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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

Citations:

Zbl 0612.30039
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References:

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