## Algebras of Toeplitz operators with oscillating symbols.(English)Zbl 1083.47023

The authors study the Banach algebras generated by Toeplitz operators $$T(a)f=P(af)$$ acting on the Hardy space $$H^2(\mathbb{R})$$ over the real line $$\mathbb{R}$$ with strongly oscillating symbols $$a=a(x),$$ $$x\in \mathbb{R}.$$ They establish conditions for normal solvability, Fredholmness, and invertibility of operators in these algebras. Here $$P$$ stands for the orthogonal projection of $$L^2(\mathbb{R})$$ onto $$H^2(\mathbb{R}).$$ The function $$a,$$ which is referred to as the symbol, is of the form $$a(x)=b(e^{i\alpha(x)}),$$ where $$\alpha:\mathbb{R}\to\mathbb{R}$$ is a homeomorphism that preserves the orientation of $$\mathbb{R}$$ and $$b$$ belongs to a given closed subalgebra of the algebra $$L^\infty(\mathbb{T})$$ of bounded functions on the unit circle $$\mathbb{T}.$$ In particular, the important case when $$\alpha(x)=\lambda x$$ of $$2\pi/\lambda$$-periodic functions $$a$$ on $$\mathbb{R}$$ is treated as well. If $$e^{i\alpha}$$ can be extended to a bounded analytic function in the upper half plane (the authors use $$H^\infty(\mathbb{R})$$ to denote the Banach algebra of all such functions from $$L^\infty(\mathbb{R})$$), then $$e^{i\alpha}$$ is an inner function, and this case is thoroughly treated in the paper. In the general case, the authors assume that $$e^{i\alpha}\in C(\dot{\mathbb{R}})+H^\infty(\mathbb{R}),$$ where $$C(\dot{\mathbb{R}})$$ is the algebra of continuous functions on the one point compactification of $$\mathbb{R},$$ factorize $$e^{i\alpha}=u_\alpha c,$$ where $$u_\alpha$$ is inner function and $$c\in C(\dot{\mathbb{R}}),$$ $$c(\infty)=1$$, and use the technique developed in [S. M. Grudsky, Math. Nachr. 129, 313–331 (1986; Zbl 0612.30039)] to extend the results for this case.

### MSC:

 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 47C05 Linear operators in algebras

Zbl 0612.30039
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### References:

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