Linear Pencil Research Articles

(N, ε)-pseudospectra of bounded linear operators on ultrametric Banach spaces

In this paper, we prove that the essential pseudospectrum of bounded linear operator pencils is invariant under perturbation of completely continuous linear operators on ultrametric Banach spaces over a spherically complete field K and we establish a characterization of the essential pseudospectrum of a bounded linear operator pencils by means of the spectra of all perturbed completely continuous operators. Furthermore, we introduce and study the notion of (n,ε)-pseudospectrum of bounded linear operators and the concept of (n,ε)-pseudospectrum of bounded linear operator pencils on ultrametric Banach spaces. We establish some results about them. Finally, several examples are provided.

Banach algebra mappings preserving the invertibility of linear pencils

Let A and B be complex unital Banach algebras, and let φ,ψ:A→B be surjective mappings. If A is semisimple with an essential socle and φ and ψ together preserve the invertibility of linear pencils in both directions, that is, for any x,y∈A and λ∈C, λx+y is invertible in A if and only if λφ(x)+ψ(y) is invertible in B, then we show that there exists an invertible element u in B and a Jordan isomorphism J:A→B such that φ(x)=ψ(x)=uJ(x) for all x∈A.

Eco-friendly fabrication of robust flexible superhydrophobic double-layer composite film by a blade coating method

Herein, a flexible superhydrophobic double-layer composite film was fabricated by a simple, low-cost, eco-friendly blade coating method under room temperature. The film can maintain superhydrophobicity after 300 bending and twisting cycles, indicating the outstanding flexibility of the film. The film can easily transfer to objects with complex shape and large curvature by using commercial double-sided tape due to its good flexibility. Moreover, the results of linear friction and pencil hardness tests show that the double-layer composite film still exhibits superhydrophobicity after 200 cycles of linear friction tests and possesses a 2H hardness, demonstrating the good mechanical robustness of the film. It can be expected that this work can provide inspiration for developing novel flexible superhydrophobic film.

The pseudospectra of linear operator pencils in a Hilbert space

The pseudospectra of linear operator pencils in a Hilbert space

Multiscale matrix pencils for separable reconstruction problems

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3–6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods.

Nontrivial invariant subspaces of linear operator pencils

AbstractIn this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair $\mathbf {T}=(T_{1},T_{2})$ and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of $\mathbf {T}$ and the linear pencil of the original pair $\mathbf {T}$ of bounded operators with dense ranges.

Spectral properties related to generalized complementary Romanovski–Routh polynomials

Complementary Romanovski–Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski–Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $$R_{II}$$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $$R_{II}$$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil are analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.

Linear pencils and quadratic programming problems with a quadratic constraint

Given bounded selfadjoint operators A and B acting on a Hilbert space H, consider the linear pencil P(λ)=A+λB, λ∈R. The set of parameters λ such that P(λ) is a positive (semi)definite operator is characterized. These results are applied to solving a quadratic programming problem with an equality quadratic constraint (or a QP1EQC problem).

Berry-Esseen bounds for the multivariate ℬ-free CLT and operator-valued matrices

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.

Ranks of linear matrix pencils separate simultaneous similarity orbits

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils L=T0+x1T1+⋯+xmTm on matrix tuples as L(X1,…,Xm)=I⊗T0+X1⊗T1+⋯+Xm⊗Tm. It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, m-tuples A and B of n×n matrices are simultaneously similar if and only if rkL(A)=rkL(B) for all linear matrix pencils L of size mn. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.

Optimum Synthesis of Pencil Beams with Constrained Dynamic Range Ratio

In antenna array design, low dynamic range ratio (DRR) of excitation coefficients is important because it simplifies array’s feeding network and enables better control of mutual coupling. Optimization-based synthesis of pencil beams allows explicit control of DRR. However, incorporating DRR into an optimization problem leads to nonconvex constraints, making its solving challenging. In this paper, a framework for global optimization of linear pencil beams with constrained DRR is presented. By using this framework, the methods for synthesis of pencil beams with minimum sidelobe level and minimum sidelobe power are developed. Both methods utilize convex problems suitable for the synthesis of pencil beams whose coefficients’ signs are known in advance. By incorporating these problems into a branch and bound algorithm, the procedures for global optimizations are formed which systematically search the space of all coefficient signs. The method for minimization of sidelobe power is further analyzed in the context of beam efficiency. It is shown that this method can be utilized in an approximate and at the same time global design of pencil beam arrays with maximum beam efficiency and constrained DRR. Based on this approach, a method for the design of pencil beam arrays with minimum DRR and specified beam efficiency is proposed.

