Günter Lumer

Bio: Günter Lumer is an academic researcher from University of Mons. The author has contributed to research in topics: Bounded function & Hilbert space. The author has an hindex of 14, co-authored 50 publications receiving 1434 citations. Previous affiliations of Günter Lumer include University of Mons-Hainaut & University of California, Los Angeles.

Semi-inner-product spaces

Abstract: function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the terminology and techniques are developed. On the other hand, this type of Hilbert space considerations find no real parallel in the general Banach space setting. Some time ago, while trying to carry over a Hilbert space argument to a general Banach space situation, we were led to use a suitable mapping from a Banach space into its dual in order to make up for the lack of an innerproduct. Our procedure suggested the existence of a general theory which it seemed should be useful in the study of operator (normed) algebras by providing better insight on known facts, a more adequate language to "classify" special types of operators, as well as new techniques. These ideas evolved into a theory of semi-inner-product spaces which is presented in this paper (together with certain applications)(1). We shall consider vector spaces on which instead of a bilinear form there is defined a form [x, y] which is linear in one component only, strictly positive, and satisfies a Schwarz inequality. Such a form induces a norm, by setting |x| = ([x, x])112; and for every normed space one can construct at least one such form (and, in general, infinitely many) consistent with the

Linear operator equations

Abstract: where / is holomorphic on a(u, Ct) — a(v, Ct), and c is a suitable contour. In the present note we study the operator SG® defined by Sx = 'YJi=iuixvi, {ui} and {vj} being commutative subsets of Ct; but it is understood that {uj} need not commute with {vj}. In a generic way, we shall refer to such an operator as an "elementary operator."2 Of course this is related to the problem of solving in Ct a general system of linear equations. We generalize the analysis made in [4] covering the ux — xv case, and extend the following unpublished theorem of D. C. Kleinecke: (1.3) If Ct is the Banach algebra of all operators on a Banach space, then "C" may be replaced by " = " in (1.1).

On the isometries of reflexive Orlicz spaces

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On the Generators of Quantum Dynamical Semigroups

Abstract: The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

The Rotation of Eigenvectors by a Perturbation. III

Abstract: When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given an estimate for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from the subspace spanned by our approximations? These questions are closely related; both are investigated here. The difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. These angles unify the treatment of natural geometric, operator-theoretic and error-analytic questions concerning those subspaces. Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation (1st question) or a computable residual (2nd question). An example is included.

Observation and Control for Operator Semigroups

Abstract: The evolution of the state of many systems modeled by linear partial difierentialequations (PDEs) or linear delay-difierential equations can be described by operatorsemigroups. The state of such a system is an element in an inflnite-dimensionalnormed space, whence the name \inflnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of flnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, difierential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Evolution Semigroups in Dynamical Systems and Differential Equations

Abstract: Introduction Semigroups on Banach spaces and evolution semigroups Evolution families and Howland semigroups Characterizations of dichotomy for evolution families Two applications of evolution semigroups Linear skew-product flows and Mather evolution semigroups Characterizations of dichotomy for linear skew-product flows Evolution operators and exact Lyapunov exponents Bibliography List of notations Index.