On algebraic and analytic core

Abstract

Let B(X) the Banach algebra of all bounded operators on a Banach space X and let T ∈ B(X). We denote by R alc (X) = {T ∈ B(X) : C(T ) = {0}} and R ac (X) = {T ∈ B(X) : K(T ) = {0}} where C(T ) and K(T ) are respectively the algebraic core and the analytic core. In this paper we show that R alc (X) and R ac (X) are a regularities in Kordula-Müller's sense. Mathematics Subject Classification : 47A10

Analytic core and quasi-nilpotent part of linear relations in Banach spaces

“…We shall also write dis(x, V) for the distance between {x} and V. The notion of analytic core and quasi-nilpotent part of a linear operator was mentioned and studied by M. Mbekhta [8,9], A. Pietro [1] and A. Tajmouati [10]. For a linear operator T, the authors have defined the analytic core by K(T) = {x ∈ X : ∃(x n ) ⊂ X and a > 0 such that x 0 = x; Tx n = x n−1 ∀n ≥ 1 and x n ≤ a n x } and the quasi-nilpotent part by H 0 (T) = {x ∈ X; lim n →+∞ T n x 1 n = 0}.…”

$C_0$ Semigroup and Local Spectral Theory

“…Denote by σ ac (T ) = {λ ∈ C : K(λ − T ) = {0}} the analytic core spectrum of T and by σ alc (T ) = {λ ∈ C : C(λ − T ) = {0}} the algebraic core spectrum of T [10,11]. As a straightforward consequence of the corollary 3.3, we have the following corollary.…”

Spectral equality for $$C_0$$ C 0 semigroups

“…K(T ) = {x ∈ X : ∃(x n ) n≥0 ⊂ X and δ > 0 such that x 0 = x and T x n = x n−1 ∀n ≥ 1 and x n ≤ δ n x }. In [15,Lemma 2.3] it is proved that if A, B ∈ B(X) are such that AB = BA, then :…”