Vol. 17, No. 2, 1966

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Which weighted shifts are subnormal

Joseph Gail Stampfli

Vol. 17 (1966), No. 2, 367–379

Let H be a Hilbert space with orthonormal basis {fj}j=1. If the operator T is defined on H by Tfi = ajfi+1 for i = 1,2, , where |ai||ai+1|M for i = 1,2, , then T will be called a monotone shift. The first section of the paper examines some of the elementary properties of such operators.

Every monotone shift is hyponormal. The central portion of the paper aims at discovering which monotone shifts are subnormal. Necessary and sufficient conditions are given in terms of the {ai}. These conditions make it easy to show that even the first four coefficients (a1 < a2 < a3 < a4) may “prevent” a shift from being subnormal. However, for any a1 < a2 < a3 there does exist a monotone shift with these as its initial terms. In fact, the unique minimal one is constructed.

A complete description is given of subnormal monotone shifts for which |aj0| = |aj0+1| for some j0. The paper concludes with counter-examples constructed from the machinery developed.

Mathematical Subject Classification
Primary: 47.40
Received: 26 January 1965
Published: 1 May 1966
Joseph Gail Stampfli