Local spectrum and local spectral radius of an operator at a fixed vector

Abstract

Abstract. Let X be a complex Banach space and e ∈ X a nonzero vector. Then the set of all operators T ∈ L(X) with σT (e) = σ δ (T ), respectively rT (e) = r(T ), is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.1. Introduction. Let X be a complex Banach space and let L(X) be the Banach algebra of all bounded linear operators on X. For T ∈ L(X), let σ(T ), σ δ (T ), and σ T (x) be the spectrum, the surjectivity spectrum, and the local spectrum of T at x ∈ X, respectively. We denote by r(T ) = lim k→∞ T k 1/k the spectral radius of T and by r T (x) = lim sup k→∞ T k x 1/k the local spectral radius of T at x. It is well known that r(T ) = max{|λ| : λ ∈ σ(T )}. On the other hand, r T (x) ≥ max{|λ| : λ ∈ σ T (x)}, and equality holds, for a given T and every nonzero x, if T has SVEP (see [6, Proposition 3.3.13 ]).For T ∈ L(X) and x ∈ X, it is easily seen that σ T (x) ⊆ σ δ (T ) ⊆ σ(T ). Moreover, the set of x ∈ X that satisfy σ δ (T ) = σ T (x) and r T (e) = r(T ) is residual (i.e., its complement is of the first category; see [8], and also [4], [6, Proposition 1.3.2]). In this paper, we are concerned with the "dual" question of how large the set D(e) = {T ∈ L(X) : σ δ (T ) = σ T (e)} is. It turns out (see Theorem 2.5) that D(e) is also residual whenever e = 0. Consequently, the set {T ∈ L(X) : r T (e) = r(T )}, which contains D(e), is residual as well.These results are later used to characterize linear mappings which preserve the local spectrum (local spectral radius) at a given vector e = 0. We show (Theorem 3.3) that a continuous linear surjection φ : L(X) → L(X) satisfies σ φ(T ) (e) = σ T (e), for every T ∈ L(X), if and only if there exists an invertible operator A ∈ L(X) such that Ae = e and φ(T ) = AT A −1 . This gives a solution to [5, Problem, p. 177].

Linear Maps Preserving Operators of Local Spectral Radius Zero

“…They showed that if ϕ : B(X) → B(X) is additive such that σ ϕ(T ) (x) = σ T (x) for all T ∈ B(X) and for all x ∈ X, then ϕ is the identity on B(X). Fixing a non-zero vector e ∈ X, the surjective linear mappings ϕ : B(X) → B(X) such that r ϕ(T ) (e) = r T (e) for all T ∈ L(X) were characterized in [5]. Different versions of this result in the case when X is finite-dimensional were considered in [7] and [3].…”

Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector

“…Also, surjective linear mappings ϕ : L (X) → L (X) such that r ϕ(T ) (e) = r T (e) for all T ∈ L (X) were characterized in [3]. These results were generalized to the infinite-dimensional case in [5]. Since the inclusions ∂σ (T ) ⊆ σ δ (e) ⊆ σ (T ) always hold, this implies that {T ∈ L (X) : r T (e) = r (T )} is also residual in L (X).…”

“…Since the inclusions ∂σ (T ) ⊆ σ δ (e) ⊆ σ (T ) always hold, this implies that {T ∈ L (X) : r T (e) = r (T )} is also residual in L (X). This is used in [5] in order to prove that, under the hypothesis of Theorem 1.1, the continuity of ϕ implies that it is a surjective spectral radius preserver, and therefore of a standard form [6]. It is asked in [5,Problem 3.5] whether the continuity assumption on ϕ may be removed from the statement of Theorem 1.1.…”

“…This is used in [5] in order to prove that, under the hypothesis of Theorem 1.1, the continuity of ϕ implies that it is a surjective spectral radius preserver, and therefore of a standard form [6]. It is asked in [5,Problem 3.5] whether the continuity assumption on ϕ may be removed from the statement of Theorem 1.1. In this paper we prove that the continuity of ϕ is automatic in the case when ϕ preserves the local spectral radius at e. …”

Surjective maps preserving local spectral radius

“…Bračič and V. Müller in [7] characterized the surjective linear map φ : B(X) → B(X) satisfies r T (x) = r φ(T ) (x) for all T ∈ B(X) and x is a nonzero vector fixed in X. Different versions of this result in the case when X is finitedimensional were considered in [4].…”