Cyclicity of reproducing kernels in weighted Hardy spaces over the bidisk (Q340970)

Let \(H\) be a Hilbert space of holomorphic functions on the unit disc, that contains the polynomials. A closed subspace \(M\) of \(H\) is called invariant if \(g(z):=zf(z)\) belongs to \(M\) for each \(f(z)\) in \(M\). In [Acta Math. 81, 239--255 (1949; Zbl 0033.37701)], \textit{A. Beurling} showed that \(M\) is an invariant subspace of the Hardy space \(H^2\) if and only if \(M= \varphi H^2\) for an inner function \(\varphi\). In [Acta Math. 177, No. 2, 275--310 (1996; Zbl 0886.30026)], \textit{A. Aleman} et al. investigated the case of the Bergman space \(H=L^2_a(\mathbb{D})\). In two variable cases, the situation is quite different and these Beurling type theorems do not hold in general. In [Trans. Am. Math. Soc. 328, No. 1, 421--432 (1991; Zbl 0749.32003)], \textit{T. Nakazi} posed a conjecture that the Beurling theorem holds for a singly generated invariant subspace. In this interesting paper, a relation between a singly generated invariant subspace and a weighted Hardy space \(H^2(d \mu)\) over the bidisk is studied. Nakazi's conjecture is characterized in terms of the cyclicity of reproducing kernels in the associated weighted Hardy spaces. It is shown that there exists a weighted Hardy space over the bidisk which has a noncyclic reproducing kernel. As a consequence, a counterexample for Nakazi's conjecture is given.