The near subnormal weighted shift and recursiveness (Q355562)

Let \(l^2(\mathbb{Z}^+)\) be the usual Hilbert space of all square-summable complex sequences and let NEWLINE\[NEWLINEW_\alpha((x_k)_{k\geq0}):=(0,\alpha_0x_0,~\alpha_1x_1,\alpha_2x_2,\dots),\quad(x_k)_{k\geq0}\in l^2(\mathbb{Z}^+),NEWLINE\]NEWLINE be a unilateral weighted shift with a bounded weight sequence \((\alpha_n)_{n\geq0}\). The authors investigate near subnormality of the weighted shift \(W_\alpha\) when its moments, defined by \(\gamma_0=1\), \(\gamma_k=\alpha_0^2\alpha_1^2\dots\alpha_{k-1}^2\), \(k>0\), satisfy a linear recursive relation. They also provide several applications and discuss, in particular, the notion of near subnormal completion.