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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.
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