Linear Maps Preserving Operators of Inner Local Spectral Radius Zero at Some Fixed Vector
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Linear Maps Preserving Operators of Inner Local Spectral Radius Zero at Some Fixed Vector Constantin Costara Abstract. Let X be a complex Banach space and let x0 ∈ X be a fixed nonzero vector. Denote by L (X) the algebra of all linear and bounded operators on X, and for T ∈ L (X) denote by σT (x0 ) the local spectrum of T at x0 . We characterize linear and surjective maps ϕ : L (X) → L (X) such that ϕ (I) ∈ L (X) is invertible and 0 ∈ σT (x0 ) ⇐⇒ 0 ∈ σϕ(T ) (x0 )
(T ∈ L (X)) .
. Mathematics Subject Classification. Primary 47B49; Secondary 47A11. Keywords. Linear preserver, inner local spectral radius, fixed vector.
1. Introduction and Statement of the Main Result Let X be a complex Banach space and denote by L (X) the space of all linear bounded operators on X. For T ∈ L (X), we shall denote by σ (T ) its classical spectrum and by r (T ) its spectral radius. For a fixed non-zero vector x0 ∈ X, the local resolvent set, ρT (x0 ), of an operator T ∈ L (X) at x0 is the union of all open subsets U ⊆ C for which there exists an analytic function f : U → X such that (T − λI)f (λ) = x0 for all λ ∈ U . The local spectrum of T at x0 is defined by σT (x0 ) = C\ρT (x0 ). While σ (T ) is always a non-empty compact subset of the complex field C, the local spectrum might be empty. The inner local spectral radius of T ∈ L (X) at x0 , usually denoted by iT (x0 ) , is the supremum of all r ≥ 0 such that there exists an analytic function f : {λ ∈ C : |λ| < r} → X such that (T − λI)f (λ) = x0
(|λ| < r).
(Throughout this paper, by I ∈ L (X) we shall denote the identity operator on X.) If T has the so-called single-valued extension property, then σT (x0 ) 0123456789().: V,-vol
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is non-empty and iT (x0 ) coincides with the minimum modulus of σT (x0 ). For an arbitrary T ∈ L (X), we still have that iT (x0 ) = 0 ⇐⇒ 0 ∈ σT (x0 ) .
(1.1)
If the interior of the point spectrum of an operator T ∈ L (X) is empty, then T has the single-valued extension property. In particular, if T is of finite rank then σT (x0 ) is non-empty. We refer the reader to the monographs [1] and [13] for definitions and background information on local spectral theory. The study of linear and non-linear local spectra preserver problems has been a very active research field in the past decade; see, for example, the last section in the survey article [6]. In this paper, we study a preserver problem of this type, which is stated in [5, Problem 5]: characterize linear and surjective maps ϕ : L (X) → L (X) which preserve operators of inner local spectral radius zero at the fixed vector x0 , that is linear and surjective maps ϕ on L (X) which satisfy iT (x0 ) = 0 ⇐⇒ iϕ(T ) (x0 ) = 0
(T ∈ L (X)) .
(1.2)
(T ∈ L (X)) .
(1.3)
By (1.1), the relations (1.2) are equivalent to 0 ∈ σT (x0 ) ⇐⇒ 0 ∈ σϕ(T ) (x0 )
Linear and non-linear maps which preserve the inner local spectral radius at each non-zero vector x ∈ X have been considered in [11], [4], [12] and [10]. In our case, the map ϕ is supposed to be linear, but the preserving property in (
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