Linear Operators In Hilbert Spaces Research Articles

On Linear Operators in Hilbert Spaces and Their Applications in OFDM Wireless Networks

This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input OFDM signal, can be described as a mapping from an input Hilbert space to an output Hilbert space, with the system response governed by linear operator theory. By employing the mathematical framework of Hilbert spaces, we formalise the representation of OFDM signals, which are interpreted as elements of an infinite-dimensional vector space endowed with an inner product. The LTI wireless channel is characterised by using bounded linear operators on these spaces, allowing for the decomposition of complex channel behaviour into a series of linear transformations. The channel’s impulse response is treated as a kernel operator, facilitating a functional analysis approach to understanding the signal transmission process. This representation enables a more profound understanding of channel effects, such as fading and interference, through the eigenfunction expansion of the operator, leading to a spectral characterization of the channel. The algebraic properties of linear operators are leveraged to develop optimal solutions for mitigating channel distortion effects.

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.

Self-adaptive CQ-type algorithms for the split feasibility problem involving two bounded linear operators in Hilbert spaces

In this article, we consider and investigate a split convex feasibility problem involving two bounded linear operators in Hilbert spaces. We introduce a self-adaptive CQ-type algorithm by selecting the stepsize which is independent of the operator norms and establish a strong convergence result of the proposed algorithm under some mild control conditions. Moreover, we propose a self-adaptive relaxed CQ-type algo- rithm for solving the problem constrained by sub-level sets of convex functions. A numerical example and an application in compressed sensing are also given to illustrate the convergence behaviour of our proposed algorithms. Our results in this paper improve and generalize some existing results in the literature

On a paradox in quantum mechanics and its resolution

Consider a free Schrödinger particle inside an interval with walls characterized by the Dirichlet boundary condition. Choose a parabola as the normalized state of the particle that satisfies this boundary condition. To calculate the variance of the Hamiltonian in that state, one needs to calculate the mean value of the Hamiltonian and that of its square. If one uses the standard formula to calculate these mean values, one obtains both results without difficulty, but the variance unexpectedly takes an imaginary value. If one uses the same expression to calculate these mean values but first writes the Hamiltonian and its square in terms of their respective eigenfunctions and eigenvalues, one obtains the same result as above for the mean value of the Hamiltonian but a different value for its square (in fact, it is not zero); hence, the variance takes an acceptable value. From whence do these contradictory results arise? The latter paradox has been presented in the literature as an example of a problem that can only be properly solved by making use of certain fundamental concepts within the general theory of linear operators in Hilbert spaces. Here, we carefully review those concepts and apply them in a detailed way to resolve the paradox. Our results are formulated within the natural framework of wave mechanics, and to avoid inconveniences that the use of Dirac’s symbolic formalism could bring, we avoid the use of that formalism throughout the article. In addition, we obtain a resolution of the paradox in an entirely formal way without addressing the restrictions imposed by the domains of the operators involved. We think that the content of this paper will be useful to undergraduate and graduate students as well as to their instructors.

Efficient Approaches for Verifying the Existence and Bound of Inverse of Linear Operators in Hilbert Spaces

This paper describes some numerical verification procedures to prove the invertibility of a linear operator in Hilbert spaces and to compute a bound on the norm of its inverse. These approaches improve on previous procedures that use an orthogonal projection of the Hilbert space and its a priori error estimations. Several verified examples which confirm the effectiveness of the new procedures are presented.

Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials.

Lebesgue type decompositions and Radon–Nikodym derivatives for pairs of bounded linear operators

For a pair of bounded linear Hilbert space operators A and Bone considers the Lebesgue type decompositions of B with respect toA into an almost dominated part and a singular part, analogous to theLebesgue decomposition for a pair of measures in which case one speaks ofan absolutely continuous and a singular part. A complete parametrizationof all Lebesgue type decompositions will be given, and the uniqueness ofsuch decompositions will be characterized. In addition, it will be shownthat the almost dominated part of B in a Lebesgue type decomposition hasan abstract Radon–Nikodym derivative with respect to the operator A.

