Israel Gohberg

Shift invariant subspaces, factorization, and interpolation for matrices. I. the canonical case

Abstract This article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper (or lower) triangular matrices. This gives a unifying point of view to several general kinds of factorization problems, including LU, QR, and Cholesky factorization. A second application is to some matrix completion problems, such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1. Included is a linear fractional parametrization of the set of all solutions of this problem.

Recent Advances in Operator Theory and Related Topics

Inverse problems associated to a canonical differential system.- Construction of Schwarz norms.- On the class of extremal extensions of a nonnegative operator.- On Livsic-Brodskii nodes with strongly regular J-inner characteristic matrix functions in the Hardy class.- Scalar perturbations of the Sz.-Nagy-Foias charac-teristic function.- Inequalities for eigenvalues of sums in a von Neu-mann algebra.- Weighted variants of the Three Chains Completion Theorem.- Semigroups in finite von Neumann algebras.- Singly generated algebras containing a coin-pact operator.- Analytic extension of vector valued functions.- On quotient modules.- On the structure of spherical contractions.- Apostols bilateral weighted shifts are hyper-reflexive.- Wielandt type extensions of the Heinz-Kato-Furuta inequality.- Logarithmic order and dual logarithmic order.- On the generalized von Neumann inequality.- Ultraproducts of C*-algebras.- Intertwining extensions and a two-sided corona problem.- On self-polar Hilbertian norms on (indefinite) inner productspaces.- Schur norms and the multivariate von Neumann inequality.- Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes.- On the hyperinvariant subspace problem for asymptotically non-vanishing contractions.- Unstable dynamics on a Markov background and stability in average.- A relation for the spectral shift function of two self-adjoint extensions.- Beppo Levi and Lebesgue type theorems for bundle convergence in noncommutative L2-spaces.- *-semigroup endomorphisms of B(H).- Spectral singularities, Szokefalvi-Nagy-Foias functional model and the spectral analysis of the Boltzmann operator.- Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces.- The Friedrichs operator of a planar domain. II.- Localization of the Wielandt-Wintner Theorem.- Order and square roots in hermitian Banach *-algebras.- Unitary dilation of several contractions.- Inequalities for semibounded operators and their applications to log-hyponormal operators.- Operator moment problems in unbounded sets.- The argument principle and boundaries of analytic varieties.- Conference Program.- List of Participants.

Contributions to operator theory in spaces with an indefinite metric

Heinz Langer and his work, A. Dijksma, I. Gohberg on the spectra of some class of quadratic operator pencils, V. Adamyan, V. Pivovarchik special realizations for Schur upper triangular operators, D. Alpay, Y. Peretz on the defect of noncontractive operators in Krein spaces -a new formula and some applications, T. Ya et al positive differential operators in the Krein space L2(Rn), B. Curgus, B. Najman singular values of positive pencils and applications, R.L. Ellis et al perturbations of Krein spaces preserving the nonsingularity of the critical point infinity, A. Fleige, B. Najman an analysis of the block structure of jqq-inner functions, B. Fritzsche et al selfadjoint extensions of the orthogonal sum of symmetric relations II, S. Hassi et al some interpolation problems of Nevanlinna-Pick type - the Krein-Langer method, s. Hassi et al on the spectral representation for singular selfadjoint boundary eigenvalue problems, D. Hinton, A. Schneider some characteristics of a linear manifold in a Krein space and their applications, E.I. Iokvidow riggings and relatively from bounded perturbations of nonnegative operators in Krein spaces, P. Jonas norm bounds for Volterra integral operators and time-varying linear systems with finite horizon, M.A. Kaashoek, A.C.M. Ran the numerical range of selfadjoint matrix polynomials, P. Lancaster et al spectral properties of a matrix polynomial connected with a component of its numerical range, A. Markus et al Lyapunov stability of a multiplication operator perturbed by a Volterra operator, S. Naboko, C. Tretter Multiplicative perturbations of positive operators in Krein spaces, B. Najman, K. Veselic on the number of negative squares of certain functions, Z. Sasvari factorization of elliptic pencils and the Mandelstam hypothesis, A.A. Shkalikov an inductive limit procedure within the quantum harmonic oscillator, F.H. Szafraniec canonical systems with a semibounded spectrum, H. Winkler.