ON THE SIMILARITY OF LINEAR OPERATORS IN Lp TO INTEGRAL OPERATORS OF THE FIRST AND SECOND KIND

摘要

We construct an example of a compact operator of the third kind in Lp (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 Lp (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.