I. V. Krasikova

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Articles: 1

Operators defined on $L_1$ which "nowhere" attain their norm

I. V. Krasikova, V. V. Mykhaylyuk, M. M. Popov

MFAT 16 (2010), no. 1, 17-27

17-27

Let $E$ be either $\ell_1$ of $L_1$. We consider $E$-unattainable continuous linear operators $T$ from $L_1$ to a Banach space $Y$, i.e., those operators which do not attain their norms on any subspace of $L_1$ isometric to $E$. It is not hard to see that if $T: L_1 \to Y$ is $\ell_1$-unattainable then it is also $L_1$-unattainable. We find some equivalent conditions for an operator to be $\ell_1$-unattainable and construct two operators, first $\ell_1$-unattainable and second $L_1$-unattainable but not $\ell_1$-unattainable. Some open problems remain unsolved.

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