Additive maps preserving the minimum and surjectivity moduli of operators

Abstract

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.

Additive maps preserving the reduced minimum modulus of Banach space operators

“…The next lemma and its proof were sitting in [16, Theorem 3.1 and its proof] and needed only a simple step to be discovered therein. It was also observed in [4] but only in the Hilbert space operators case; see [4, Corollary 2.3]. Lemma 2.2.…”

“…The obtained result, which extends [5,Theorem 7.1], improves Skhiri's result and shows that the condition that ϕ(1) is invertible in the above theorem is also superfluous even for the Banach space operators case. Our proof is simple and self-contained and also works to recapture and extend, to Banach space operators case, the recent results from [4,6,22] which describe linear and additive maps preserving the minimum modulus, the surjectivity modulus, and the maximum modulus of Hilbert space operators. Unlike in [4], we avoid using several deep results such as Herstein theorem's [13] and the celebrate Theorem of Kadison [18].…”

“…Our proof is simple and self-contained and also works to recapture and extend, to Banach space operators case, the recent results from [4,6,22] which describe linear and additive maps preserving the minimum modulus, the surjectivity modulus, and the maximum modulus of Hilbert space operators. Unlike in [4], we avoid using several deep results such as Herstein theorem's [13] and the celebrate Theorem of Kadison [18].…”