## Invertibility of functional operators with slowly oscillating non-Carleman shifts

Citation:
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Invertibility of functional operators with slowly oscillating non-Carleman shifts." Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, 142. Eds. Albrecht Böttcher, Marinus A. Kaashoek, Amarino Brites Lebre, António Ferreira dos Santos, and Frank-Olme Speck. Basel: Birkhäuser, 2003. 147-174.

### 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 $$\log\alpha'$$ is bounded and continuous on $$(0,1)$$ and that $$a,b,\alpha'$$ slowly oscillate at $$0$$ and $$1$$. The main difficulty connected with slow oscillation is overcome by using the method of limit operators.