Articles written in Proceedings – Mathematical Sciences
Volume 117 Issue 1 February 2007 pp 13-30
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
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
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.