Non-compact Operators Research Articles

Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces

Abstract We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.

On the convergence of Fourier spectral methods involving noncompact operators

Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

Diagonals of self-adjoint operators II: non-compact operators

Diagonals of self-adjoint operators II: non-compact operators

Randomized approximation of summable sequences — adaptive and non-adaptive

We prove lower bounds for the randomized approximation of the embedding ℓ1m↪ℓ∞m based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix N∈Rn×m. These lower bounds reflect the increasing difficulty of the problem for m→∞, namely, a term logm in the complexity n. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity n only exhibits a (loglogm)-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order n1/2(logn)−1/2.

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components

An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components

Perturbation of singular integral operators with piecewise continuous coefficients

In the paper it is shown that the property of singular integral operators with piecewise continuous coefficients to be Noetherian is stable with respect to their perturbation with certain non-compact operators. An example is constructed showing that the corner points of the integration contour significantly affect the Noetherian property of singular operators with translations. These results are obtained using the symbol of the singular operators on contours with corner points, symbol, which is also determined.

Perturbation of singular integral operators with piecewise continuous coefficients

In the paper it is shown that the property of singular integral operators with piecewise continuous coefficients to be Noetherian is stable with respect to their perturbation with certain non-compact operators. An example is constructed showing that the corner points of the integration contour significantly affect the Noetherian property of singular operators with translations. These results are obtained using the symbol of the singular operators on contours with corner points, symbol, which is also determined.

Discontinuous Galerkin methods for the acoustic vibration problem

In two and three dimension we analyze discontinuous Galerkin methods (DG) for the acoustic vibration problem. Through all our study we consider an inviscid fluid, leading to a linear eigenvalue problem. The acoustic problem is written, in first place, in terms of the displacement. Under the approach of the non-compact operators theory, we prove convergence and error estimates for the method when the displacement formulation is considered. We analyze the influence of the stabilization parameter on the computation of the spectrum, where spurious eigenmodes arise when this parameter is not correctly chosen. Alternatively we present the formulation depending only on the pressure, comparing the performance of the DG methods with the pure displacement formulation. Computationally, we study the influence of the stabilization parameter on the arising of spurious eigenvalues when the spectrum is computed. Also, we report tests in two and three dimensions where convergence rates are reported, together with a comparison between the displacement and pressure formulations for the proposed DG methods.

Joint k-numerical ranges of operators

Let \(\mathcal{H} \) be a complex Hilbert space and \(\mathcal{B}(\mathcal{H}) \) the algebra of all bounded linear operators on \( \mathcal{H} \). For a positive integer k less than the dimension of \(\mathcal{H} \) and \({\bf A}=(A_1,...,AM)\in\mathcal{B}(\mathcal{H})^{m} \), the joint k-numerical range \(W_k(\bf A) \) is the set of \((\alpha_1,..,\alpha_m)\in \mathbb{C^m}\) such that \(\alpha_{i}={\Sigma^{k}_{j=1}} \langle A_{i}x{j},x{j }\rangle \) for an orthonormal set \(\{x_{1},...,x_{k}\} \) in \(\mathcal{H} \). The geometrical properties of \( W_k(\bf A) \) and their relations with the algebraic properties of \(\{A_{1},...,A_{m}\} \) are investigated in this paper. First, conditions for\(W_k(\bf A) \) to be convex are studied, and some results for finite dimensional operators are extended to the infinite dimensional case. An example of A is constructed such that \(W_k(\bf A) \) is not convex, but \(W_r(\bf A) \) is convex for all positive integer r not equal to k. Second, descriptions are given for the closure of \(W_k(\bf A) \) and the closure of conv\(W_k(\bf A) \) in terms of the joint essential numerical range of A for infinite dimensional operators\(A_{1},...,A_{m} \) . These lead to characterizations of \(W_k(\bf A) \)Wk(A) or conv\(W_k(\bf A) \) to be closed. Moreover, it is shown that conv \(W_k(\bf A) \) is closed whenever \(W_{k+1}(\bf A) \) or conv\(W_{k+1}(\bf A) \) is. These results are used to study the connection between the geometric properties of \( W_k(\bf A) \) and algebraic properties of \(A_{1},...,A_{m} \). For instance, \(W_k(\bf A) \) is a polyhedral set, i.e., the convex hull of a finite set, if and only if \(A_{1},...,A_{k} \) have a common reducing subspace V of finite dimension such that the compression of \(A_{1},...,A_{m} \) on the subspace V are diagonal operators \( D_{1},...,D_{m} \) and \( W_k(A)=W_{k}(D_1,...,D_m) \). Characterization is also given to A such that the closure of \(W_k(\bf A) \) is polyhedral. For finite rank operators the following two condition are equivalent: (1) \(A_{1},...,A_{m} \) is a commuting family of normal operators. (2) \(W_k(A)=W_{k}(A_1,...,A_m) \) is polyhedral for every positive integer k less than dim\(\mathcal{H} \). However, the two conditions are not equivalent for compact operators. Characterizations are given for compact operators \( A_{1},...,A_{m} \) satisfying (1) and (2), respectively. Results are also obtained for general non-compact operators. Open problems and future research topics are presented.

