Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $S{p}_u(T,S)$. For $S={\mathbb{Z}}_{+}^n$ or ${ℝ}_{+}^n$, we establish connections with polynomial convexity.

A new version of the maximum principle is presented. The classical Kantorovich-Rubinstein principle gives necessary conditions for the maxima of a linear functional acting on the space of Lipschitzian functions. The maximum value of this functional defines the Hutchinson metric on the space of probability measures. We show an analogous result for the Fortet-Mourier metric. This principle is then applied in the stability theory of Markov-Feller semigroups.

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first...

In a Banach space $E$, let $U\left(t\right)$$(t>0)$ be a $C_0$-semigroup with generating operator $A$. For a cone $K\subseteq E$...

Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R\left(A\right)$ of $A$ as follows. Let $0\ne {\lambda }_n\in \rho \left(A\right)$ for all $1\le n<\infty$, where $\rho \left(A\right)$ denotes the resolvent set of $A$, and assume that ${lim}_{n\to \infty }{\lambda }_n=0$ and ${sup}_{n\ge 1}\parallel {\lambda }_n{({\lambda }_n-A)}^{-1}\parallel <\infty$. Furthermore, assume that there exists ${\lambda }_{\infty }\in \rho \left(A\right)$ such that $\parallel {\lambda }_{\infty }{({\lambda }_{\infty }-A)}^{-1}\parallel \le 1$. Then $f\in R\left(A\right)$ is equivalent to ${sup}_{n\ge 1}{\parallel {({\lambda }_n-A)}^{-1}f\parallel }_1<\infty$. This generalizes Shaw’s result for scalar-valued functions.

We prove pointwise lower bounds for the heat kernel of Schrödinger semigroups on Euclidean domains under Dirichlet boundary conditions. The bounds take into account non-Gaussian corrections for the kernel due to the geometry of the domain. The results are applied to prove a general lower bound for the Schrödinger heat kernel in horn-shaped domains without assuming intrinsic ultracontractivity for the free heat semigroup.