Bi-parameter Semigroups of linear operators

摘要

Let X be a Banach space. We define the concept of a bi-parameter semigroupon X and its first and second generators. We also study bi-parameter semigroups on Banachalgebras. A relation between uniformly continuous bi-parameter semigroups and σ-derivations isalso established. It is proved that if {αt,s}t,s0 is a uniformly continuous bi-parameter semigroupon a Banach algebra X , whose first and second generators are d and σ, respectively, and if d isalso a σ-derivation then dn(ab) = (d + σ)n(a) (d + σ)n(b) and αt,0(ab) = αt,1(a) αt,1(b) forall a, b ∈ X .