Berger measure for \(S(a,b,c,d)\)
Given a bounded sequence of positive numbers \(\alpha = (\alpha_0, \alpha_1, \alpha_2,\dots),\) the weighted shift \(W_\alpha\) is defined by \(W_\alpha e_k := \alpha_k e_{k+1} ~(k \geq 0)\). The moments of \(\alpha\) are given as \(\gamma_k(\alpha) = 1\) if \(k=0\) and \(\gamma_k(\alpha) = \alpha_0^2 \cdot\dots \cdot \alpha_{k-1}^2\) if \(k>0\). It is well known that \(W_\alpha\) is subnormal if and only if there exists a positive Borel measure \(\xi\) on \([0, \|W_\alpha\|^2]\) such that \(\gamma_k(\alpha) = \int t^k \,d\xi(t)\) (\(k \geq 0\)), where \(\xi\) is the Berger measure of \(W_\alpha\). The authors get the Berger measure of the weighted shift \(S(a,b,c,d)\) with weights \(\alpha_n := \sqrt{\frac{an+b}{cn+d}}\) (\(a,b,c,d>0\) and \(n \geq 0\)) as well as its \(p\)-subshift. Then they give examples from analytic function theory to illustrate that the necessary condition given by \textit{R. E. Curto} and \textit{J. Yoon} [Proc. Lond. Math. Soc. (3) 92, No. 2, 381--402 (2006; Zbl 1092.47025)] for subnormality of 2-variable weighted shift is not sufficient. Furthermore, they do some calculations about subnormal backward extension of \(S(a,b,c,d)\).
scientific article; zbMATH DE number 6420179
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