By Robin Harte
The remedy develops the idea of open and virtually open operators among incomplete areas. It builds the expansion of a normed area and of a bounded operator and units up an simple algebraic framework for Fredholm concept. The method extends from the definition of a normed area to the perimeter of contemporary multiparameter spectral idea and concludes with a dialogue of the kinds of joint spectrum. This variation incorporates a short new Prologue via writer Robin Harte in addition to his long new Epilogue, "Residual Quotients and the Taylor Spectrum."
Dover republication of the variation released by way of Marcel Dekker, Inc., big apple, 1988.
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Extra resources for Invertibility and Singularity for Bounded Linear Operators
Example text
Die Auswahl an Begriffen und Sätzen ist jedoch so gewählt, dass auch ohne weiteres Literaturstudium die nachfolgenden Kapitel verstanden werden können. Wir betrachten lineare Abbildungen T : X → Y . Hierbei bezeichnen (X, · X ) und (Y, · Y ) normierte Räume über demselben Körper K (= R ∨ C). T heißt beschränkt := ∃ C ≥ 0 ∀ u ∈ X : Tu Y ≤C u X . 14) Mit diesem Begriff kann die Stetigkeit linearer Operatoren charakterisiert werden. 24 Für lineare Operatoren T : X → Y sind die folgenden drei Bedingungen äquivalent: (i) T ist stetig auf ganz X.
47 X := Y := l2 und T (x1 , x2 , x3 , . ) := (0, x1 , x2 , . ) die Rechtsverschiebung. Da T injektiv ist, folgt dim ker(T ) = 0. Andererseits ist codim R(T ) = 1 und somit ind(T ) = −1. Man überlegt sich leicht, dass ind(T n ) = −n (n ∈ N) gilt. Wir betrachten ein Beispiel eines linearen stetigen Operators, der kein FredholmOperator ist. 52 Es seien X = Y := C 0 [−1, 1] und beide Räume mit der Max-Norm ausgestattet. Der Operator T sei definiert durch s T :X→Y , (T x)(s) := x(t) dt −1 (s ∈ [−1, 1]) .
Iii) T ist stetig in x0 ∈ X. Damit haben wir ein bemerkenswertes Ergebnis: Ein linearer Operator ist entweder in jedem Punkt stetig oder in jedem Punkt unstetig. Wir setzen L(X, Y ) := { T : X → Y | T ist linear und beschränkt } und nennen L(X, Y ) den Vektorraum der linearen stetigen Operatoren. Es lässt sich nämlich leicht zeigen, dass L(X, Y ) mit den üblichen Operationen der Addition „+“ und der skalaren Multiplikation „·“ ein Vektorraum ist. Wir führen nun in L(X, Y ) eine geeignete Norm ein.