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E Sanchez A Perez - One of the best experts on this subject based on the ideXlab platform.
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strong extensions for q Summing Operators acting in p convex banach function spaces for 1 le p le q
Positivity, 2016Co-Authors: O Delgado, E Sanchez A PerezAbstract:Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-Summing Operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-Summing Operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).
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strong extensions for q Summing Operators acting in p convex banach function spaces for 1 le p le q
arXiv: Functional Analysis, 2015Co-Authors: O Delgado, E Sanchez A PerezAbstract:Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces $L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{\,q}(\xi)$, where $\xi$ is a probability Radon measure on a certain compact set associated to $X$. We show some of its properties, and the relevant fact that every $q$-Summing Operator $T$ defined on $X$ can be continuously (strongly) extended to $S_{X_p}^{\,q}(\xi)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factorizations for $q$-Summing Operators through $L^q$-spaces when $1 \le q \le p$. Thus, our result completes the picture, showing what happens in the complementary case $1\le p\le q$, opening the door to the study of the multilinear versions of $q$-Summing Operators also in these cases.
Feyzi Basar - One of the best experts on this subject based on the ideXlab platform.
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Operator valued series almost summability of vector valued multipliers and weak compactness of Summing Operator
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Mahmut Karakus, Feyzi BasarAbstract:Abstract In this study, we introduce the vector valued multiplier spaces M f ∞ ( ∑ k T k ) and M w f ∞ ( ∑ k T k ) by means of almost summability and weak almost summability, and a series of bounded linear Operators. Since these multiplier spaces are equipped with the sup norm and are subspaces of l ∞ ( X ) , we obtain the completeness of a normed space via the multiplier spaces which are complete for every c 0 ( X ) -multiplier Cauchy series. We also characterize the continuity and (weakly) compactness of the Summing Operator S from the multiplier spaces M f ∞ ( ∑ k T k ) or M w f ∞ ( ∑ k T k ) to an arbitrary normed space Y through c 0 ( X ) -multiplier Cauchy and l ∞ ( X ) -multiplier convergent series, respectively. Finally, we show that if ∑ k T k is l ∞ ( X ) -multiplier Cauchy, then the multiplier spaces of almost convergence and weak almost convergence are identical. These results are more general than the corresponding consequences given by Swartz [20] , and are analogues given by Altay and Kama [6] .
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Operator valued series almost summability of vector valued multipliers and weak compactness of Summing Operator
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Mahmut Karakus, Feyzi BasarAbstract:Abstract In this study, we introduce the vector valued multiplier spaces M f ∞ ( ∑ k T k ) and M w f ∞ ( ∑ k T k ) by means of almost summability and weak almost summability, and a series of bounded linear Operators. Since these multiplier spaces are equipped with the sup norm and are subspaces of l ∞ ( X ) , we obtain the completeness of a normed space via the multiplier spaces which are complete for every c 0 ( X ) -multiplier Cauchy series. We also characterize the continuity and the compactness, and the weakly compactness of the Summing Operator S from the multiplier space M f ∞ ( ∑ k T k ) or M w f ∞ ( ∑ k T k ) to an arbitrary normed space Y through c 0 ( X ) -multiplier Cauchy or l ∞ ( X ) -multiplier convergent series, respectively. Finally, we show that if ∑ k T k is l ∞ ( X ) -multiplier Cauchy, then the multiplier spaces of almost convergence and weak almost convergence are identical. These results are more general than the corresponding consequences given by Swartz [20] , and are analogues given by Altay and Kama [6] .
Tradacete Pérez Pedro - One of the best experts on this subject based on the ideXlab platform.
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Maurey-Rosenthal factorization for p-Summing Operators and Dodds-Fremlin domination
The Theta Foundation, 2012Co-Authors: Palazuelos Cabezón Carlos, Sánchez Pérez, Enrique A., Tradacete Pérez PedroAbstract:We characterize by means of a vector norm inequality the space of Operators that factorize through a p-Summing Operator from an L-r-space to an L-s-space. As an application, we prove a domination result in the sense of Dodds-Fremlin for p-Summing Operators on Banach lattices with cotype 2, showing moreover that this cannot hold in general for spaces with higher cotype. We also present a new characterization of Banach lattices satisfying a lower 2-estimate in terms of the order properties of 2-Summing Operators
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Maurey-Rosenthal factorization for p-Summing Operators and Dodds-Fremlin domination
'Theta Foundation', 2012Co-Authors: Palazuelos Carlos, Sánchez Pérez, Enrique Alfonso, Tradacete Pérez PedroAbstract:We characterize by means of a vector norm inequality the space of Operators that factorize through a p-Summing Operator from anLr-space to an Ls-space. As an application, we prove a domination result in the sense of Dodds-Fremlin for p-Summing Operators on Banach lattices with cotype 2, showing moreover that this cannot hold in general for spaces with higher cotype. We also present a new characterization of Banach lattices satisfying a lower 2-estimate in terms of the order properties of 2-Summing Operators. © by THETA, 2012.First author partially supported by Spanish grants MTM2008-01366, I-MATH and CCG08- UCM / ESP-4394. Second author supported by the Ministerio de Educacion y Ciencia, under project MTM2009-14483-C02-02 (Spain), and FEDER. Third author partially supported by Spanish MICINN through Juan de la Cierva program and grant MTM2008-02652, Santander/Complutense PR34/07-15837 and Generalitat Valenciana grant Prometeo/2008/101.Palazuelos, C.; Sánchez Pérez, EA.; Tradacete Pérez, P. (2012). Maurey-Rosenthal factorization for p-Summing Operators and Dodds-Fremlin domination. Journal of Operator Theory. 68(1):205-222. http://hdl.handle.net/10251/56997S20522268
Daniel Pellegrino - One of the best experts on this subject based on the ideXlab platform.
