Roots of invertibly weighted shifts with finite defect

Author:
Gerard E. Keough

Journal:
Proc. Amer. Math. Soc. **91** (1984), 399-404

MSC:
Primary 47B37

MathSciNet review:
744638

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Abstract: Let be a unilateral invertibly weighted shift; i.e., maps a square-summable vector sequence from a Hilbert space to the sequence , where is a uniformly bounded sequence of invertible operators on . If is the identity operator on , and for , then is unitarily equivalent to multiplication by the variable on the space consisting of formal series having coefficients which satisfy . The commutant of this multiplication consists of formal series which define bounded operators on --where each is an operator on , and the action of such a series on an element of is given by the Cauchy product of the two series. Using these characterizations, it is shown that if has finite dimension , then has an th root only if divides . Examples are given of shifts with (a) , but has no square root, and (b) , has a square root, but no fourth root.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1984-0744638-1

Keywords:
Invertibly weighted shift,
commutant,
root of an operator

Article copyright:
© Copyright 1984
American Mathematical Society