Operators In Infinite-dimensional Spaces Research Articles

Method of Mathematical Theory of Moments

Infinite matrices play an important role in many aspects of analysis, algebra, differential equations, andthetheory of mechanical vibrations.Jacobi matrices are interestingbecause they are the simplest representatives of symmetric operators in infinite-dimensional space. they are used in interpolation theory, quantum physics, moment problem.In this paper, basedon the elements of Jacobi matrix, itwill be determinedthe type of the operator that occurs when processing the results of measurements of random variables. The first type of operators arematrices,for which the moment problem has a unique solution, and Jacobi matrix generates a specific moment problem. The second type of operators arematrices, for which the moment problem has many solutions, and Jacobi matrix is said to generate an indeterminate moment problem.

Shift-invert Rational Krylov method for an operator $$\phi $$ ϕ -function of an unbounded linear operator

The product of a matrix function and a vector is used to solve evolution equations numerically. Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016) proposed the Shift-invert Rational Krylov method for computing these products. However, since matrices produced by evolution equations behave like unbounded operators in infinite-dimensional spaces, an analysis with the unbounded operator is essential. In this paper, the Shift-invert Rational Krylov method is extended to be applied to unbounded operators.

Dynamical sampling

Let Y={f(i),Af(i),…,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(I) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states Alf. We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, although Y can be complete, using the Müntz–Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H2(D). Finally, using the recently proved Kadison–Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite.

Multiplicative perturbation theory of the Moore–Penrose inverse and the least squares problem

Bounds for the variation of the Moore–Penrose inverse of general matrices under multiplicative perturbations are presented. Their advantages with respect to classical bounds under additive perturbations and with respect to other bounds under multiplicative perturbations available in the literature are carefully studied and established. Closely connected to these developments a complete multiplicative perturbation theory for least squares problems, valid for perturbations of any size, is also presented, improving in this way recent multiplicative perturbation bounds which are valid only to first order in the size of the perturbations. The results in this paper are mainly based on exact expressions of the perturbed Moore–Penrose inverse in terms of the unperturbed one, the perturbation matrices, and certain orthogonal projectors. Such feature makes the new results amenable to be generalized in the future to linear operators in infinite dimensional spaces.

Stability of positive linear switched systems on ordered Banach spaces

We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas in the continuous-time case an appropriate interior point of the dual cone is sufficient for stability. This is an extension of the concept of linear Lyapunov functions for positive systems to the setting of infinite-dimensional partially ordered spaces. We illustrate our results with examples.

Variation of the Spectrum of Operators in Infinite Dimensional Spaces

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.

Smooth decomposition of random fields

The Smooth Decomposition (SD) method was introduced to analyze discrete-time signals and generalized to continuous-time vector-valued random processes. The SD is obtained solving a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process which defines the decomposition basis. This paper presents a new extension of the SD to continuous-time and continuous-space vector-valued random processes, classically named random fields. This generalization is a major step since one now deals with operators in infinite-dimensional spaces and not matrices. It is shown that in this new context the main properties of the SD are preserved. Applied to the responses of randomly excited continuous mechanical systems, the SD can be considered as an output-only analysis tool. Moreover, two natural orderings are defined to classify the decomposition terms which permit to interpret the SD in terms of modal analysis or in terms of Karhunen–Loève analysis.

Splitting methods with complex times for parabolic equations

Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations. We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.

A Parametric Newton Method for Optimization Problems in Hilbert Spaces

The main purpose of the present work is to modify the Newton method for solving the minimization problems, avoiding the calculation of the inverse operators in infinite dimensional spaces. The article deals with a parametric optimization problem in real Hilbert spaces, assuming that the gradient of the cost functional is Lipschitz continuous, but not a smooth function. The supergeometric rate of convergence, as well as the monotonicity, of the obtained minimizing sequences is proved. Two examples are given.

Regularized Products and Determinants

Zeta-regularized products are used to define determinants of operators in infinite dimensional spaces. This article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination.

Extension of generalized multi-valued variational inequalities

In this paper, we derive the existence result for a new class of generalized multi-valued variational inequality (1.1). Theorem 3.3 of this paper seems to be a natural extension of the celebrated result of Browder, Hartman and Stampacchia [1,2] variational inequalities for multi-valued operator in infinite-dimensional space.

Self-Adjoint Operators in Indefinite Metric Spaces

Some properties of self-adjoint operators in indefinite metric spaces are explored, with emphasis on the problem of the completeness of the set of eigenvectors. For operators in spaces of finite dimension, some simple criteria are deduced regarding the existence of such a complete set. Implications of completeness of eigenvectors for operators in infinite-dimensional spaces are discussed, and some partial extensions of the results for finite dimensions given.