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Some inclusion results for interpolated Summing Operator ideals and integrability improvement of vector valued functions
arXiv: Functional Analysis, 2016Co-Authors: Daniel Pellegrino, Pilar Rueda, Enrique A. Sánchez-pérezAbstract:Consider a Banach space valued measurable function $f$ and an Operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely Summing. In a previous work we have shown that a well-known interpolation technique for Operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-Summing Operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusion theorem for classes of {abstract} Summing Operators is shown and applied to the class of interpolated Operator ideals. Together with the results of the {aforementioned} paper, it provides more results on the relation about the integrability of the function $u \circ f$ and the summability properties of $u$.
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surveying the spirit of absolute summability on multilinear Operators and homogeneous polynomials
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2016Co-Authors: Daniel Pellegrino, Pilar Rueda, E A SanchezperezAbstract:We draw a fundamental compendium of the most valuable results of the theory of Summing linear Operators and detail those that are not shared by known multilinear and polynomial extensions of absolutely Summing linear Operators. The lack of such results in the theory of non-linear Summing Operators justifies the introduction of a class of polynomials and multilinear Operators that satisfies at once all related non-linear results. Surprisingly enough, this class, defined by means of a Summing inequality, happens to be the well known ideal of composition with a Summing Operator.
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Improving integrability via absolute summability: a general version of Diestel's Theorem
Positivity, 2015Co-Authors: Daniel Pellegrino, Pilar Rueda, Enrique A. Sánchez-pérezAbstract:A classical result by J. Diestel establishes that the composition of a Summing Operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the Operator. After proving a general result, we center our attention in the particular case given by the \((p,\sigma )\)-absolutely continuous Operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), \(L^p\) and Hilbert spaces—and Operators—p-Summing, (q, p)-Summing and p-approximable Operators.
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INCLUSION THEOREMS FOR ABSOLUTELY Summing HOLOMORPHIC MAPPINGS
2010Co-Authors: Heinz Junek, Daniel Pellegrino, Mário C. Matos, Communicated N. Tomczak-jaegermannAbstract:Abstract. For linear Operators, if 1 ≤ p ≤ q < ∞, then every absolutely p-Summing Operator is also absolutely q-Summing. On the other hand, it is well known that for n ≥ 2, there are no general “inclusion theorems ” for absolutely Summing n-linear mappings or n-homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely p-Summing and absolutely q-Summing linear Operators coincide, and prove that for 1 ≤ p ≤ q ≤ 2andn ≥ 2, we have inclusion theorems for absolutely Summing n-linear mappings/n-homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as p increases, the classes of absolutely p-Summing mappings becomes smaller. 1
O Delgado - One of the best experts on this subject based on the ideXlab platform.
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strong extensions for q Summing Operators acting in p convex banach function spaces for 1 le p le q
Positivity, 2016Co-Authors: O Delgado, E Sanchez A PerezAbstract:Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-Summing Operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-Summing Operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).
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strong extensions for q Summing Operators acting in p convex banach function spaces for 1 le p le q
arXiv: Functional Analysis, 2015Co-Authors: O Delgado, E Sanchez A PerezAbstract:Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces $L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{\,q}(\xi)$, where $\xi$ is a probability Radon measure on a certain compact set associated to $X$. We show some of its properties, and the relevant fact that every $q$-Summing Operator $T$ defined on $X$ can be continuously (strongly) extended to $S_{X_p}^{\,q}(\xi)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factorizations for $q$-Summing Operators through $L^q$-spaces when $1 \le q \le p$. Thus, our result completes the picture, showing what happens in the complementary case $1\le p\le q$, opening the door to the study of the multilinear versions of $q$-Summing Operators also in these cases.
