We propose a new general method of estimating Schrödinger perturbations of transition densities using an auxiliary transition density as a majorant of the perturbation series. We present applications to Gaussian bounds by proving an optimal inequality involving four Gaussian kernels, which we call the 4G Theorem. The applications come with honest control of constants in estimates of Schrödinger perturbations of Gaussian-type heat kernels and also allow for specific non-Kato perturbations.

The convexity and compactness in the weak operator topology of the image and pre-image of a generalized fractional linear transformation is established. As an application the exponential dichotomy of solutions to evolution problems of the parabolic type is proved.

The Sova-Kurtz approximation theorem for semigroups is applied to prove convergence of solutions of the telegraph equation with small parameter. Convergence of the solutions of the diffusion equation with varying boundary conditions is also considered.

We prove that Brownian motion on an abstract Wiener space B generates a locally equicontinuous semigroup on $C_b\left(B\right)$ equipped with the $T_t$-topology introduced by L. Le Cam. Hence we obtain a “Laplace operator” as its infinitesimal generator. Using this Laplacian, we discuss Poisson’s equation and heat equation, and study its properties, especially the difference from the Gross Laplacian.

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ\left(t\right)=A^{-1}x\left(t\right)$, and the difference equation $x_d(n+1)=(A+I){(A-I)}^{-1}{x}_d\left(n\right)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ${\left(e^{A^{-1}t}\right)}_{t\ge 0}$ is O(∜t), and for $\left((A+I){(A-I)}^{-1}\right)ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions...

It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma$-th order Cesàro mean $C_t^{\gamma }:=\gamma t^{-\gamma }{\int }_0^t{(t-s)}^{\gamma -1}T\left(s\right)ds$ and Abel mean $A_{\lambda }:=\lambda {\int }_0^{\infty }{e}^{-\lambda s}T\left(s\right)ds$ of the uniformly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in ℝ$, satisfy that (a) $\parallel C_t^{\gamma }\parallel \sim t^{k-\gamma }(t\to \infty )$ for all $0<\gamma \le k+1$; (b) $\parallel C_t^{\gamma }\parallel \sim t^{-1}(t\to \infty )$ for all $\gamma \ge k+1$; (c) $\parallel A_{\lambda }\parallel \sim \lambda (\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function ${\left(C\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by ${(iaI+N)}^2$.