Showing papers in "Mathematische Annalen in 1966"

Wave Operators and Similarity for Some Non-selfadjoint Operators

Abstract: The purpose of the present paper is to develop a new method for establishing the similarity of a perturbed operator T(ϰ), formally given by T + ϰV, to the unperturbed operator T. It is basically a “small perturbation” theory, since the parameter ϰ is assumed to be sufficiently small. Otherwise the setting of the problem is rather general; T or V need not be symmetric or bounded, although they are assumed to act in a separable Hilbert space r and ϰ need not be real. The basic assumptions are that the spectrum of T is a subset of the real axis, that V can be written formally as V = B* A, where A is T-smooth and B is T*-smooth (see Definition 1.2), and that A(T − ζ)−1 B* is uniformly bounded for nonreal ζ. Here A and B are (in general unbounded) operators from r to another Hilbert space r′ (r′ = r is permitted). Strictly speaking, we are dealing with a certain extension T(ϰ) of T + ϰB* A, which is uniquely determined by T, A, and B; T(ϰ) = T + ϰB* A is true if A and B are bounded1).

Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II

Abstract: Diese Arbeit erscheint in zwei Teilen. Die Aufteilung ist aus der nachfolgenden Inhaltsubersicht zu ersehen. Die Einleitung bezieht sich auf beide Teile zusammen. Eine Liste haufig benutzter Bezeichnungen und das Literaturverzeichnis stehen am Ende des ersten Teiles der Arbeit. Zahlen in eckigen Klammern stellen Verweise auf das Literaturverzeichnis dar. Formeln, Definitionen und Satze sind paragraphenweise numeriert — Satz 3.1 bedeutet z. B. den ersten Satz in § 3. Das Ende des Beweises eines Satzes oder Lemmas ist stets durch einen fetten senkrechten Strich markiert.

Nonlinear operators in Banach spaces

Abstract: From the point of view of its applications to nonlinear boundary value problems for partial differential equations (as well as to other problems in nonlinear analysis) the principal result of the Leray-Schauder theory [9] of nonlinear functional equations is embodied in the following theorem: L-S Theorem: Let G be an open subset of the Banach space X, C a mapping of\( \overline G \times [0,1] \)into X whose image is precompact and such that if C t (x) = C(x, t), each mapping C t of\( \overline G \)into X has no fixed points on the boundary of G. Then if I — C0is a homeomorphism of G on an open set of X containing 0, the equation (I — C1) u = 0 has a solution u in G (i.e, C1has a fixed point in G).

Some Examples of Normed Köthe Spaces

Abstract: Let X be a non-empty point set, and μ a countably additive and non-negative measure in X. We assume that the Caratheodory extension procedure has already been applied to μ, so that the σ-field Λ on which μ is defined cannot be enlarged by another application of the Caratheodory procedure. Furthermore, it will be assumed that μ is (totally) (σ-finite, i.e., X is the union of a finite or countable number of sets of finite measure. Hence, the triple (X, Λ, μ) is a (totally) σ-finite measure space in the usual terminology. The notation ∫ d μ will denote integration (with respect to μ) over the whole set X, and χ E = χ E (x) will stand for the characteristic function of the set E ⊂ X.

Über die Einbettung der nuklearen Räume in (s)A

Abstract: In dieser Arbeit geben wir eine neue Charakterisierung der nuklearen Raume an: Ein lokal konvexer Raum ist dann und nur dann nuklear, wenn er isomorph zu einem Teilraum des Produkts (s) A des Raumes (s) aller schnell fallenden Folgen ist. (A bezeichnet eine geeignete Indexmenge.) Damit haben wir gleichzeitig bewiesen, das jeder nukleare (F)-Raum in den Raum E (R 1) aller beliebig oft differenzierbaren Funktionen auf R 1 isomorph eingebettet werden kann. Dieses Problem stammt von Grothendieck [3], und in einigen speziellen Fallen waren positive Antworten bekannt, insbesondere im Fall von nuklearen (F)-Raumen mit Basis, d. h. im Fall von nuklearen, vollkommenen (F)-Raumen (vgl. [1], [3]).

A Representation Theorem for Positive Functionals on Involution Algebras

Abstract: D. Raikov has shown [6] that for a commutative Banach algebra A with symmetric involution, the set p of positive linear functionals on A having norm at most one is isometrically isomorphic to the set of positive measures (of norm at most one) defined on the maximal ideal space of A. Raikov’s proof of this theorem depends on the Gelfand theory of commutative Banach algebras and the Riesz-Markov Theorem (see also [8; p. 230]). Here we shall give a new and elementary proof of Raikov’s result by first proving a Radon-Nikodym type theorem for positive functionals (Theorem 1) and then showing directly that the extreme points of the compact convex set of positive linear functionals in the unit ball of A′ are exactly the set M of positive multiplicative linear functionals (Theorem 2). An application of the Krein-Milman Theorem makes possible the representation of every element of p as the centroid of a positive measure on M (Theorem 3) and uniqueness of this representation is a consequence of the Stone-Weierstrass Theorem.