Dumitru Popa
Articles written in Proceedings – Mathematical Sciences
Volume 117 Issue 1 February 2007 pp 13-30
Khinchin's Inequality, Dunford-Pettis and Compact Operators on the Space $C([0, 1], X)$
We prove that if $X, Y$ are Banach spaces, 𝛺 a compact Hausdorff space and $U:C(\Omega,X)\to Y$ is a bounded linear operator, and if 𝑈 is a Dunford–Pettis operator the range of the representing measure $G(\Sigma)\subseteq D P(X, Y)$ is an uniformly Dunford–Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_0$ or $l_p,(1\leq p < \infty)$ be Dunford–Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.
Volume 119 Issue 2 April 2009 pp 221-230
2-Summing Operators on $C([0, 1], l_p)$ with Values in $l_1$
Let 𝛺 be a compact Hausdorff space, 𝑋 a Banach space, $C(\Omega,X)$ the Banach space of continuous 𝑋-valued functions on 𝛺 under the uniform norm, $U:C(\Omega,X)\to Y$ a bounded linear operator and $U^\#,U_\#$ two natural operators associated to 𝑈. For each $1\leq s <\infty$, let the conditions $(\alpha)U\in \Pi_s(C(\Omega, X), Y);(\beta)U^\#\in \Pi_s(C(\Omega), \Pi_s(X, Y));(\gamma)U_\#\in \Pi_s(X, \Pi_s(C(\Omega), Y))$. A general result, [10,13], asserts that $(\alpha)$ implies (𝛽) and (𝛾). In this paper, in case $s=2$, we give necessary and sufficient conditions that natural operators on $C([0,1],l_p)$ with values in $l_1$ satisfies (𝛼), (𝛽) and (𝛾), which show that the above implication is the best possible result.
Volume 123 Issue 3 August 2013 pp 415-426
Direct and Reverse Inclusions for Strongly Multiple Summing Operators
We prove some direct and reverse inclusion results for strongly summing and strongly multiple summing operators under the assumption that the range has finite cotype.
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