We determine the smallest Schatten class containing all integral operators with kernels in$L$_{p}(L_{p', q})^{symm}, where 2 <$p$∞ and 1≦$q$≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].
Daniel CarandoDepartamento de Matemática y Ciencias, Universidad de San AndrésSilvia LassalleDepartmento de Matematica-Pab I Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires
We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on$E$and$F$, we show that the spaces of$X$-valued$n$-homogeneous polynomials and analytic functions of bounded type on$E$and$F$are isomorphic whenever$X$is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on$E$and$F$are isomorphic without conditions on the involved spaces.