Matrix pencils with the numerical range equal to the whole complex plane

The main purpose of this article is to show that the numerical range of a linear pencil λA+B is equal to C if and only if 0 belongs to the convex hull of the joint numerical range of A and B. We also prove that if the numerical range of a linear pencil λA+B is equal to C and A+A⁎,B+B⁎⩾0, then A and B have a common isotropic vector.Moreover, we improve the classical result which describes Hermitian linear pencils.

On a conjecture of Davies and Levitin

Let be a symmetric tridiagonal matrix with diagonal elements and off‐diagonal elements one, and be a diagonal matrix with the first diagonal elements being plus ones and the last being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil as approaches to infinity. It was conjectured by DL that for any the non‐real eigenvalues of satisfy both and . The conjecture has been verified numerically for a wide range of and , but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non‐real eigenvalues of . We provide a proof of the conjecture for , and also in the case where . In addition, numerics indicate that some phenomena may occur for more general linear pencils.

Some spectral sets of linear operators pencils on non-archimedean Banach spaces

In this paper, we define the notions of trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Many results are proved about trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Examples are given to support our work.

On the condition spectrum of linear operator pencils

On the condition spectrum of linear operator pencils

A Note on Essential Spectra of Linear Operator Pencils

A Note on Essential Spectra of Linear Operator Pencils

COINTEGRATION AND REPRESENTATION OF COINTEGRATED AUTOREGRESSIVE PROCESSES IN BANACH SPACES

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in$C[0,1]$equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

Distinguished varieties through the Berger-Coburn-Lebow theorem

A distinguished algebraic variety in $\mathbb{C}^2$ has been the focus of much research in recent years because of good reasons. This note gives a different perspective. (1) We find a new characterization of an algebraic variety $\mathcal W$ which is distinguished with respect to the bidisc. It is in terms of the joint spectrum of a pair of commuting linear matrix pencils. (2) There is a characterization known of $\mathbb{D}^2\cap\mathcal{W}$ due to a seminal work of Agler and McCarthy. We show that Agler--McCarthy characterization can be obtained from the new one and vice versa. (3) En route, we develop a new realization formula for operator-valued contractive analytic functions on the unit disc. (4) There is a one-to-one correspondence between operator valued contractive holomorphic functions and {\em canonical model triples}. This pertains to the new realization formula mentioned above. (5) Pal and Shalit gave a characterization of an algebraic variety, which is distinguished with respect to the symmetrized bidisc, in terms of a matrix of numerical radius no larger than $1$. We refine their result by making the class of matrices strictly smaller. (6) In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc. At the root of our work is the Berger--Coburn--Lebow theorem characterizing a commuting tuple of isometries.

Invariance of the essential spectra of closed linear operator pencils on a Banach space

Invariance of the essential spectra of closed linear operator pencils on a Banach space

Inverse numerical range and Abel-Jacobi map of Hermitian determinantal representation

Let A be an n×n matrix. The Hermitian parts of A are denoted by ℜ(A)=(A+A⁎)/2 and ℑ(A)=(A−A⁎)/(2i). The kernel vectors of the linear pencil xℜ(A)+yℑ(A)+zIn play a role for the inverse numerical range of A. This kernel vector technique was applied to perform the inverse numerical range of 3×3 symmetric matrices. In this paper, we follow the kernel vector method and apply the Abel theorem for 3×3 Hermitian matrices. We present the elliptic curve group structure of the cubic curve associated to the ternary form of the matrix, and characterize the Abel type additive structure of the divisors of the cubic curve. A numerical example is given to illustrate the characterization related to the Riemann theta representation.