A Collection of Problems on Spectrally Bounded Operators ∗

Let A be a unital complex Banach algebra. A linear mapping T : E → B from a subspace E ⊆ A into another unital complex Banach algebra B is called spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E. Here, and in what follows, r(x) stands for the spectral radius of a Banach algebra element x. This concept evolved in Banach algebra theory, and especially automatic continuity, over time in the 1970’s and 1980’s but the terminology was only introduced in [16], together with its companions spectrally infinitesimal : M = 0; spectrally contractive: M = 1; and spectrally isometric: r(Tx) = r(x) for all x. It follows from [1], see also [4], Lemma A, that the separating space of every surjective spectrally bounded operator T on a closed subspace E is contained in the radical of B; thus T is bounded if B is semisimple. This was used by Aupetit in [1] to give a new proof of