We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.

Let ϰ be a positive, continuous, submultiplicative function on ${ℝ}^{+}$ such that $li{m}_{t\to \infty }{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a$ for some ω ∈ ℝ, α ∈ $\overline{{ℝ}^{+}}$ and $a\in {ℝ}^{+}$. For every λ ∈ (ω,∞) let ${\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}$ for $t\in {ℝ}^{+}$. Let $L_{\varkappa }^1\left({ℝ}^{+}\right)$ be the space of functions Lebesgue integrable on ${ℝ}^{+}$ with weight $\varkappa$, and let E be a Banach space. Consider the map ${\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)$. Theorem 5.1 of the present paper characterizes the range of the linear map $T\to T{\varphi }_{•}$ defined on $L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)$, generalizing a result established by B. Hennig and F. Neubrander for $\varkappa \left(t\right)=e^{\omega t}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...

We extend the notion of ascent and descent for an operator acting on a vector space to sets of operators. If the ascent and descent of a set are both finite then they must be equal and give rise to a canonical decomposition of the space. Algebras of operators, unions of sets and closures of sets are treated. As an application we construct a Browder joint spectrum for commuting tuples of bounded operators which is compact-valued and has the projection property.

Let T be a Fredholm operator on a Banach space. Say T is rootless if there is no bounded linear operator S and no positive integer m ≥ 2 such that $S^m=T$. Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for T with examples having infinite ascent and descent.

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if ${\sigma }_{asc}^e(T+F)={\sigma }_{asc}^e\left(T\right)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued extension...

We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra.

In this work we study the multivalued complementarity problem on the non-negative orthant. This is carried out by describing the asymptotic behavior of the sequence of approximate solutions to its multivalued variational inequality formulation. By introducing new classes of multifunctions we provide several existence (possibly allowing unbounded solution set), stability as well as sensitivity results which extend and generalize most of the existing ones in the literature. We also present some kind...

We study the existence of global in time and uniform decay of weak solutions to the initial-boundary value problem related to the dynamic behavior of evolution equation accounting for rotational inertial forces along with a linear nonlocal frictional damping arises in viscoelastic materials. By constructing appropriate Lyapunov functional, we show the solution converges to the equilibrium state polynomially in the energy space.