Sesión Análisis Funcional y ComplejoWeak Minimizing Property and Compact Perturbation
Geivison dos Santos Ribeiro GG
Universidade Federal do Maranhão, Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
For a bounded linear operator \(T:X\to Y\), put \(m(T)=\inf \{\lVert Tx\rVert:x \in S_X \}\). A sequence \((x_n)\) in \(S_X\) is said to be minimizing for \(T\) if \(\lVert Tx_n\rVert\to m(T)\). A pair \((X,Y)\) has the weak minimizing property (WmP), introduced by Chakraborty in [2], if every operator \(T:X\to Y\) admitting a non-weakly null minimizing sequence attains its minimum modulus \(m(T)\). More recently, Han in [5] introduced the Compact Perturbation Property for the minimum modulus (CPPm), which requires that, for every operator \(T:X\to Y\) that does not attain its minimum modulus, \(\sup_{K\in\mathcal{K}(X,Y)}m(T+K)=m(T)\). In [5], it is shown that \((\ell_1,\ell_1)\) fails both properties, while \((c_0,c_0)\) fails the WmP. However, whether \((c_0,c_0)\) has the CPPm was left open in Problem 3.6 of [5]. In this work, we answer this question negatively by proving that \((c_0,c_0)\) does not have the CPPm. Our proof is constructive: we exhibit a non-minimum-attaining operator whose minimum modulus is strictly increased by a rank-one compact perturbation. Moreover, we show that this phenomenon is not specific to \(c_0\): if \(X=\mathbb{K}\oplus_\infty Y\), where \(Y\) is non-reflexive, then the pair \((X,X)\) fails the CPPm.
Trabajo en conjunto con: Vinícius Vieira Fávaro (Universidade Federal de Uberlândia, Brasil) y Anselmo Raposo Jr. (Universidade Federal do Maranhão, Brasil).
Referencias
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