Cauchy transforms of measures and weighted shift operators on the disc algebra
Let \(H(U)\) denote the space of holomorphic functions on the unit disk \(U= \{z \in \mathbb{C}:|z|<1\}\), and denote by \(M (\Gamma)\) the Banach space of complex Borel measures \(\mu\) on the unit circle \(\Gamma=\{z\in\mathbb{C}:|z|=1\}\) with the total variation norm \(|\mu |=\int_\Gamma|d\mu|\). For \(\alpha >0\), the \(\alpha\)-Cauchy transform \({\mathcal C}_\alpha\mu\) of \(\mu\in M (\Gamma)\) is given by \[ ({\mathcal C}_\alpha,\mu) (z)=\int_\Gamma (1-\zeta z)^{-\alpha} d\mu (\zeta), \quad z \in U, \] and we let \({\mathcal F}_\alpha={\mathcal C}_\alpha (M (\Gamma))\). The latter is a subspace of \(H(U)\) and is a Banach space with the norm \(|f|_\alpha=\min \{|\mu |:\mu\in M (\Gamma)\), \(f= {\mathcal C}_\alpha\mu\}\) for \(f\in {\mathcal F}_\alpha\). Moreover, for each \(\alpha >0\), the space \({\mathcal F}_\alpha\) is isometrically isomorphic to the dual space \(A(U)^*\) of the disk algebra \(A(U)\). Denote by \({\mathcal F}_{\alpha a}\) the subspace of \({\mathcal F}_\alpha\) consisting of all \({\mathcal C}_\alpha\mu\) such that \(\mu\in M (\Gamma)\) is absolutely continuous with respect to the Lebesgue measure on \(\Gamma\). Then \({\mathcal F}_{\alpha a}\) is the norm closure of the complex polynomials in \({\mathcal F}_\alpha\). \textit{T. H. MacGregor} [Indiana Univ. Math. J. 36, 109-130 (1987; Zbl 0644.30009)] has shown that for \(0<\alpha < \beta\), \({\mathcal F}_\alpha\subset {\mathcal F}_\beta\). In the present paper, the authors establish the stronger inclusion \({\mathcal F}_\alpha\subset {\mathcal F}_{\beta a}\) with the inclusion map being a compact operator of norm one. MacGregor has also shown that \({\mathcal F}_\alpha {\mathcal F}_\beta\subset {\mathcal F}_\alpha +_\beta\) for \(\alpha, \beta >0\). Using different methods, the authors prove the more precise result, namely that \(|fg |_{\alpha + \beta} \leq|f |_\alpha |g |_\beta\) whenever \(f\in {\mathcal F}_\alpha\) and \(g\in {\mathcal F}_\beta\), \(\alpha, \beta>0\). Let \({\mathcal M}_\alpha\) be the multipliers space of \({\mathcal F}_\alpha\), i.e. \({\mathcal M}_\alpha=\{\varphi \in H(U):\varphi {\mathcal F}_\alpha\subset {\mathcal F}_\alpha\}\). Each multiplier \(\varphi\in {\mathcal M}_\alpha\) of \({\mathcal F}_\alpha\) determines a multiplication operator \(M_\varphi\), given by \(M_\varphi f=\varphi f\), \(f\in {\mathcal F}_\alpha\), which belongs to \({\mathcal B} ({\mathcal F}_\alpha)\), the Banach algebra of bounded linear operators on \({\mathcal F}_\alpha\) with the operator-norm \(|\cdot |_{\infty,\alpha}\). In particular, \({\mathcal M}_\alpha\subset {\mathcal F}_\alpha\) with \(|\varphi |_\alpha \leq|M_\varphi |_{\infty,\alpha}\) for \(\varphi \in {\mathcal M}_\alpha\) and \({\mathcal M}_\alpha = \{\varphi \in H(U): M_\varphi \in {\mathcal B} ({\mathcal F}_\alpha)\}\) is itself a Banach algebra with identity and with the norm \(|\cdot |_{\infty, \alpha}\). Using the inclusion \({\mathcal F}_\alpha {\mathcal F}_\beta \subset {\mathcal F}_{\alpha + \beta}\), \(\alpha, \beta>0\), the first author and \textit{T. H. Mac Gregor} [Trans. Am. Math. Soc. 331, No. 1, 377-394 (1992; Zbl 0769.30029)] have shown that \({\mathcal M}_\alpha \subset {\mathcal M}_\beta\) when \(0< \alpha< \beta\). Here, the authors show that this inclusion is contractive, i.e. that \(|M_\varphi |_{\infty, \beta} \leq |M_\varphi |_{\infty, \alpha}\) for \(\varphi \in {\mathcal M}_\alpha (0< \alpha< \beta)\). They also show that, for \(\alpha >0\), each \(M_\varphi\), \(\varphi \in {\mathcal M}_\alpha\), is the adjoint of the commutant of a certain weighted shift operator on the disk algebra \(A(U)\). When \(\alpha =1\), this result is quite classical and is due to \textit{S. A. Vinogradov} [Proc. Seventh Winter School, Drogobych, 1974; also in Transl., II. Ser., Am. Math. Soc. 115, 1-31 (1980; Zbl 0449.30027)]. The other results of the paper concern coefficients conditions on a function \(\varphi \in H(U)\), with \(\varphi (z) = \sum^\infty_{n=0} a_nz^n\), \(z \in U\), which imply \(\varphi \in {\mathcal M}_\alpha\), \(\alpha > 0\). One, however, should compare these coefficients conditions with the apparently sharper conditions found in \textit{D. J. Hallenbeck}, \textit{T. H. MacGregor} and \textit{K. Samotij} [Proc. Lond. Math. Soc., III. Ser. 72, No. 1, 157-187 (1996; Zbl 0842.30028)] and \textit{A. P. Dansereau} [Houston J. Math. 22, No. 2, 357-374 (1996; reviewed above)]. Such conditions are used to show that the inclusion \({\mathcal M}_\alpha \subset {\mathcal M}_\beta\) for \(0< \alpha< \beta\) is proper in a number of cases but, in general, this question remains open.
scientific article; zbMATH DE number 944411
scientific article