Roots of invertibly weighted shifts with finite defect

Author:
Gerard E. Keough

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

MSC:
Primary 47B37

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

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.

**[1]**John B. Conway and Pei Yuan Wu,*The structure of quasinormal operators and the double commutant property*, Trans. Amer. Math. Soc.**270**(1982), 641-657. MR**645335 (83c:47042)****[2]**James A. Deddens,*Another description of nest algebras*, Hilbert Space Operators, Lecture Notes in Math., vol. 693, Springer-Verlag, Berlin and New York, pp. 77-85. MR**526534 (80f:47033)****[3]**James A. Deddens and Tin Kin Wong,*The commutant of analytic Toeplitz operators*, Trans. Amer. Math. Soc.**184**(1973), 261-273. MR**0324467 (48:2819)****[4]**P. R. Halmos,*A Hilbert space problem book*, Van Nostrand, Princeton, N. J., 1967. MR**0208368 (34:8178)****[5]**Alan Lambert,*Unitary equivalence and reducibility of invertibly weighted shifts*, Bull. Austral. Math. Soc.**5**(1971), 157-173. MR**0295128 (45:4196)****[6]**A. L. Lambert and T. R. Turner,*The double commutant of invertibly weighted shifts*, Duke Math. J.**39**(1972), 385-389. MR**0310683 (46:9781)****[7]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Math. Surveys, no. 13 (C. Pearcy, ed.), Amer. Math. Soc., