Condition Spectrum of Rank One Operators and Preservers of the Condition Spectrum of Skew Product of Operators
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Complex Analysis and Operator Theory
Condition Spectrum of Rank One Operators and Preservers of the Condition Spectrum of Skew Product of Operators Zine El Abidine Abdelali1 · Hamid Nkhaylia1 Received: 28 February 2020 / Accepted: 25 August 2020 © Springer Nature Switzerland AG 2020
Abstract Let L (H ) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H ≥ 3, and let A and B be two subsets of L (H ) containing all operators of rank at most one. For ε ∈ (0, 1) the ε-condition spectrum of any A ∈ L (H ) is defined by 1 −1 , σ (A) := σ (A) ∪ λ ∈ C \ σ (A) : (λI − A) λI − A ≥ ε where σ (A) is the spectrum of A. The ε-condition spectral radius of A is given by rε (A) := sup {|z| : z ∈ σε (A)} . We compute the ε-condition spectrum of any operator of rank at most one, and give an explicit formula for its ε-condition spectral radius. It is then illustrated that the results can be applied to characterize surjective mappings φ : A −→ B satisfying δ(φ(A)∗ φ(B)) = δ(A∗ B) for all A, B ∈ A where δ stands for σε (·) or rε (·).
Communicated by Daniel Aron Alpay. “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.
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Zine El Abidine Abdelali [email protected]; [email protected] Hamid Nkhaylia [email protected]
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Laboratory of Mathematics, Statistics and Applications, Department of Mathematics, Mathematical Research Center of Rabat, Faculty of Sciences, Mohammed-V University in Rabat, Rabat, Morocco 0123456789().: V,-vol
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Page 2 of 29
Z. E. A. Abdelali
Keywords Condition spectrum · Condition spectral radius · Nonlinear preserver Mathematics Subject Classification 46H05 · 47B49
1 Introduction and Preliminaries Throughout this paper, let L (H ) be the algebra of all bounded linear operators on a complex Hilbert space (H , ·, · ) with dim H ≥ 3. Let F1 (H ) be the set of all operators of rank at most one in H . This means that F1 (H ) := {x ⊗ y : x, y ∈ H }, where (x ⊗ y)(z) := z, y z for all x, y, z ∈ H . If H has dimension n < ∞, we identify L (H ) with the algebra Mn (C) of n × n-complex matrices. According to the above identifications we have x, y = x1 y1 + · · · + xn yn and x ⊗ y = x t y, where x t stands for the transpose of x, for all x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in Cn . For every ring automorphism τ on C and every x = (x1 , . . . , xn ) ∈ Cn we set xτ = (τ (x1 ), τ (x2 ), . . . , τ (xn )). The identity operator of L (H ) will be denoted by I , and A∗ will stand for the adjoint of any operator A ∈ L (H ). Fix an arbitrary , and define the conjugate linear operator J on H by orthogonal basis (ei )i∈ of H J (x) = i∈ ξi ei for all x = i∈ ξi ei ∈ H . Let A ∈ L (H ) and denote by A the bounded linear operator J A J . Notice that Aei , e j = Aei , e j for all i, j ∈ . Let ε be a fixed positive real number such that 0 < ε < 1. The ε-condition spectrum of an operator A ∈ L (H ) is 1 , σε (A) := σ (A) ∪ λ ∈ C \ σ (A) : (λI − A)−1 λI − A ≥ ε where σ (A) is the spec
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