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Let H be a complex Hilbert space and let L(H) be the algebra of all bounded linear operators on H. We characterize additive maps from L(H) onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Mathematics Subject Classification (2000). Primary 47B48; Secondary 47A10, 46A05.

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Abstract. Let ℒ(ℋ) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space ℋ. We characterize essentially spectrally bounded linear maps from ℒ(ℋ) onto ℒ(ℋ) itself. As a consequence, we characterize linear maps from ℒ(ℋ) onto ℒ(ℋ) itself that compress different essential spectral sets such as the the essential spectrum, the (left, right) essential spectrum, and the semi-Fredholm spectrum.