spectrum and the formation of a polaron. An impact of this effect on resonant inelastic light scattering is studied theoretically. The polaron spectrum is obtained by numerical diagonalisation of the exciton,phonon interaction Hamiltonian in a truncated Hilbert space of the non-interacting excitons and phonons. Based on this spectrum, the probability of the multi-phonon Raman scattering is calculated, which is compared to that obtained within the standard perturbation theory approach (where phonon emission and absorption are irreversible). It is shown that there are two major effects of the polaron formation: (i) the intensity of the two-phonon (2 LO) peak, relative to that of the fundamental 1 LO one is strongly increased and (ii) the resonant behaviour of the 1 LO peak differs considerably from the perturbation theory predictions. With the correct theoretical interpretation, resonant Raman scattering in SAQD's opens the possibility of accessing the (renormalised) exciton spectrum and exciton,phonon coupling constants. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Projection estimators of Pickands dependence functions

Amélie Fils-Villetard
Abstract The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme-value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite-dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite-dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one. Estimation par projection de la fonction de dépendance de Pickands Les auteurs s'intéressent à la construction d'estimateurs intrinsèques de la fonction de dépendance de Pickands d'une copule des valeurs extrêmes. Ils montrent comment un estimateur initial quelconque peut être modifié pour satisfaire les contraintes de forme voulues. Leur solution consiste à projeter cet estimateur dans l'espace des fonctions de Pickands, qui forme un sous-ensemble convexe fermé d'un espace de Hilbert. Comme la solution n'est pas explicite, ils remplacent cet espace paramétrique fonctionnel par une succession d'approximations de dimension finie. Ils établissent la distribution asymptotique de la projection de l'estimateur et de ses approximations de dimension finie, ce qui leur permet de conclure que l'estimateur projeté est au moins aussi efficace que l'estimateur initial. [source]

A review on the use of the adjoint method in four-dimensional atmospheric-chemistry data assimilation

K.-Y. Wang
Abstract In this paper we review a theoretical formulation of the adjoint method to be used in four-dimensional (4D) chemistry data assimilation. The goal of the chemistry data assimilation is to combine an atmospheric-chemistry model and actual observations to produce the best estimate of the chemistry of the atmosphere. The observational dataset collected during the past decades is an unprecedented expansion of our knowledge of the atmosphere. The exploitation of these data is the best way to advance our understanding of atmospheric chemistry, and to develop chemistry models for chemistry-climate prediction. The assimilation focuses on estimating the state of the chemistry in a chemically and dynamically consistent manner (if the model allows online interactions between chemistry and dynamics). In so doing, we can: produce simultaneous and chemically consistent estimates of all species (including model parameters), observed and unobserved; fill in data voids; test the photochemical theories used in the chemistry models. In this paper, the Hilbert space is first formulated from the geometric structure of the Banach space, followed by the development of the adjoint operator in Hilbert space. The principle of the adjoint method is described, followed by two examples which show the relationship of the gradient of the cost function with respect to the output vector and the gradient of the cost function with respect to the input vector. Applications to chemistry data assimilation are presented for both continuous and discrete cases. The 4D data variational adjoint method is then tested in the assimilation of stratospheric chemistry using a simple catalytic ozone-destruction mechanism, and the test results indicate that the performance of the assimilation method is good. [source]

Semiclassical expansion of quantum characteristics for many-body potential scattering problem

M.I. Krivoruchenko
Abstract In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ,. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p -forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior. [source]

A representation of acoustic waves in unbounded domains,

Bradley K. Alpert
Compact, time-harmonic, acoustic sources produce waves that decay too slowly to be square-integrable on a line away from the sources. We introduce an inner product, arising directly from Green's second theorem, to form a Hilbert space of these waves and present examples of its computation.1 © 2005 Wiley Periodicals, Inc. [source]

Landweber scheme for compact operator equation in Hilbert space and its applications

