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A note on common range of a class of co-analytic Toeplitz operators. (English) Zbl 1177.47035

Let \(\mathbf D\) denote the open unit circle in the complex plane and let \(\mathbf T\) denote the boundary of \(\mathbf D\). Let \( H^{2}\subset L^{2}(\mathbf T) \) be the standard Hardy space. For \( m\in L^{\infty}(\mathbf T)\), denote by \(T_{m}\) the Toeplitz operator \(T_{m}f=P(mf)\), \(f\in H^{2} \), where \(P\) is the orthogonal projection from \(L^{2}(\mathbf T)\) onto \(H^{2}\). If \(m\) is the complex conjugate of a function in \(H^{\infty}\), \(T_{m}\) is called a co-analytic Toeplitz operator.
For \(p>1\), denote by \( {\mathcal W}^{p} \) the set of all positive functions in \(L^{1}(\mathbf T)\) such that \( \int_{0}^{2\pi}|\log w(e^{i\theta})|^{p}\,d\theta<\infty \). Consider a class of co-analytic Toeplitz operators \(T_{\bar h}\) with a function \(h\in H^{\infty }\) such that \( |h^{\ast}|^{2}\in {\mathcal W}^{p} \), where \(h^{\ast}\) denotes the radial limits of \(h\) on \(\mathbf T\). It is shown that the intersection of the ranges of all Toeplitz operators belonging to this class consists of all analytic functions \(f\) on \(\mathbf D\) whose Taylor coefficients \(\hat{f}(n)\) decay like \( O(\exp (-cn^{1/(p+1)})) \) for a positive constant \(c\). There is given another characterization of this intersection in terms of multipliers of de Branges spaces.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H10 Hardy spaces
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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References:

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