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Sibirskii Matematicheskii Zhurnal, 2009, Volume 50, Number 1, Pages 118–122 (Mi smj1942)  

$\langle2,1\rangle$-Compact operators

V. B. Korotkov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
References:
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
English version:
Siberian Mathematical Journal, 2009, Volume 50, Issue 1, Pages 96–99
DOI: https://doi.org/10.1007/s11202-009-0011-x
Bibliographic databases:
UDC: 517.983
Language: Russian
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
Citation in format AMSBIB
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\paper $\langle2,1\rangle$-Compact operators
\jour Sibirsk. Mat. Zh.
\yr 2009
\vol 50
\issue 1
\pages 118--122
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\jour Siberian Math. J.
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\vol 50
\issue 1
\pages 96--99
\crossref{https://doi.org/10.1007/s11202-009-0011-x}
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