## Invertibility in Banach algebras of functional operators with non-Carleman shifts

Citation:
Karlovich, Alexei Yu., and Yuri I. Karlovich. "Invertibility in Banach algebras of functional operators with non-Carleman shifts." Ukrains'kyj matematychnyj kongres -- 2001. Pratsi. Sektsiya 11. Funktsional'nyj analiz. Kyiv: Instytut Matematyky NAN Ukrainy, 2002. 107-124.

### Abstract:

We prove the inverse closedness of the Banach algebra $$\mathfrak{A}_p$$ of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on $$L^p$$. We suppose that $$1 \le p \le \infty$$ and the generators of the algebra $$\mathfrak{A}_p$$ have essentially bounded data. An invertibility criterion for functional operators in $$\mathfrak{A}_p$$ is obtained in terms of
the invertibility of a family of discrete operators on $$l^p$$. An effective invertibility criterion is established for binomial difference operators with $$l^\infty$$ coefficients on the spaces $$l^p$$. Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces $$L^p$$.

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