We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let {Tt}t>0 be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space HA1 by means of a maximal function associated with the semigroup {Tt}t>0. Atomic and Riesz transforms characterizations of HA1 are shown.

A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.

Motivated by a great deal of interest recently in operators that may not be densely defined, we deal with regularized semigroups and integrated semigroups that are either not exponentially bounded or not defined on [0,∞) and generated by closed operators which may not be densely defined. Some characterizations and related examples are presented. Our results are extensions of the corresponding results produced by other authors.

Our aim in this paper is to prove that every separable infinite-dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both a chaotic operator and an operator which...

Hypercyclicity of C0-semigroups is a very unstable property: We give examples to show that adding arbitrary small constants or a bounded rank one operator to the generator of a hypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (even in operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limit of nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, the restriction of a hypercyclic...