We examine various types of $ℱ$-hypercyclic ($ℱ$-topologically transitive) and disjoint $ℱ$-hypercyclic (disjoint $ℱ$-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.

We show that for a linear space of operators $ℳ\subseteq ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi }_1,{\psi }_2)$ on a bilattice $\mathrm{Bil}\left(ℳ\right)$ of subspaces determined by $ℳ$ with $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$, and such that an operator $T\in ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ lies in $ℳ$ if and only if ${\psi }_2(P,Q)T{\psi }_1(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

We characterize some S-essential spectra of a closed linear relation in terms of certain linear relations of semi-Fredholm type.

In this article the essential spectrum of closed, densely defined linear operators is characterized on a large class of spaces, which possess the Dunford-Pettis property or which isomorphic to one of the spaces $L_p\left(\mathrm{\Omega }\right)$$p>1$. A practical...

Given a domain $\Omega$ of class $C^{k,1}$, $k\in \mathbb{N}$, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega$ in the sense that $(\partial -\partial x_n)\alpha (x^{\text{'}},0)=-N\left(x^{\text{'}}\right)$ and that still is of class $C^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class $C^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${\left(Tⁿ\right)}_{n=1,2,...}$ by the continuous semigroup ${\left(e^{-t(I-T)}\right)}_{t\ge 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $supn\parallel Tⁿ-T^{n+1}\parallel :n=1,2,...<\infty$. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

We consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N....

A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1/2}$. It also depends on a certain angle between A() and the orthogonal of .