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Recursively generated weighted shifts and the subnormal completion problem. (English) Zbl 0804.47028
Summary: For $$m\geq 0$$, let $$\alpha_ 0,\dots,\alpha_ m$$ be a finite sequence of positive numbers. We find necessary and sufficient conditions for the existence of a subnormal unilateral weighted shift whose first $$(m+1)$$ weights are $$\alpha_ 0,\dots,\alpha_ m$$. We explicitly construct such a subnormal “completion” $$W_{\widehat\alpha}$$, with weights $$\{\widehat\alpha_ n\}^ \infty_{n=0}$$ and “moments” $$\gamma_ 0= 1$$, $$\gamma_ n= \alpha^ 2_ 0\cdot\cdots\cdot \alpha^ 2_{n-1}$$ $$(n\geq 1)$$, such that:
(i) There exists a finitely atomic probability measure $$\mu$$ on $$[0,+\infty)$$ satisfying $$\int t^ n d\mu= \gamma_ n$$ $$(0\leq n\leq m+1)$$;
(ii) $$W_{\widehat\alpha}$$ is the unique minimal-norm subnormal completion of $$\alpha$$, and the moments of $$W_{\widehat\alpha}$$ are minimal relative to the moments of any other subnormal completion;
(iii) The moments of $$W_{\widehat\alpha}$$ satisfy a linear recursion relation.
Moreover, every subnormal shift is the norm limit of recursively generated ones.

##### MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B20 Subnormal operators, hyponormal operators, etc. 47A57 Linear operator methods in interpolation, moment and extension problems 44A60 Moment problems 47B47 Commutators, derivations, elementary operators, etc.
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