Abstract
We construct an example of a compact operator of the third kind in L p (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in L p (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in L p (p ≠ 2) in terms of the spectrum and its components.
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Original Russian Text Copyright © 2014 Korotkov V.B.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 124–130, January–February, 2014.
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Korotkov, V.B. On the similarity of linear operators in L p to integral operators of the first and second kind. Sib Math J 55, 100–104 (2014). https://doi.org/10.1134/S0037446614010121
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DOI: https://doi.org/10.1134/S0037446614010121