The notion of an alternatingly hyperexpansive operator on a Hilbert space is generalized to that of an alternatingly hyperexpansive operator tuple, which necessitates exploring the theory of absolutely monotone functions as defined on the m-fold product N m of the semi-group N of non-negative integers and as defined on semi-open cubes in the m-dimensional real Euclidean space R m. The multi-variable Laplace transform and the Stieltjes Moment Problem make a natural appearance in the development of the relevant theory, which also highlights the close connections of alternatingly hyperexpansive operator tuples with completely hyperexpansive and subnormal ones. In particular, if T is subnormal and the joint (Taylor) spectrum of its minimal normal extension is contained in a certain subset of the Hermitian space C m, then T turns out to be alternatingly hyperexpansive. In the context of multi-variable weighted shifts, the last assertion can be related to the notion of a Stieltjes Moment Net. The general characterization of an alternatingly hyperexpansive m-variable weighted shift T, however, requires a certain net of (positive) numbers associated with T to be absolutely monotone on N m and allows for such a T to be non-subnormal.
Positivity – Springer Journals
Published: Oct 19, 2004
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