J. Korean Math. Soc. 40 (2003), No. 2, pp. 307–317 ON SPECTRAL BOUNDEDNESS

Abstract. For linear operators between Banach algebras “spectral boundedness ” is derived from ordinary boundedness by substituting spectral radius for norm. The interplay between this concept and some of its near relatives is conspicuous in a result of Curto and Mathieu. It is familiar that a linear operator T: X → Y between Banach spaces is continuous if and only if it is “bounded ” in the usual sense, that for some k> 0 there is inequality ‖Tx ‖ ≤ k‖x ‖ for every x ∈ X. If X = A and Y = B are Banach algebras, and we replace norms by spectral radii, then we are looking at something new: “spectral boundedness”. In this note we look at spectral boundedness and some of its relatives. Suppose A is a complex linear algebra, with identity 1 6 = 0 and invertible group A−1: we write (0.1) σ(x) = {λ ∈ C: x − λ / ∈ A−1} for the spectrum of x ∈ A and (0.2) |x|σ = sup{|λ | : λ ∈ σ(x)} for the spectral radius. Recall the quasinilpotents in A (0.3) QN(A) = {x ∈ A: σ(x) = {0}} = {x ∈ A: |x|σ = 0} and the radical of