More classes of non-orbit-transitive operators

Carl Pearcy; Lidia Smith

Displaying similar documents to “More classes of non-orbit-transitive operators”

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

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Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

$ℱ$ -hypercyclic and disjoint $ℱ$ -hypercyclic properties of binary relations over topological spaces

Marko Kostić (2020)

Mathematica Bohemica

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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 orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi (2010)

Fundamenta Mathematicae

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The Hurwitz action of the n-braid group Bₙ on the n-fold direct product $(B_{m}) ⁿ$ of the m-braid group $B_{m}$ is studied. We show that the orbit of any n- tuple of the n standard generators of $B_{n + 1}$ consists of the (n-1)th powers of n+1 elements.

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

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We prove that on $ℝ^{N}$ , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if $ℝ^{N}$ has an n-dimensional subspace whose orbit under $T \in (ℝ^{N})$ is dense in $ℝ^{N}$ , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator $T \in (ℝ^{N})$ is strongly n-supercyclic if $ℝ^{N}$ has an n-dimensional subspace whose orbit under T is dense in $ℙ ₙ (ℝ^{N})$ , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially...

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius $ϱ_{e} (T)$ of T ∈ B(H) can be calculated by the formula $ϱ_{e} (T)$ = inf $ℱ_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator, where $ℱ_{♯ \cdot ♯} (T)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if $_{♯ \cdot ♯} (T)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $d i s t (0, σ_{e} (T))$ = sup $_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator.

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂 = 𝕂^{+} \oplus 𝕂^{-}$ of the Krein space $𝕂$ with $𝕂^{+}$ and $𝕂^{-}$ invariant under $T$ .

A note on solvable vertex stabilizers of $s$ -transitive graphs of prime valency

Song-Tao Guo, Hailong Hou, Yong Xu (2015)

Czechoslovak Mathematical Journal

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A graph $X$ , with a group $G$ of automorphisms of $X$ , is said to be $(G, s)$ -transitive, for some $s \geq 1$ , if $G$ is transitive on $s$ -arcs but not on $(s + 1)$ -arcs. Let $X$ be a connected $(G, s)$ -transitive graph of prime valency $p \geq 5$ , and $G_{v}$ the vertex stabilizer of a vertex $v \in V (X)$ . Suppose that $G_{v}$ is solvable. Weiss (1974) proved that $| G_{v} {| ∣ p (p - 1)}^{2}$ . In this paper, we prove that $G_{v} ≅ (ℤ_{p} ⋊ ℤ_{m}) \times ℤ_{n}$ for some positive integers $m$ and $n$ such that $n div m$ and $m ∣ p - 1$ .

Tetravalent half-arc-transitive graphs of order $p^{2} q^{2}$

Hailin Liu, Bengong Lou, Bo Ling (2019)

Czechoslovak Mathematical Journal

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We classify tetravalent $G$ -half-arc-transitive graphs $Γ$ of order $p^{2} q^{2}$ , where $G \leq 𝖠𝗎𝗍 Γ$ and $p$ , $q$ are distinct odd primes. This result involves a subclass of tetravalent half-arc-transitive graphs of cube-free order.

Invariant symbolic calculus for semidirect products

Benjamin Cahen (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $π$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$ . We construct an invariant symbolic calculus for $π$ , under some technical hypothesis. We give some examples including the Poincaré group. ...

Construction methods for gaussoids

Tobias Boege, Thomas Kahle (2020)

Kybernetika

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The number of $n$ -gaussoids is shown to be a double exponential function in $n$ . The necessary bounds are achieved by studying construction methods for gaussoids that rely on prescribing $3$ -minors and encoding the resulting combinatorial constraints in a suitable transitive graph. Various special classes of gaussoids arise from restricting the allowed $3$ -minors.

Finite orbit decomposition of real flag manifolds

Bernhard Krötz, Henrik Schlichtkrull (2016)

Journal of the European Mathematical Society

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Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$ . It is shown that $H$ has an open orbit on the flag manifold $G / P$ if and only if it has finitely many orbits on $G / P$ . This confirms a conjecture by T. Matsuki.

A characterization of reflexive spaces of operators

Janko Bračič, Lina Oliveira (2018)

Czechoslovak Mathematical Journal

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We show that for a linear space of operators $ℳ \subseteq ℬ (ℋ_{1}, ℋ_{2})$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $Ψ = (ψ_{1}, ψ_{2})$ on a bilattice $Bil (ℳ)$ of subspaces determined by $ℳ$ with $P \leq ψ_{1} (P, Q)$ and $Q \leq ψ_{2} (P, Q)$ for any pair $(P, Q) \in Bil (ℳ)$ , and such that an operator $T \in ℬ (ℋ_{1}, ℋ_{2})$ lies in $ℳ$ if and only if $ψ_{2} (P, Q) T ψ_{1} (P, Q) = 0$ for all $(P, Q) \in Bil (ℳ)$ . This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

Non-orbit equivalent actions of $𝔽_{n}$

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

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For any $2 \leq n \leq \infty$ , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group $𝔽_{n}$ . These actions arise as diagonal products between a generalized Bernoulli action and the action $𝔽_{n} ↷ (𝕋^{2}, λ^{2})$ , where $𝔽_{n}$ is seen as a subgroup of ${SL}_{2} (ℤ)$ .

Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

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Let $f = a x + b x^{q} + x^{2 q - 1} \in_{q} [x]$ . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of $_{q ²}$ . This result allows us to solve a related problem: Let $g_{n, q} \in_{p} [x]$ (n ≥ 0, $p = c h a r_{q}$ ) be the polynomial defined by the functional equation $\sum_{c \in_{q}} {(x + c)}^{n} = g_{n, q} (x^{q} - x)$ . We determine all n of the form $n = q^{α} - q^{β} - 1$ , α > β ≥ 0, for which $g_{n, q}$ is a permutation polynomial of $_{q ²}$ .

A map maintaining the orbits of a given $ℤ^{d}$ -action

Bartosz Frej, Agata Kwaśnicka (2016)

Colloquium Mathematicae

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Giordano et al. (2010) showed that every minimal free $ℤ^{d}$ -action of a Cantor space X is orbit equivalent to some ℤ-action. Trying to avoid the K-theory used there and modifying Forrest’s (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map F on X∖one point such that for a residual subset of X the orbits of F are the same as the orbits of a given minimal free $ℤ^{d}$ -action.