Hamburger-type weighted shifts: Jumping flatness and Aluthge transforms

Abstract In this paper, we introduce a new kind of flatness, namely “jumping flatness,” of a Hamburger-type weighted shift; we characterize jumping flatness of a nonsubnormal Hamburger-type weighted shift W α and obtain the expression of the associated Hamburger moment measure. Also we discuss a relationship between jumping flatness and subnormality of the Aluthge transform W ˜ α . Precisely, we show that if W α is a nonsubnormal Hamburger-type weighted shift, then W α has jumping flatness if and only if there exist positive real numbers ϕ , φ , ρ , p , and q such that μ = ϕ δ − p + φ δ 0 + ρ δ q (or μ = ϕ δ − p + ρ δ q ) and W ˜ α is subnormal, where δ x is the usual Dirac measure.