Pairs of fixed points for a class of operators on Hilbert spaces

The work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces. The need to study these equations is related to numerous ones applications of integro- differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere. There are different approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coefficient conditions of solvability and analytical presentation of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods. In this connection, Fredholm’s integro-differential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.

The paper examines an algorithm for finding approximate sparse solutions of convex cardinality constrained optimization problem in Hilbert spaces. The proposed algorithm uses the penalty decomposition (PD) approach and solves sub-problems on each iteration approximately. We examine the convergence of the algorithm to a stationary point satisfying necessary optimality conditions. Unlike other similar works, this paper discusses the properties of PD algorithms in infinite-dimensional (Hilbert) space. The results showed that the convergence property obtained in previous works for cardinality constrained optimization in Euclidean space also holds for infinite-dimensional (Hilbert) space. Moreover, in this paper we established a similar result for convex optimization problems with cardinality constraint with respect to a dictionary (not necessarily the basis).

In this study, reconstruction of the normalized radar cross-section (NRCS) image from noisy Global Navigation Satellite System Reflectometry (GNSS-R) Delay-Doppler Maps (DDMs) is addressed in both Hilbert and Banach spaces. The proposed approach, which is based on the Landweber regularization method, appropriately specialized to account for spatial invariance, is first developed in Hilbert space and then extended to the Banach space. The reconstruction performance of the methods is discussed using simulated DDMs and contrasted with a deconvolution technique based on constrained least squares (CLS). Experimental results demonstrate that the Landweber method in Banach space outperforms the Landweber method in Hilbert space and the CLS. In fact, these latter methods exhibit both over-smoothing and typical Gibbs-related oscillations.

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

Inverse problems can usually be modelled as operator equations in infinite-dimensional spaces with a forward operator acting between Hilbert or Banach spaces—a formulation which quite often also serves as the basis for defining and analyzing solution methods. The additional amount of structure and geometric interpretability provided by the concept of an inner product has rendered these methods amenable to a convergence analysis, a fact which has led to a rigorous and comprehensive study of regularization methods in Hilbert spaces over the last three decades. However, for numerous problems such as x-ray diffractometry, certain inverse scattering problems and a number of parameter identification problems in PDEs, the reasons for using a Hilbert space setting seem to be based on conventions rather than an appropriate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, non-Hilbertian regularization and data fidelity terms incorporating a priori information on solution and noise, such as general Lp-norms, TV-type norms, or the Kullback–Leibler divergence, have recently become very popular. These facts have motivated intensive investigations on regularization methods in Banach spaces, a topic which has emerged as a highly active research field within the area of inverse problems. Meanwhile some of the most well-known regularization approaches, such as Tikhonov-type methods requiring the solution of extremal problems, and iterative ones like the Landweber method, the Gauss–Newton method, as well as the approximate inverse method, have been investigated for linear and nonlinear operator equations in Banach spaces. Convergence with rates has been proven and conditions on the solution smoothness and on the structure of nonlinearity have been formulated. Still, beyond the existing results a large number of challenging open questions have arisen, due to the more involved handling of general Banach spaces and the larger variety of concrete instances with special properties.

The solution of the customary adjoint Boltzmann equation for linear transport of particles and radiation, referred to as the “adjoint flux,” plays a prominent role in reactor physics, shielding, control, and optimization as a weighting function for cross-section processing, optimization, Monte Carlo acceleration procedures, and sensitivity and uncertainty analyses. All of the textbooks and scientific works published thus far use the same procedure to derive “the” adjoint Boltzmann operator, thereby conveying inadvertently the misleading impression that this traditional procedure is the only way to obtain “the” adjoint Boltzmann operator, and that the form of “the” adjoint operator thus derived is universally unique. None of the works published in the literature thus far touches on the fact that the customary textbook-form of the adjoint Boltzmann operator is actually derived in a particular Hilbert space, which is endowed with a specific inner product that is based on integrating spatially over the domain’s spatial volume such that Gauss’ divergence theorem holds. As this work will show, however, the Hilbert space that has been implicitly used in all of the published works thus far is not the only possible Hilbert space for deriving operators that are adjoint, in the respective Hilbert space, to the forward Boltzmann operator. Alternative Hilbert spaces may be used just as legitimately, and may actually be more suitable than the customary Hilbert space for computing adjoint functions to be used in inner products involving various forward and/or adjoint fluxes and forward and/or adjoint source terms.By presenting paradigm illustrative examples in three-dimensional spherical coordinates, this work shows that although a unique form of the adjoint Boltzmann operator is obtained for each Hilbert space in which the respective adjoint operator is constructed, distinct Hilbert spaces will produce distinct adjoint Boltzmann operators accompanied by distinct forms of the corresponding bilinear concomitants on the respective spatial domain’s boundary. The fundamental practical reason for using alternative Hilbert spaces is to obtain alternative adjoint functions and/or Green’s functions that may be less singular than the customary adjoint function and/or Green’s functions (in the customary Hilbert space) and would consequently be computable numerically. As this work shows, such situations arise when attempting to use the adjoint sensitivity analysis methodology in the conventional Hilbert space for computing sensitivities to cross sections, isotopic number densities, etc., of responses of flux and/or power detectors placed near or at the center of the spherical coordinates. In such sensitivity analysis problems, the singularities of the conventional adjoint Boltzmann equation in the conventional Hilbert space may preclude its use, but the requisite sensitivities can nevertheless be computed efficiently using an alternative adjoint Boltzmann equation in an alternative Hilbert space. The consequences of this powerful breakthrough new concept of using alternative adjoint operators in alternative Hilbert spaces are highlighted by presenting a paradigm benchmark problem that admits a closed-form exact solution. This benchmark problem shows that the customary adjoint equation becomes singular at the sphere’s center, so the conventional adjoint flux is therefore noncomputable there, but the alternative adjoint transport equation in a judiciously chosen alternative Hilbert space is everywhere nonsingular and can therefore be used to compute the requisite sensitivities. By indicating the path for using alternative Hilbert spaces, this work opens new conceptual procedures for solving problems that have hitherto been unsolvable in the framework of the conventional adjoint particle transport equation.

