Bernstein functions, complete hyperexpansivity and subnormality. I.
The notion of subnormal operator was introduced in [Summa Brasil. Math. 2, 125--134 (1950; Zbl 0041.23201)] by \textit{P. R. Halmos}, while the notion of a completely hyperexpansive operator was introduced in [Proc. Am. Math. Soc. 124, 3745--3752 (1996; Zbl 0863.47017)] by \textit{A. Athavale}. Let us recall that by definition, a bounded operator \(T\in B(H)\) is completely hyperexpansive if \( \sum_{0\leq p\leq n}(-1)^p \binom np T^{*p}T^p\leq 0\) for all \(n\geq 1\). It is well-known that subnormal operators are closely related to the theory of positive definite functions on the abelian semigroup \(({\mathbb{N}},+,n^*=n)\), while completely hyperexpansive operators correspond to negative definite functions on \(({\mathbb{N}},+,n^*=n)\). In the paper under review, the authors characterize minimal Levy sequences and prove that the composition of a completely alternating function with a Bernstein function is a completely alternating function. Using this, the authors show that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. They also observe that the weighted sequence of a completely hyperexpansive weighted shift with the weight sequence \(\{\alpha_n\}_{n=0}^\infty\) gives rise to a subnormal weighted shift with weight sequence \(\{\alpha_{n+1}/\alpha_n\}_{n=0}^\infty\).
scientific article; zbMATH DE number 1802505
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