### $\langle 2,1\rangle $-compact operators.

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We examine various types of $\mathcal{F}$-hypercyclic ($\mathcal{F}$-topologically transitive) and disjoint $\mathcal{F}$-hypercyclic (disjoint $\mathcal{F}$-topologically transitive) properties of binary relations over topological spaces. We pay special attention to finite structures like simple graphs, digraphs and tournaments, providing a great number of illustrative examples.

The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its ${}^{\infty}$-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Let X and Y be complex Banach spaces of dimension greater than 2. We show that every 2-local Lie isomorphism ϕ of B(X) onto B(Y) has the form ϕ = φ + τ, where φ is an isomorphism or the negative of an anti-isomorphism of B(X) onto B(Y), and τ is a homogeneous map from B(X) into ℂI vanishing on all finite sums of commutators.

Let 1 ≤ p < ∞, $={\left(X\u2099\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and ${l}_{p}\left(\right)$ the coresponding vector valued sequence space. Let $={\left(X\u2099\right)}_{n\in \mathbb{N}}$, $={\left(Y\u2099\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(V\u2099\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator ${M}_{}:{l}_{p}\left(\right)\to {l}_{q}\left(\right)$ by ${M}_{}\left({\left(x\u2099\right)}_{n\in \mathbb{N}}\right):={\left(V\u2099\left(x\u2099\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for ${M}_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

In this paper we consider the map L defined on the Bergman space [...] of the right half plane [...] .

Let $X$ be a quasicomplete locally convex Hausdorff space. Let $T$ be a locally compact Hausdorff space and let ${C}_{0}\left(T\right)=\{f\phantom{\rule{0.222222em}{0ex}}T\to I$, $f$ is continuous and vanishes at infinity$\}$ be endowed with the supremum norm. Starting with the Borel extension theorem for $X$-valued $\sigma $-additive Baire measures on $T$, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map $u\phantom{\rule{0.222222em}{0ex}}{C}_{0}\left(T\right)\to X$ to be weakly compact.

Let $\U0001d504$ be a ${C}^{*}$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of $\U0001d504,{\U0001d504}^{\tau}$ the fixed point algebra of $\tau $ and ${\U0001d504}_{F}$ the dense sub-algebra of $G$-finite elements in $\U0001d504$. Further let $H$ be a linear operator from ${\U0001d504}_{F}$ into $\U0001d504$ which commutes with $\tau $ and vanishes on ${\U0001d504}^{\tau}$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a ${C}_{0}$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite...

In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space $X$. Our main results assert, in particular, that the generators of such semigroups are either polynomially Riesz (then bounded) or there exist two closed infinite dimensional invariant subspaces ${X}_{0}$ and ${X}_{1}$ of $X$ with $X={X}_{0}\oplus {X}_{1}$ such that the part of the generator in ${X}_{0}$ is unbounded with resolvent of Riesz type while its part in ${X}_{1}$ is a polynomially Riesz operator.