We use the functorial properties of Rieffel’s pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are extracted from the quasi-orbit structure of the dynamical system. The semi-classical behavior of the families of spectra is also studied.

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Let $L$, $M$ be Archimedean Riesz spaces and ${ℒ}_b(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of ${ℒ}_b(L,M)$ to be an ordered vector subspace $ℒ$ of ${ℒ}_b(L,M)$ such that the elements of $ℒ$ preserve disjointness and any pair of operators in $ℒ$ has a supremum in ${ℒ}_b(L,M)$ that belongs to $ℒ$. It turns out that the lattice operations in any Lamperti Riesz subspace $ℒ$ of ${ℒ}_b(L,M)$ are given pointwise, which leads to a generalization of the classic Radon-Nikod’ym theorem for Riesz homomorphisms....

Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, ${\mathscr{H}}_{Nb}\left(E\right)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on ${\mathscr{H}}_{Nb}\left(E\right)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\phi \left(D\right){\equiv }^t\phi$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on ${\mathscr{H}}_{Nb}\left(E\right)$ is PDE-preserving for a set ℙ ⊆ Exp(F) if it...

Under suitable conditions we prove the wellposedness of small time-varied delay equations and then establish the robust stability for such systems on the phase space of continuous vector-valued functions.