Dissipative operators on Banach spaces

Abstract

A bounded linear operator T on a Banach space is said to be dissipative if e tT 1 for all t 0. We show that if T is a dissipative operator on a Banach space, then: (a) lim t→∞ e tT T = sup{|λ|: λ ∈ σ (T ) ∩ iR}. (b) If σ (T ) ∩ iR is contained in [−iπ/2, iπ/2], then lim t→∞ e tT sin T = sup | sin λ|: λ ∈ σ (T ) ∩ iR .Some related problems are also discussed.

Asymptotic behavior of polynomially bounded operators

“…The similar result holds for polynomially bounded operators [3] (for related results see also [2,5,8,10]). We see that under the assumptions of Esterle-Strouse-Zouakia Theorem the Lebesgue measure of σ u (T ) is necessarily zero.…”

Local Spectrum, Local Spectral radius,and Growth Conditions

“…Proof of Theorem 2.11. We basically follow the proof in [14,Lemma 3.4]. Let a > 0 be such that iσ T (x) ⊂ (−a, a) .…”

Operators Whose Conjugation Orbits Satisfy Polynomial Growth Conditions

“…Note that in the preceding lemma, the weight function ω Proof. We basically follow the proof of Lemma 3.4 in [12]. Let an arbitrary a > r T (x) be fixed.…”