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Monday, March 12, 2018 - 11:40 to 12:40
SMS Seminar Hall
IIT Kanpur
Approximation Property and Its Variants in Weighted Spaces of Holomorphic Functions on Banach Spaces
Abstract:- Let $E$ be a complex Banach space and $U$ be an open subset of $E$. Corresponding to a weight $w$(a strictly positive continuous function) defined on U, the weighted space of holomorphic functions is defined as $$\mathcal{H}_w(U) = \{ f\in \mathcal{H}(U) : \|f\|_w = \sup_{x\in U}w(x)\|f(x)\|\leq 1\}.$$ The space $(\mathcal{H}_w(U),~ \|\cdot\|_w)$ is a dual Banach space and its predual $\mathcal{G}_w(U)$ is given by \begin{align*} \mathcal{G}_w(U)=\{\phi \in \mathcal{H}_w(U)^\prime: \phi |B_w ~\textrm{is}~ \tau_0 -\textrm{continuous}\} \end{align*} where $B_w$ denotes the closed unit ball of the weighted space $(\mathcal{H}_w(U),~ \|\cdot\|_w)$. In this talk, we consider the approximation property and its variants for $\mathcal{H}_w(U)$ and its predual $\mathcal{G}_w(U)$. After introducing a locally convex topology $\tau_\mathcal{M}$ on these spaces, we show that the weighted space equipped with the topology $\tau_\mathcal{M}$ becomes topologically isomorphic to the class of linear operators endowed with the topology of uniform convergence on compact sets. This leads us to characterize the approximation property for a complex Banach space $E$ in terms of the approximation property for these spaces. Further, we characterize the bounded approximation property for weighted Fr\'{e}chet and (LB)-spaces using the techniques of $\mathcal{S}$-absolute decompositions

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