Pointwise limit theorem for a class of unbounded operators in $^{r}$-spaces

Ryszard Jajte

Displaying similar documents to “Pointwise limit theorem for a class of unbounded operators in $^{r}$ -spaces”

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M ₙ^{α} = {(A ₙ^{α})}^{- 1} \sum_{k = 0}^{n} A_{n - k}^{α - 1} T^{k}$ . For α = 1, we find $M ₙ ¹ = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

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Given two n-tuples $a = (a_{1}, . . ., a_{n})$ and $b = (b_{1}, . . ., b_{n})$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E a_{j} = b_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^{n}$ .

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 the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new $(H_{p}, L_{p})$ -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients ${q_{k} : k \geq 0}$ . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...

Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

On the $C^{α} | C^{β}$ convergence

L. Jeśmanowicz (1962)

Annales Polonici Mathematici

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On the inclusion relation between strong $(J, p_{n})$ and strong $(\bar{N}, p_{n})$ summability methods

R. K. Jain, A. Ganguly (1978)

Matematički Vesnik

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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 $T$ .

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

On a proof of the boundedness and nuclearity of pseudodifferential operators in $R^{n}$

Jan A. Rempała (1990)

Annales Polonici Mathematici

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Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

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Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators $^{b, c}$ is bounded on the Dirichlet spaces $_{p, a}$ . We also give a short and direct proof of boundedness of $^{b, c}$ on the Hardy space $H^{p}$ for 1 < p < ∞.

Relations among K $S (λ)$ and certain other methods for evaluations of sequences and series

B. Martić (1964)

Matematički Vesnik

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Example of a mean ergodic L¹ operator with the linear rate of growth

Wojciech Kosek (2011)

Colloquium Mathematicae

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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty$ . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $l i m s u p | | T ⁿ | | / n^{1 - γ ₀} = 0 .$ In this note we construct an example of a nonpositive L¹ operator with the...

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^{p} (ν)$ , $T ₁, . . ., T_{k}$ , $n ̅ = (n ₁, . . ., n_{k}) \in ℕ^{k}$ and $α ̅ = (α ₁, . . ., α_{k})$ with $0 < α_{j} \leq 1$ , we define the ergodic Cesàro-α̅ averages $_{n ̅, α ̅} f = 1 / (\prod_{j = 1}^{k} A_{n_{j}}^{α_{j}}) \sum_{i_{k} = 0}^{n_{k}} \dots \sum_{i ₁ = 0}^{n ₁} \prod_{j = 1}^{k} A_{n_{j} - i_{j}}^{α_{j} - 1} T_{k}^{i_{k}} \dots T ₁^{i ₁} f$ . For these averages we prove the almost everywhere convergence on X and the convergence in the $L^{p} (ν)$ norm, when $n ₁, . . ., n_{k} \to \infty$ independently, for all $f \in L^{p} (d ν)$ with p > 1/α⁎ where $α ⁎ = m i n_{1 \leq j \leq k} α_{j}$ . In the limit case p = 1/α⁎, we prove that the averages $_{n ̅, α ̅} f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ (1 / α ⁎, φ_{m - 1})$ with $φ ₘ (t) = t {(1 + l o g ⁺ t)}^{m}$ . To obtain the result in the limit case we need...

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages $A_{k} f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_{k}} f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k} f (x) = 1 / (2^{k}) \sum_{j = 4^{k} + 1}^{4^{k} + 2^{k}} f (T^{j} x)$ , then the subsequence $A_{k ²} f$ will not be pointwise good even on $L^{\infty}$ , but the subsequence $A_{2^{k}} f$ will be pointwise good on L¹. Understanding when the hyperexponential...

On extensions of families of operators

Oleg Lihvoinen (2023)

Commentationes Mathematicae Universitatis Carolinae

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The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

The natural linear operators $T * ⇝ T T^{(r)}$

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators $T * \times_{ℳ f ₙ} T^{(0, 0)} ⇝ T^{(0, 0)} T^{(r)}$ is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators $T *_{| ℳ f ₙ} ⇝ T T^{(r)}$ is obtained.

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, Wojciech Kosek (2003)

Colloquium Mathematicae

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $| | T ⁿ | | \sim n^{1 / 4}$ ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty . I n t h e c l a s s o f p o s i t i v e o p e r a t o r s t h e s e e x a m p l e s a r e t h e b e s t p o s s i b l e i n t h e s e n s e t h a t f o r e v e r y s u c h o p e r a t o r T t h e r e e x i s t s a γ ₀ > 0 s u c h t h a t$ lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 $.$ A class of numerical sequences αₙ, intimately...

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_{G}$ . We consider the map $τ_{φ}$ defined on X × G by $τ_{φ} : (x, g) \mapsto (τ x, φ (x) g)$ and the cocycle ${(φ ₙ)}_{n \in ℤ}$ generated by φ. Using a characterization of the ergodic invariant measures for $τ_{φ}$ , we give the form of the ergodic decomposition of $μ (d x) \otimes m_{G} (d g)$ or more generally of the $τ_{φ}$ -invariant measures $μ_{χ} (d x) \otimes χ (g) m_{G} (d g)$ , where $μ_{χ} (d x)$ is χ∘φ-conformal for an exponential χ on G.

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

Displaying similar documents to “Pointwise limit theorem for a class of unbounded operators in r -spaces”

Displaying similar documents to “Pointwise limit theorem for a class of unbounded operators in $^{r}$ -spaces”