Spectrum-Preserving Linear Maps on the Algebra of Regular Operators

Abstract

Publisher Summary This chapter describes the spectrum-preserving linear maps on the algebra of regular operators. A bounded operator on a complex Banach lattice X is called “regular” if it is a linear combination of positive operators. The algebra of all regular operators on X is denoted by L r ( X ). For a regular operator T , the spectrum of T in the algebra L r ( X ) is called the “ o -spectrum of T ” and is denoted by σ o ( T ). The algebra of all bounded operators on X is denoted by L ( X ), and the usual spectrum of T —that is, its spectrum in L ( X )—is denoted by σ( T ). The chapter reviews a few observations and some questions about spectrum-preserving (or O -spectrum-preserving) linear maps from L r ( X ) to L r ( Y ) for Banach lattices X and Y . A linear map ϕ among algebras is called “spectrum-preserving” if σ(ϕ( a )) = σ ( a ) for every a . In particular, every (ordered algebra) automorphism of L r ( X ) is inner. The definitions of Jordan homomorphism and anti-isomorphism are also described in the chapter.