Formulae for joint spectral radii of sets of operators

Abstract

The formula (M) = max{ χ (M), r(M)} is proved for precompact sets M of weakly compact operators on a Banach space. Here (M) is the joint spectral radius (the Rota-Strang radius), χ (M) is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

Topological Radicals, II. Applications to Spectral Theory of Multiplication Operators

“…In [25] the authors showed that the same is true if M is a precompact set of compact operators on a Banach space. In the further works [26,27,30] there were obtained several extensions of this result. Here we need the following consequence of [30,Theorem 4.11] (where only Banach algebras were considered).…”

Topological radicals and joint spectral radius

“…A characteristic operator close to T e is the Hausdorff noncompactness measure T χ of the image of the unit ball X under the map T (recall that this is the greatest lower bound of the set of those ε > 0 for which T X has a finite ε-net and that T χ T e ). The advantage of the seminorm T χ is that it well agrees with the restriction of the operator to an invariant subspace and with its induced action on a quotient space (see, e.g., [17,Lemma 2.5]):…”

“…In [17], where formula (1.4) was proved for operators on any reflexive Banach space, we called it the generalized Berger-Wang formula, or the generalized BW-formula; but it shortly turned out that there are several types of formulas generalizing (1.3), and a more specific term is required. Note that (1.4) implies (1.3) for precompact families of operators of the form λ1 + K , where λ ∈ C and K is a compact operator; in [16] this observation served as a basis for the proof that any Lie algebra of Volterra operators has an invariant subspace.…”

15. From Lomonosov lemma to radical approach in joint spectral radius theory

“…In [ST02] the following extension of (1.1) to precompact sets of general (not necessarily compact) operators was obtained:…”