# 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.