We consider one-parameter (C₀)-semigroups of operators in the space ${}^{\text{'}}(ℝⁿ;{ℂ}^m)$ with infinitesimal generator of the form ${(G*)|}_{{}^{\text{'}}(ℝⁿ;{ℂ}^m)}$ where G is an $M_{m\times m}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;{ℂ}^m)$, ${}_{L^p}(ℝⁿ;{ℂ}^m)$, p ∈ [1,∞], ${(}_a)(ℝⁿ;{ℂ}^m)$, a ∈ ]0,∞[, or the spaces ${}_{L^q}^{\text{'}}(ℝⁿ;{ℂ}^m)$, q ∈ ]1,∞], of bounded distributions.

2000 Mathematics Subject Classification: 47A45.An estimation of the growth of a non-contracting semigroup Zt = exp(itA) where A is a non-dissipative operator with a two-dimensional imaginary component is given. Estimation is given in terms of the functional model in de Branges space.

A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results...

A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on...

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${x^n(x-1)}_{n\in \mathbb{N}}$ for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of ${x^n}_{n\in \mathbb{N}}$ and ${1/n{\sum }_{k=0}^{n-1}{x}^k}_{n\in \mathbb{N}}$.

We prove existence and uniqueness of classical solutions for an incomplete second-order abstract Cauchy problem associated with operators which have polynomially bounded resolvent. Some examples of differential operators to which our abstract result applies are also included.

A characterization of exponentially dichotomic and exponentially stable $C_0$-semigroups in terms of solutions of an operator equation of Lyapunov type is presented. As a corollary a new and shorter proof of van Neerven’s recent characterization of exponential stability in terms of boundedness of convolutions of a semigroup with almost periodic functions is given.

We construct two Banach algebras, one which contains analytic semigroups ${\left(a^z\right)}_{Rez>0}$ such that $|a^{1+iy}|\to \infty$ arbitrarily slowly as $\left|y\right|\to \infty$, the other which contains ones such that $|a^{1+iy}|\to \infty$ arbitrarily fast