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Sibirskii Matematicheskii Zhurnal, 2009, Volume 50, Number 1, Pages 118–122
(Mi smj1942)
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$\langle2,1\rangle$-Compact operators
V. B. Korotkov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
We consider the class of the continuous $L_{2,1}$ linear operators in $L_2$ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of $L_2$ into a compact subset of $L_1$. We prove that a functional equation with an operator from $L_{2,1}$ is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if $T\in L_{2,1}$ and $VTV^{-1}\in L_{2,1}$ for all unitary operators $V$ in $L_2$ then $T=\alpha1+C$, where $\alpha$ is a scalar, 1 is the identity operator in $L_2$, and $C$ is a compact operator in $L_2$.
Keywords:
compact operator, $\langle2,1\rangle$-compact operator, multiplication operator, integral operator, Carleman integral operator, integral equation.
Received: 10.01.2008
Citation:
V. B. Korotkov, “$\langle2,1\rangle$-Compact operators”, Sibirsk. Mat. Zh., 50:1 (2009), 118–122; Siberian Math. J., 50:1 (2009), 96–99
Linking options:
https://www.mathnet.ru/eng/smj1942 https://www.mathnet.ru/eng/smj/v50/i1/p118
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