On the joint spectral radius

Abstract

We prove the `p-spectral radius formula for n-tuples of commuting Banach algebra elements. This generalizes results of some earlier papers. Let A be a Banach algebra with the unit element denoted by 1. Let a = (a1, . . . , an) be an n-tuple of elements of A. Denote by σ(a) the Harte spectrum of a, i.e. λ = (λ1, . . . , λn) 6∈ σ(a) if and only if there exist u1, . . . , un, v1, . . . , vn ∈ A such that n ∑ j=1 (aj − λj)uj = n ∑ j=1 vj(aj − λj) = 1. Let 1 ≤ p ≤ ∞. The (geometric) spectral radius of a is defined by rp(a) = max{‖λ‖p : λ ∈ σ(a)}, where ‖λ‖p = { max1≤j≤n |λj | (p =∞), ( ∑n j=1 |λj |) (1 ≤ p <∞); see [10], cf. also [4]. If σ(a) is empty we put formally rp(a) = −∞. Clearly, rp(a) depends on p. On the other hand, instead of the Harte spectrum we can take any other reasonable spectrum (e.g. the left, right, approximate point, defect, Taylor etc.) without changing the value of rp(a); see [4], [9]. For a single Banach algebra element the just defined spectral radius rp(a) does not depend on p and coincides with the ordinary spectral radius r(a1) = max{|λ1| : λ1 ∈ σ(a1)}. By the well-known spectral radius formula 1991 Mathematics Subject Classification: Primary 46H05, 46J05.