Spectra

  • Francis Hirsch
  • Gilles Lacombe
Part of the Graduate Texts in Mathematics book series (GTM, volume 192)

Abstract

We fix here a Banach space E over K ℝ or ℂ, and we wish to study the (noncommutative) Banach algebra L(E) of continuous linear maps from E to E, the product operation being composition. We use the same notation ∥·∥ for the norm on E and the associated norm on L(E), and we denote by I the identity map on E. Thus, I is the unity of the algebra L(E). An element TL(E) is called invertible if it has an inverse in L(E); that is, if there exists a continuous linear map S such that TS = ST = I. Because composition is associative, T has an inverse in L(E) if and only if it has a right inverse (an element U such that TU = I) and a left inverse (an element V such that VT = I) in L(E). Clearly, if T is invertible, it is bijective and its inverse in L(E) is unique and equals the inverse map T-1 Thus, for TL(E), the following properties are equivalent:
  • T is invertible.

  • T is bijective and T-1 is continuous.

  • ker T = {0}, im T=E, and T-1 is continuous.

Keywords

Radon Toll Prool 

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Francis Hirsch
    • 1
  • Gilles Lacombe
    • 1
  1. 1.Département de MathématiquesUniversité d’Évry-Val d’EssonneÉvry CedexFrance

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