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Let B(H ) the algebra of all bounded linear operators on a complex Hilbert space H with dimH 3 . Let W , V be subsets of B(H ) which contain all rank-one operators. Denote by rε (A) the condition spectral radius of A ∈ B(H ) . We determine the form of surjective maps φ : W → V satisfying rε (AB∗A) = rε (φ(A)φ(B)∗φ(A)) for all A , B in W , we characterize also the structure of surjective maps φ : B(H ) → B(H ) with σε (AB∗A) = σε (φ(A)φ(B)∗φ(A)) for all A , B in B(H ) where σε (A) is the ε -condition spectrum of an operator A in B(H ) . Mathematics subject classification (2010): Primary 47B49, Secondary 47B48, 47A10, 46H05.

Nonlinear maps preserving condition spectrum of Jordan skew triple product of operators