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Register a new user$C_0$-sequentially equicontinuous semigroups: theory and applications
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We present and apply a theory of one parameter $C_0$-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of $C_0$-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille-Yosida theorem. Then, we particularize the theory in some functional spaces and identify two locally convex topologies that allow to gather under a unified framework various notions $C_0$-semigroup introduced by some authors to deal with Markov transition semigroup. Finally, we apply the results to transition semigroups.
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The dual space of the C*-algebra of bounded uniformly continuous functions on a uniform space carries several natural topologies. One of these is the topology of uniform convergence on bounded uniformly equicontinuous sets, or the UEB topology for short. In the particular case of a topological group and its right uniformity, the UEB topology plays a significant role in the continuity of convolution. In this paper we derive a useful characterisation of bounded uniformly equicontinuous sets on locally compact groups. Then we demonstrate that for every locally compact group G the UEB topology on the space of finite Radon measures on G coincides with the right multiplier topology. In this sense the UEB topology is a generalisation to arbitrary topological groups of the multiplier topology for locally compact groups. In the final section we prove results about UEB continuity of convolution.
We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions ${T_t}_{t>0}$ on $L_p(Omega)$ for a fixed $1<p<infty$. We prove that if a Banach space $X$ is of martingale cotype $q$, then there is a constant $C$ such that $$ left|left(int_0^inftybig|tfrac{partial}{partial t}P_t (f)big|_X^q,frac{dt}tright)^{frac1q}right|_{L_p(Omega)}le C, big|fbig|_{L_p(Omega; X)},, quadforall, fin L_p(Omega; X),$$ where ${P_t}_{t>0}$ is the Poisson semigroup subordinated to ${T_t}_{t>0}$. Let $mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$mathsf{L}^{P}_{c,q, p}(X)lesssim maxbig(p^{frac1{q}},, pbig) mathsf{M}_{c,q}(X).$$ Moreover, the order $maxbig(p^{frac1{q}},, pbig)$ is optimal as $pto1$ and $ptoinfty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally ${T_t}_{t>0}$ is analytic on $L_p(Omega; X)$, the semigroup ${P_t}_{t>0}$ in these results can be replaced by ${T_t}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one is more powerful in several aspects: a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; b) it yields the optimal orders of growth on $p$ for most of the relevant constants; c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood-Paley-Stein inequalities for symmetric submarkovian semigroups are better than the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and ${P_t}_{t>0}$ is the classical Poisson and heat semigroups on $mathbb{R}^d$.
We study polynomial and exponential stability for $C_{0}$-semigroups using the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.
The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of Hermitean matri ces of the same dimension. Moreover, in this identification, the multidimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic clas sification of probability measures on $mathbb{R}^d$ with finite moments of any order. In this classification the usual Boson Fock space over $mathbb{C}^d$ is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.
In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $ell^1_+(S)$ have a common nonzero right multiple, where $ell^1_+(S)$ is the positive part of the Banach algebra $ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.
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