Kernels of bounded operators on transfinite Banach sequence spaces

Abstract: For a Banach space $E$, we ask the following question: "Is it true that for every closed subspace $Y$ of $E$, there exists some bounded linear operator $T : E \to E$ for which $Y= \ker T$?"

In a recent paper by Niels Laustsen and Jared White, it was proved that every separable Banach space answers the question in the positive, and that there exists a reflexive Banach space which answers the question in the negative.

Let $\Gamma$ be an uncountable cardinal. In this talk we will investigate the above question for the transfinite Banach sequence spaces $\ell_p(\Gamma)$ for $1\leq p <\infty$, and $c_0(\Gamma)$. The question is answered in the negative for $\ell_1(\Gamma)$, and in the positive for $\ell_p(\Gamma)$ for $1 < p <\infty$ and $c_0(\Gamma)$.

functional analysisgroup theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Groups, Operators, and Banach Algebras Webinar

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