Linear maps between Banach algebras compressing certain spectral functions

In this paper, we discuss the linear maps between semi-simple Banach algebras which compress any one of the spectrum, the left spectrum, the right spectrum, the intersection of left spectrum and right spectrum, the boundary of spectrum and the full spectrum. We prove that such linear maps are idempotent preserving. As applications, we characterize such maps in terms of Jordan homomorphisms on C*-algebras of real rank zero. In particular, we give several characterizations of isomorphisms between standard operator algebras by using such spectral function compressing linear maps and surjectivity spectrum compressing or approximate point spectrum compressing linear maps.