This chapter discusses compact operators in detail. Section 1 concerns convergence properties for sequences of bounded operators. We introduce compact operators in Section 2. The main result of the chapter is the spectral theorem for compact Hermitian operators. We provide additional discussion about integral operators, and we close the chapter with a glimpse at singular integral operators. This chapter prepares us for the applications in Chapter VII, where we use facts about compact operators to study positivity conditions for polynomials. Convergence properties for bounded linear transformations Completeness of the real number system is crucial for doing analysis; without it limits would be a useless concept. Similarly, completeness for metric spaces (such as Hilbert and Banach spaces) is needed for doing analysis in more general settings. We therefore begin this chapter with a short discussion of three possible notions for convergence of sequences of bounded linear transformations between Hilbert spaces. We write L ( H , H ′) for the vector space of bounded linear transformations between Hilbert spaces H and H ′. We mentioned in Section II.1 that L ( H , H ′) is a Banach space; in particular this space is complete. See Theorem V.1.4. We next consider three reasonable definitions of convergence in L ( H , H ′). Definition V.1.1. ( Notions of convergence for bounded linear transformations ) Let { L n } be a sequence in L ( H , H ′), let L ∈ L ( H , H ′). 1) { L n } converges to L in norm if the sequence of real numbers ∥ L n − L ∥ converges to 0. 2) { L n } converges to L in the strong operator topology if, for all z ∈ H the sequence { L n ( z )} converges to L ( z ) in H ′. […]

Interference in the classical probabilistic model and its representation in complex Hilbert space

To sample from complex, high-dimensional distributions, one may choose algorithms based on the Hybrid Monte Carlo (HMC) method. HMC-based algorithms generate nonlocal moves alleviating diffusive behavior. Here, I build on an already defined HMC framework, hybrid Monte Carlo on Hilbert spaces (Beskos, et al. Stoch. Proc. Applic. 2011), that provides finite-dimensional approximations of measures , which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert (path) space. In all HMC algorithms, one has some freedom to choose the mass operator. The novel feature of the algorithm described in this article lies in the choice of this operator. This new choice defines a Markov Chain Monte Carlo (MCMC) method that is well defined on the Hilbert space itself. As before, the algorithm described herein uses an enlarged phase space having the target as a marginal, together with a Hamiltonian flow that preserves . In the previous work, the authors explored a method where the phase space was augmented with Brownian bridges. With this new choice, is augmented by Ornstein–Uhlenbeck (OU) bridges. The covariance of Brownian bridges grows with its length, which has negative effects on the acceptance rate in the MCMC method. This contrasts with the covariance of OU bridges, which is independent of the path length. The ingredients of the new algorithm include the definition of the mass operator, the equations for the Hamiltonian flow, the (approximate) numerical integration of the evolution equations, and finally, the Metropolis–Hastings acceptance rule. Taken together, these constitute a robust method for sampling the target distribution in an almost dimension-free manner. The behavior of this novel algorithm is demonstrated by computer experiments for a particle moving in two dimensions, between two free-energy basins separated by an entropic barrier.

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Hilbert spaces of various types provide the natural setting for much of the sampling theory that we shall be studying.The reader is assumed to be familiar with the foundations of Hilbert and Banach spaces. We review here some of the important facts, mostly concerning Hilbert spaces, that will be needed throughout the text. Of particular importance is the rather more specialized topic of bases in Hilbert space. An elementary account can be found in the book by Higgins (1977). A very readable introduction to the theory of bases is in the book by Young (1980); the book of Marti (1969) has a more advanced treatment, while a vastly comprehensive treatment is to be found in the book of Singer (1970).Apart from the last section this chapter is intended as a review only, and almost no proofs are included. However, ample references to the texts mentioned above are given.

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.