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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

Hilbert Space Banach Space Spectral Image Spectral Radius Continuous Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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