Gangrong Qu
Abstract We study the Landweber scheme for linear compact operator equation in infinite Hilbert spaces. Using the singular value decomposition for compact operators, we obtain a formula for the Landweber scheme after n iterations and iterative truncated error and consequently establish its convergence conditions. Our results extend known results on convergence conditions. As applications, we apply the Landweber scheme to the X-ray tomography and extrapolation of band-limited functions, and establish accelerated strategies for each application. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Selection of the relevant information set for predictive relationships analysis between time series

Umberto Triacca
Abstract In time series analysis, a vector Y is often called causal for another vector X if the former helps to improve the k -step-ahead forecast of the latter. If this holds for k=1, vector Y is commonly called Granger-causal for X. It has been shown in several studies that the finding of causality between two (vectors of) variables is not robust to changes of the information set. In this paper, using the concept of Hilbert spaces, we derive a condition under which the predictive relationships between two vectors are invariant to the selection of a bivariate or trivariate framework. In more detail, we provide a condition under which the finding of causality (improved predictability at forecast horizon 1) respectively non-causality of Y for X is unaffected if the information set is either enlarged or reduced by the information in a third vector Z. This result has a practical usefulness since it provides a guidance to validate the choice of the bivariate system {X, Y} in place of {X, Y, Z}. In fact, to test the ,goodness' of {X, Y} we should test whether Z Granger cause X not requiring the joint analysis of all variables in {X, Y, Z}. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Bayesian classification of tumours by using gene expression data

Bani K. Mallick
Summary., Precise classification of tumours is critical for the diagnosis and treatment of cancer. Diagnostic pathology has traditionally relied on macroscopic and microscopic histology and tumour morphology as the basis for the classification of tumours. Current classification frameworks, however, cannot discriminate between tumours with similar histopathologic features, which vary in clinical course and in response to treatment. In recent years, there has been a move towards the use of complementary deoxyribonucleic acid microarrays for the classi-fication of tumours. These high throughput assays provide relative messenger ribonucleic acid expression measurements simultaneously for thousands of genes. A key statistical task is to perform classification via different expression patterns. Gene expression profiles may offer more information than classical morphology and may provide an alternative to classical tumour diagnosis schemes. The paper considers several Bayesian classification methods based on reproducing kernel Hilbert spaces for the analysis of microarray data. We consider the logistic likelihood as well as likelihoods related to support vector machine models. It is shown through simulation and examples that support vector machine models with multiple shrinkage parameters produce fewer misclassification errors than several existing classical methods as well as Bayesian methods based on the logistic likelihood or those involving only one shrinkage parameter. [source]

Second-Order Noncausality in Multivariate GARCH Processes

Fabienne Comte
Typical multivariate economic time series may exhibit co-behavior patterns not only in the conditional means, but also in the conditional variances. In this paper we give two new definitions of variance noncausality in a multivariate setting a Granger-type noncausality and a linear Granger noncausality through projections on Hilbert spaces. Both definitions are related to a previous second-order noncausality concept defined by Granger et al. in a bivariate setting. The implications of second-order noncausality on multivariate ARMA processes with GARCH-type errors are investigated. We derive exact testable restrictions on the parameters of the processes considered, implied by this type of noncausality. Conditions for the finiteness of the fourth-order moment of the multivariate GARCH process are derived and related to earlier results in the univariate framework. We include an illustration of second-order noncausality in a trivariate model of daily financial returns. [source]

Starlike and convex rational mappings on infinite dimensional domains

Cho-Ho Chu
Abstract We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Computability of solutions of operator equations

Volker Bosserhoff
Abstract We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs (T, u) (where T is given in form of a program) to solutions of Tx = u ? Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T,u, which is the standard concept of solution in the theory of inverse problems. Typically, T, is discontinuous (i. e. the equation Tx = u is ill-posed) and hence no computable mapping. However, we will use effective versions of theorems from the theory of regularization to show that the mapping (T, T *, u, ,T,u ,) , T,u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T, is considered as an element of an L2 -space. We define suitable representations for such spaces and use the results from the first part of the paper to show that (T, T *, ,T,,) , T, is computable. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Computability of compact operators on computable Banach spaces with bases

Vasco Brattka
Abstract 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. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]