Spectral Representation of Local Semigroups.
To any bounded analytic semigroup on Hilbert space or on -space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative -spaces, Banach lattices, and their subspaces. We give some applications to functional calculus, similarity problems, multiplier theory, and control theory.
Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).
Let be an open subset of , the linear space of -vector valued functions defined on , a group of orthogonal matrices mapping onto itself and a linear representation of order of . A suitable group of linear operators of is introduced which leads to a general definition of -invariant linear operator with respect to . When is a finite group, projection operators are explicitly obtained which define a "maximal" decomposition of the function space into a direct sum of subspaces...