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Prolongations and cyclic vectors. (English) Zbl 1062.30005
Let \(X\) be a Banach space of analytic functions on the open disc \({\mathbb D}=\{z\in{\mathbb C}: | z| <1\}\) which is invariant under the backward shift operator \(Bf:=\frac{f-f(0)}{z}\). A vector \(f\in X\) is called cyclic if \(\overline{\text{span}}\,\{B^nf:\,n\in{\mathbb Z}_+\}=X\). The well known Douglas-Shapiro-Shields result says that a vector \(f\) in the Hardy space \(H^2\) is non-cyclic if and only if it admits a pseudocontinuation \(\widetilde f\) to \({\mathbb D}_e\cup\{\infty\}\), where \({\mathbb D}_e=\{z\in{\mathbb C}: | z| >1\}\), (see R.G Douglas, H.S. Shapiro, A.L. Shields [Ann. Inst. Fourier, 20, 37–76 (1970; Zbl 0186.45302)]). The latter means that there is a meromorphic function \(\widetilde f\) of bounded type on \({\mathbb D}_e\cup\{\infty\}\) such that \(\widetilde f(\zeta)=f(\zeta)\) a.e. on \({\mathbb T}=\{\zeta:| \zeta| =1\}\). Another continuation \(\widetilde f_L\) of \(f\) to \({\mathbb D}_e\), given by the formula \(\widetilde f_L(w):=L\left(\frac{f}{z-w}\right)/L\left(\frac{1}{z-w}\right)\), where \(L\) is a bounded linear functional on \(X\) annihilating on \(f\), is called an \(L\)-prolongation of \(f\). In some spaces, such as Hardy and Bergman spaces, \(L\)-prolongation \(\widetilde f_L\) of \(f\) coincides with its pseudocontinuation and does not depend on the choice of \(L\), in others, like Dirichlet space, a non cyclic vector \(f\) may have an \(L\)-prolongation without admitting a pseudocontinuation.
In the present paper the authors investigate compatibility of \(L\)-prolongations with ordinary analytic continuation for a wide class of Banach spaces of analytic functions. In the case when the \(L\)-prolongation \(\widetilde f_L\) of \(f\) has an analytic continuation across a boundary point \(\zeta\in{\mathbb T}\) the following question is studied: Is this continuation equal to \(f\) on a neighborhood of \(\zeta\)? As was shown by H. S. Shapiro [Ark. Mat. 7, 343–349 (1968; Zbl 0159.41903)] the concept of \(L\)-prolongation is related to the problem of “overconvergence” of rational functions from the \(B\)-invariant subspace \({\mathcal M}\) of \(X\) with the spectral synthesis property. The authors give a generalization (and a new proof) of a result from that paper saying that any sequence \(\{f_n\}_{n\geq 1}\) of finite linear combinations of root vectors from \({\mathcal M}\) which converges in norm to \(f\in{\mathcal M}\) converges also to \(\widetilde f_L\) uniformly on compact subsets of \({\mathbb D}_e\setminus\{a_n\}_{n\geq 1}\), where \(L\in{\mathcal M}^\perp\) and \(a_n^{-1}\) are eigenvalues of \(B| {\mathcal M}\). The problem of “overconvergence” of rational functions from \(B\)-invariant subspaces with the approximate spectral synthesis property is also examined.
30B40 Analytic continuation of functions of one complex variable
47B38 Linear operators on function spaces (general)
Full Text: DOI
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