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Square functions, bounded analytic semigroups, and applications

Christian Le Merdy (2007)

Banach Center Publications

To any bounded analytic semigroup on Hilbert space or on L p -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 L p -spaces, Banach lattices, and their subspaces. We give some applications to H functional calculus, similarity problems, multiplier theory, and control theory.

Strong continuity of semigroup homomorphisms

Bolis Basit, A. Pryde (1999)

Studia Mathematica

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).

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