On the nonlinear perturbations of self-adjoint operators

Abstract Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=N\left(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {\mathcal{ {\mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

Core–EP orthogonal operators

The concept of the core orthogonality (CO) was recently introduced for two square matrices A and B of the same size and of index at most 1 as follows: A is core orthogonal to B if and , where is the core inverse of A. Using the core–EP inverse instead of the core inverse, we extend the concept of the CO and present the new concept of the core–EP orthogonality (CEPO) for two Drazin invertible bounded linear Hilbert space operators A and B as follows: A is said to be core–EP orthogonal to B if and , where is the core–EP inverse of A. A number of characterizations for CEPO are presented as well as the relation between the CEPO and core–EP additivity. Applying CEPO, the concept of strong core–EP orthogonality is defined and characterized.

The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

Quasiaffine Inverses of Linear Operators in Hilbert Spaces

Quasiaffine Inverses of Linear Operators in Hilbert Spaces

Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law A B C † = C † B † A † . Moreover, several equivalent statements of ℛ A A ∗ A B C = ℛ A B C and ℛ C ∗ C A B C ∗ = ℛ A B C ∗ are also deducted by the theory of operators.

Hilbert space operators with two-isometric dilations

A continuous linear Hilbert space operator S is said to be a 2-isometry if the operator S and its adjoint S∗ satisfy the relation S∗2S2−2S∗S+I=0. We study operators having liftings or dilations to 2-isometries. The adjoint of an operator which admits such liftings is the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators and to operators similar to contractions. Two types of liftings to 2-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.

Frame decompositions of bounded linear operators in Hilbert spaces with applications in tomography

We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.

Is it possible to give a more precise formulation of the criterion of maximal accretivity for one extension of nonnegative operator?

The conditions being necessary and sufficient for maximal accretivity and maximal nonnegativity of some closed linear operators in Hilbert space are announced. The following problem is proposed: write down these conditions in more convenient form (one of the admissible variants is indicated).

A Note on Real Operator Monotone Functions

Abstract In this paper, we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

M-Isometric operators and their local properties

In this paper we give necessary and sufficient conditions for a bounded linear Hilbert space operator to be an m-isometry for an unspecified m written in terms of conditions that are applied to “one vector at a time”. We provide criteria for orthogonality of generalized eigenvectors of an (a priori unbounded) linear operator T on an inner product space that correspond to distinct eigenvalues of modulus 1. We also discuss a similar question of when Jordan blocks of T corresponding to distinct eigenvalues of modulus 1 are “orthogonal”.

Spatio-angular fluorescence microscopy I. Basic theory.

We introduce the basic elements of a spatio-angular theory of fluorescence microscopy, providing a unified framework for analyzing systems that image single fluorescent dipoles and ensembles of overlapping dipoles that label biological molecules. We model an aplanatic microscope imaging an ensemble of fluorescent dipoles as a linear Hilbert-space operator, and we show that the operator takes a particularly convenient form when expressed in a basis of complex exponentials and spherical harmonics-a form we call the dipole spatio-angular transfer function. We discuss the implications of our analysis for all quantitative fluorescence microscopy studies and lay out a path toward a complete theory.

Regular factorizations and perturbation analysis for the core inverse of linear operators in Hilbert spaces

ABSTRACTThis paper concerns the regular factorization and expression of the core inverse in a Hilbert space. Utilizing the regular factorization, we first give some characterizations for the existence and the expression of the group inverse and core inverse. Based on these, we prove that the core inverse of the perturbed operator has the simplest possible expression if and only if the perturbation is range-preserving, and derive an explicit expression under the rank-preserving perturbation. Thus we can conclude that both the range-preserving perturbation and the rank-preserving perturbation are all continuous perturbations. The obtained results extend and improve many recent ones in matrix theory and operator theory.

An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces

In the present paper, we propose a computer-assisted procedure to prove the invertibility of a linear operator in a Hilbert space and to compute a verified norm bound of its inverse. A number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates. The approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications. Several verification examples that confirm the actual effectiveness of the proposed procedure are reported.