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On the numerical range of operators on some special Banach spaces (2005.01288v1)
Published 4 May 2020 in math.FA
Abstract: The numerical range of a bounded linear operator on a complex Banach space need not be convex unlike that on a Hilbert space. The aim of this paper is to study operators $T$ on $ \ell2_p $ for which the numerical range is convex. We also obtain a nice relation between $V(T)$ and $ V(Tt)$ considering $ T \in \mathbb{L} (\ell_p2) $ and $ Tt \in \mathbb{L} (\ell_q2) ,$ where $Tt$ denotes the transpose of $T$ and $p$ and $q$ are conjugate real numbers i.e., $ 1 <p,q< \infty $ and $ \frac{1}{p}+\frac{1}{q}=1.$