scholarly journals Linear Sequences and Weighted Ergodic Theorems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Tanja Eisner
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Return Time ◽  
Linear Operators ◽  
Full Measure ◽  
Ergodic Theorems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Time Sequences

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form for a measure preserving system and , avoiding in the latter case the problem of finding the full measure set of appropriate points .

scholarly journals Generalized Dobrushin Coefficients on Banach Spaces

2021 ◽  
Author(s):  
Wojciech Bartoszek ◽  
Marek Beśka ◽  
Wiktor Florek
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
Positive Operators ◽  
Linear Operators ◽  
Ergodic Theorems ◽  
Markov Semigroups ◽  
Bounded Linear Operators ◽  
State Spaces ◽  
Bounded Linear

AbstractThe asymptotic behavior of iterates of bounded linear operators (not necessarily positive), acting on Banach spaces, is studied. Through the Dobrushin ergodicity coefficient, we generalize some ergodic theorems obtained earlier for classical Markov semigroups acting on $$L^1$$ L 1 (or positive operators on abstract state spaces).

Duality of the distance to closed operator ideals

2003 ◽  
Vol 133 (1) ◽  
pp. 197-212 ◽  
Author(s):  
Hans-Olav Tylli
Keyword(s):  
Banach Spaces ◽  
Closed Operator ◽  
Operator Ideal ◽  
Linear Operators ◽  
Distance Functions ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Special Operator

Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

scholarly journals On Riesz operators

1969 ◽  
Vol 16 (3) ◽  
pp. 227-232 ◽  
Author(s):  
J. C. Alexander
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators ◽  
Image Position ◽  
Schauder’S Theorem

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

On the Non-Existence of a Projection onto the Space of Compact Operators

1982 ◽  
Vol 25 (1) ◽  
pp. 78-81 ◽  
Author(s):  
Moshe Feder
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Space Of Compact Operators

AbstractLet X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {Ti} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

scholarly journals Deviation from weak Banach–Saks property for countable direct sums

2014 ◽  
Vol 68 (2) ◽  
Author(s):  
Andrzej Kryczka
Keyword(s):  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Direct Sums ◽  
Bounded Linear

AbstractWe introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (X

scholarly journals MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

2014 ◽  
Vol 57 (3) ◽  
pp. 709-718 ◽  
Author(s):  
ABDELLATIF BOURHIM ◽  
JAVAD MASHREGHI
Keyword(s):  
Banach Spaces ◽  
Linear Mapping ◽  
Linear Operators ◽  
Local Spectrum ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Formula Group ◽  
Image Position

AbstractLet X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map ϕ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies $ \sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)), $ if and only if there exists a bijective bounded linear mapping A from X into Y such that Ax0 = y0 and either ϕ(T) = ATA−1 or ϕ(T) = -ATA−1 for all T ∈ ${\mathcal B}$(X).

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