A curious functional equation

For Israel Gohberg, outstanding analyst, with affection and admiration.     

A completeness theorem for the linear stability problem of nearly parallel flows

A completeness/expansion theorem, analogous to that of DiPrima and Habetler (D-H), is proved for the equation governing the linear stability of nearly parallel flows, to which the D-H theorem does not apply. It is also proved that only a finite number of eigenvalues with negative real parts can occur. Both results are based on a theorem of Gohberg and Kreǐn.     

Parametrization of minimal rank block Hankel matrix extensions and minimal partial realizations

A parametrization of allm-rowsn-columns extensions of block Hankel matrices with minimal rank is presented. As a particular case, a new parametrization of the minimal partial realizations of a sequence of blocks is given. In this way results of O.H. Bosgra [2], I. Gohberg/M.A. Kaashoek/L. Lerer [3], and H. Woerdeman [13] are extended.     

Singular integral operators on curves with corners

The symbol map of Gohberg and Krupnik [6] for the closed algebra generated by singular integral operators with piecewise continuous coefficients is extended to the case of curves with corners. This algebra includes the operator of the double layer potential on such curves.     

The Gohberg Lemma,compactness, and essential spectrum of operators on compact Lie groups

We prove a version of the Gohberg Lemma on compact Lie groups giving an estimate from below for the distance from a given operator to the set of compact operators. As a consequence, we obtain several results on bounds for the essential spectrum and a criterion for an operator to be compact. The conditions are given in terms of the matrix-valued symbols of operators.     

On complementary matrix algebras

It is proved that complementary algebras of linear operators on ⁿ do not necessarily have nontrivial complementary invariant subspaces. This settles a conjecture of Gohberg, Lancaster, and Rodman in the negative. A positive result is also proved under certain additional hypotheses.This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.     

A general look at local principles with special emphasis on the norm computation aspect

Local principles associate with every element of a Banach algebra a family of local objects in terms of which the properties of the original element can be studied. In this paper some general relations between three such principles, usually affiliated with the names of Simonenko, Allan/Douglas, and Gohberg/Krupnik, are discussed. Special attention is paid to the question on how the norm of an element can be expressed in terms of the norms of the local objects associated with it. The general theory is illustrated by some concrete results on singular integral operators and Toeplitz operators.     

On the reconstruction of Toeplitz matrix inverses from columns

In this paper we discuss the problem whether and how the inverse of a Toeplitz matrix can be recovered from some of its columns or parts of columns under the requirement that only 2n−1 parameters are involved. The results generalize and strengthen earlier findings by Trench, Gohberg, Semencul, Krupnik, Ben-Artzi, Shalom, Labahn, Rodman and others. Special attention is paid to symmetric, skewsymmetric and hermitian Toeplitz matrix inverses and the question whether such a matrix can be retrieved from a single column.     

Rational contractive and unitary interpolants in realized form

A solution of the rational Nehari problem is given in terms of a realization. Other aspects of this problem, like one step extension, maximum entropy interpolants and unitary interpolants, are also analyzed for the rational case. The results are based on earlier work of H. Dym and I. Gohberg.Dedicated to M.S. Livic on the occasion of his seventieth birthday, with admiration     

Shift‐invert Lanczos method for the symmetric positive semidefinite Toeplitz matrix exponential

The Lanczos method with shift‐invert technique is exploited to approximate the symmetric positive semidefinite Toeplitz matrix exponential. The complexity is lowered by the Gohberg–Semencul formula and the fast Fourier transform. Application to the numerical solution of an integral equation is studied. Numerical experiments are carried out to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

In memoriam Israel Gohberg August 23, 1928-October 12, 2009

This obituary for Israel Gohberg consists of a general introduction, separate contributions of the six authors, all of whom worked closely with him, and a final note. The material gives an impression of the life of this great mathematician, of his monumental impact in the areas he worked in, of how he cooperated with colleagues, and of his ability to stimulate people in their mathematical activities.     

Local theory of the collocation method for the approximate solution of singular integral equations,I

A new approach for investigating the collocation method for singular integral equations is given. Using Banach algebra techniques and a local principle established by I. Gohberg and N. Krupnik under suitable conditions the convergence of the Lagrange- and Mul thopp-collocation method is proved for one-dimensional singular integral equations and systems of such equations with piecewise continuous coefficients having a countable infinite number of discontinuities.     

Minnaert resonances for acoustic waves in bubbly media

Through the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.     

On pasting -weights

Theorem 3 gives a condition when two -weights can be ``pasted' together to yield another -weight. It is subsequently used in Example 6 to give an example that shows that a necessary condition by Gohberg, Krupnik, and Spitkovsky is not sufficient.

     

Algorithms for fast multidimensional signal extraction

In recent studies operator factorization techniques have been developed for problems associated with multidimensional control and image processing. In particular the “special factorization” of Gohberg and Krein and the Schur-Coleski factorization have both been extended to the multivariate setting. The present study focuses on computational aspects of the earlier results. Also of interest are implications for designing computer architectures which facilitate high-speed, online computation of the operator factors.     

Stable factorizations of monic matrix polynomials and stable invariant subspaces

The stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties. Proofs are based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman. A large part of the paper is devoted to a detailed analysis of stable invariant subspaces of a matrix. The results are also used to describe all stable solutions of the operator Riccati equation.     

Boundedness and spectral invariance for standard pseudodifferential operators on anisotropically weighted LP-Sobolev spaces

It is shown that pseudodifferential operators with symbols in the standard classes S _, ^m (ⁿ) define bounded maps between large classes of weighted L^P-Sobolev spaces where the growth of the weight does not have to be isotropic. Moreover, the spectrum is independent of the choice of the space.Dedicated to Professor Israel Gohberg on the occasion of his sixtieth birthday     

Matrix representations of split Bezoutians

Inverses of symmetric (or skewsymmetric) Toeplitz matrices as well as of centrosymmetric (or centro-skewsymmetric) Toeplitz-plus-Hankel matrices can be represented as sums of two split Bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors. Thus it is desirable to find matrix representations for split Bezoutians B. This is the main aim of the present paper.Recursion formulas for the entries of B are presented, bases of very simple split Bezoutians or of sparse matrices are constructed, and B is represented as a corresponding linear combination. Moreover, matrix representations of Gohberg/Semencul type are established.