@inbook {7485, title = {Invertibility of functional operators with slowly oscillating non-Carleman shifts}, booktitle = {Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, 142}, year = {2003}, pages = {147-174}, publisher = {Birkh{\"a}user}, organization = {Birkh{\"a}user}, address = {Basel}, abstract = {

We prove criteria for the invertibility of the binomial functional operator
\[
A=aI-bW_\alpha
\]
in the Lebesgue spaces \(L^p(0,1)\), \( 1 \< p \< \infty\), where \(a\) and \(b\) are continuous functions on \((0,1)\), \(I\) is the identity operator, \(W_\alpha\) is the shift operator, \(W_\alpha f=f\circ\alpha\), generated by a non-Carleman shift \(\alpha:[0,1]\to[0,1]\) which has only two fixed points \(0\) and \(1\). We suppose that \(\log\alpha{\textquoteright}\) is bounded and continuous on \((0,1)\) and that \(a,b,\alpha{\textquoteright}\) slowly oscillate at \(0\) and \(1\). The main difficulty connected with slow oscillation is overcome by using the method of limit operators.

}, url = {http://link.springer.com/chapter/10.1007/978-3-0348-8007-7_9}, author = {Karlovich, Alexei Yu. and Karlovich, Yuri I. and Lebre, Amarino B.}, editor = {B{\"o}ttcher, Albrecht and Marinus A. Kaashoek and Amarino Brites Lebre and Ferreira dos Santos, Ant{\'o}nio and Frank-Olme Speck} }