An Operator Theoretic Approach to Truncated Moment Problems∗

We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together with the construction of a “functional calculus” for the columns of the associated moment matrix, our operator-theoretic approach allows us to obtain existence theorems for the truncated complex moment problem, ∗This survey is based on four lectures given at the Stefan Banach International Mathematical Center during April 5 8, 1994, as part of the Semester on Linear Operators. The author is deeply indebted to the organizers, Professors J. Janas, F.H. Szafraniec and J. Zemánek, for their invitation and for the warm hospitality during his stay in Warsaw. †Research partially supported by a grant from the National Science Foundation