scholarly journals Mean ergodic operators and reflexive Fréchet lattices

2011 ◽  
Vol 141 (5) ◽  
pp. 897-920 ◽  
Author(s):  
José Bonet ◽  
Ben de Pagter ◽  
Werner J. Ricker
Keyword(s):  
Bounded Operator ◽  
Banach Lattices ◽  
Positive Power ◽  
Bounded Linear ◽  
Mean Ergodic Operators ◽  
Power Bounded Operator ◽  
Complemented Lattice ◽  
Locally Convex ◽  
Strong Dual ◽  
Power Bounded

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

scholarly journals On power-bounded prespectral operators

1976 ◽  
Vol 20 (2) ◽  
pp. 173-175
Author(s):  
H. R. Dowson
Keyword(s):  
Growth Condition ◽  
Bounded Operator ◽  
Spectral Operator ◽  
Scalar Type ◽  
Image Position ◽  
Power Bounded Operator ◽  
Power Bounded

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

scholarly journals Boundedly Spaced Subsequences and Weak Dynamics

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
C. S. Kubrusly ◽  
P. C. M. Vieira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Bounded Operator ◽  
Weak Stability ◽  
Weakly Stable ◽  
Power Bounded Operator ◽  
Power Bounded

Weak supercyclicity is related to weak stability, which leads to the question that asks whether every weakly supercyclic power bounded operator is weakly stable. This is approached here by investigating weak l-sequential supercyclicity for Hilbert-space contractions through Nagy–Foliaş–Langer decomposition, thus reducing the problem to the quest of conditions for a weakly l-sequentially supercyclic unitary operator to be weakly stable, and this is done in light of boundedly spaced subsequences.

Two Consequences of Brunel's Theorem

1991 ◽  
Vol 34 (1) ◽  
pp. 105-108
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Bounded Operators ◽  
Positive Power ◽  
Multiple Sequence ◽  
Power Bounded Operators ◽  
Recent Theorem ◽  
The Individual ◽  
Power Bounded

AbstractIn this note we observe two consequences of Brunei's recent theorem. If T1,..., Tn are majorized by positive power-bounded operators S1,..., Sn of Lp, 1 < p < ∞, for which the ergodic theorem holds, then a multiple sequence ergodic theorem holds for T1,....,Tn. Further, the individual convergence for each Tk can be taken along uniform sequences.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Separable Banach lattices on which every bounded linear operator is regular

2011 ◽  
Vol 150 (3) ◽  
pp. 557-560
Author(s):  
A. W. WICKSTEAD
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Linear ◽  
The Difference

AbstractWe give a complete description of those separable Banach lattices E with the property that every bounded linear from E into itself is the difference of two positive operators.

Ranges of Products of Operators

1974 ◽  
Vol 26 (6) ◽  
pp. 1430-1441 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Space ◽  
Dual Problem ◽  
Bounded Operator ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Operators

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.

On the convergence of iterates of convolution operators in Banach spaces

2020 ◽  
Vol 126 (2) ◽  
pp. 339-366
Author(s):  
Heybetkulu Mustafayev
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Continuous Representation ◽  
Convolution Operators ◽  
Locally Compact ◽  
Bounded Linear ◽  
Almost Everywhere ◽  
Power Bounded

Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}<\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.

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