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The Hardy operator has all the monomial functions as eigenvectors. We studybounded operators on L^2 that take monomial functions to multiples of othermonomials, with a shifted exponent. We prove that they all leave the space offunctions vanishing on [0,s] invariant. We prove an asymptotic Muntz-Szasztheorem, characterizing the set of functions that are limits of linearcombinations of monomials with exponents between n and 2n.

A generalization of Hardy's operator and an asymptotic Muntz-Szasz Theorem