Extensions and extremality of recursively generated weighted shifts (Q2758995)

A weight sequence \(\widehat\alpha= (\widehat\alpha_i)^\infty_{i=0}\) is said to be recursively generated by an initial segment \((\alpha_0,\dots, \alpha_k)\) of weights with the property \(0< \alpha_0<\cdots< \alpha_k\) if there exist \(r\geq 1\) and coefficients \(\varphi_0,\dots, \varphi_{r-1}\in \mathbb{R}\) such that \(\gamma_{i+r}= \varphi_0\gamma_i+\cdots+ \varphi_{r-1}\gamma_{i+r-1}\) for each \(i\geq 0\) where \(\gamma_0:= 1\), \(\gamma_i:= \widehat\alpha^2_0\cdots \widehat\alpha^2_{i-1}\) for each \(i\geq 1\); in terms \(\widehat\alpha= (\alpha_0,\dots, \alpha_k)^\wedge\). Subnormality is investigated for unilateral weighted shifts \(W_\alpha\), where the weight sequence \(\alpha\) is obtained from a recursively generated weight sequence by putting \(n\) elements in front \(\alpha= (x_n,\dots, x_1,(\alpha_0,\dots, \alpha_k)^\wedge)\). \(W_\alpha\) is subnormal in case \(n=1\) if and only if \(W_\alpha\) is \(([{k+1\over 2}]+ 1)\)-hyponormal and in case \(n> 1\) if and only if \(W_\alpha\) is \(([{k+1\over 2}]+ 2)\)-hyponormal. (An operator \(T\) is \(m\)-hyponormal if the operator matrix \((A_{ij})^m_{i,j=0}\) with entries \(A_{ij}:= T^{*j}T^i\) is positive.) Conversely let \(\alpha(x)\) be a canonical rank-one perturbation of a weight sequence. If \((k+1)\)- and \(k\)-hyponormality coincide for \(W_{\alpha(x)}\) and some \(k\), then \(\alpha(x)\) is recursively generated.