Strictly singular non-compact operators between $L_p$ spaces

We study the structure of strictly singular non-compact operators between L_p spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators T , for which the set of points (1/p,1/q)\in(0,1)\times (0,1) such that T\colon L_p\rightarrow L_q is strictly singular but not compact contains a line segment in the triangle \{(1/p,1/q):1<p<q<\infty\} of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of L_p is also explored.

A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM

Abstract We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function x ∈ L 2 (0, 1) from elements of the infinite dimensional sequence space ` 2 that characterize moments applied to the antiderivative of x. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator A, which is the composition of the compact simple integration operator mapping in L 2 (0, 1) and the non-compact Hausdorff moment operator B(H) mapping from L 2 (0, 1) to ` 2 . There is a seeming contradiction be- tween (a) numerical computations, which show (even for large n) an exponential decay of the singular values for n-dimensional matrices obtained by discretizing the operator A, and (b) a strongly limited smoothness of the well-known kernel k of the Hilbert-Schmidt operator A∗A. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness oc- curs in common just for C∞-kernels k. We recall arguments for the possible occurrence of a polynomial decay of the singular values of A, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.

Sum of generalized weighted composition operators between weighted spaces of analytic functions

Let [Formula: see text] be the space of analytic functions on the unit disc [Formula: see text] and let [Formula: see text] denote the set of analytic self-maps of [Formula: see text]. Let [Formula: see text] be such that [Formula: see text] and [Formula: see text]. We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators [Formula: see text] between weighted Banach spaces of analytic functions [Formula: see text] and [Formula: see text] which unifies the study of products of composition operators, multiplication operators and differentiation operators. As applications, we obtain the boundedness and compactness of the generalized weighted composition operators [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators [Formula: see text] and [Formula: see text] are given. Examples of bounded, unbounded, compact and non-compact operators [Formula: see text] and [Formula: see text] are given to explain the role of inducing functions [Formula: see text], [Formula: see text] and the weights [Formula: see text], [Formula: see text] of the underlying weighted spaces.

A note on numerical singular values of compositions with non-compact operators

Linear non-compact operators are difficult to study because they do not exist in the finite-dimensional world. Recently, Hofmann and Mathé [Electron. Trans. Numer. Anal., 57 (2022), pp. 1–16] studied the singular values of the compact composition of the non-compact Hausdorff moment operator and the compact integral operator and found credible arguments, but no strict proof, that those singular values fall only slightly faster than those of the integral operator alone. However, the fact that numerically the singular values of the combined operator fall exponentially fast was not mentioned. In this note, we supply the missing numerical results and provide an explanation why the two seemingly contradictory results may both be true.

Super-convergence analysis of collocation methods for linear and nonlinear third-kind Volterra integral equations with non-compact operators

This paper concerns iterated collocation methods for linear and nonlinear Volterra integral equations (VIEs) of the third kind with noncompact operators. The optimal global and local super-convergence orders are determined under a new mesh. Finally, some numerical examples are provided to verify the convergence results.

Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

A priori error analysis for a mixed VEM discretization of the spectral problem for the Laplacian operator

The aim of the present work is to derive error estimates for the Laplace eigenvalue problem in mixed form, implementing a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is spurious free and convergent. Optimal order of convergence for the eigenvalues and eigenfunctions are derived. Finally, we report numerical tests to confirm the theoretical results together with a rigorous computational analysis of the effects of the stabilization parameter, inherent for the virtual element methods, in the computation of the spectrum.

Spectral properties of a non-compact operator in ecology.

Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate [Formula: see text], or a stable-stage distribution [Formula: see text]. In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector [Formula: see text] corresponding to its spectral radius [Formula: see text], this [Formula: see text] is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population [Formula: see text], there is a [Formula: see text] such that [Formula: see text].

A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

In this paper we analyze a virtual element method (VEM) for a pseudostress formulation of the Stokes eigenvalue problem. This formulation allows to eliminate the velocity and the pressure, leading to an elliptic formulation where the only unknown is the pseudostress tensor. Adapting the non-compact operator theory, we prove that our method provides a correct approximation of the spectrum and is spurious free. We derive optimal a priori error estimates, which we confirm with some numerical tests. Also we present a computational spurious analysis of the proposed method.

Collocation methods for third-kind Volterra integral equations with proportional delays

In this paper, a class of Volterra delay integral equations (VDIEs) with noncompact operators is approximated by collocation methods. The properties of corresponding operators as well as existence, uniqueness and regularity of exact solution are discussed. The existence and uniqueness of collocation solutions are proved under two special graded meshes. Moreover, we present the convergence conditions and convergence order. Finally, some numerical examples are given to verify the validity of the theoretical orders of convergence.