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Register a new userPower bounded weighted composition operators on function spaces defined by local properties
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Authors
Thomas Kalmes
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We study power boundedness and related properties such as mean ergodicity for (weighted) composition operators on function spaces defined by local properties. As a main application of our general approach we characterize when (weighted) composition operators are power bounded, topologizable, and (uniformly) mean ergodic on kernels of certain linear partial differential operators including elliptic operators as well as non-degenrate parabolic operators. Moreover, under mild assumptions on the weight and the symbol we give a characterisation of those weighted composition operators on the Frechet space of continuous functions on a locally compact, $sigma$-compact, non-compact Hausdorff space which are generators of strongly continuous semigroups on these spaces.
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We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of $mathbb{R}^d$, as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of $mathbb{R}^d$.
We study topologizability and power boundedness of weigh-ted composition operators on (certain subspaces of) $mathscr{D}(X)$ for an open subset $X$ of $mathbb{R}^d$. For the former property we derive a characterization in terms of the symbol and the weight of the weighted composition operator, while for the latter property necessary and sufficient conditions on the weight and the symbol are presented. Moreover, for an unweighted composition operator a characterization of power boundedness in terms of the symbol is derived for the special case of a bijective symbol.
We characterize the (essentially) decreasing sequences of positive numbers $beta$ = ($beta$ n) for which all composition operators on H 2 ($beta$) are bounded, where H 2 ($beta$) is the space of analytic functions f in the unit disk such that $infty$ n=0 |c n | 2 $beta$ n < $infty$ if f (z) = $infty$ n=0 c n z n. We also give conditions for the boundedness when $beta$ is not assumed essentially decreasing.
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. cite{AG1} and K{o}nig, W. cite{K} and answers a question of Siskakis, A. G. proposed in cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator $M_z$.
Suppose $ngeq 3$ and let $B$ be the open unit ball in $mathbb{R}^n$. Let $varphi: Bto B$ be a $C^2$ map whose Jacobian does not change sign, and let $psi$ be a $C^2$ function on $B$. We characterize bounded weighted composition operators $W_{varphi,psi}$ acting on harmonic Hardy spaces $h^p(B)$. In addition, we compute the operator norm of $W_{varphi,psi}$ on $h^p(B)$ when $varphi$ is a Mobius transformation of $B$.
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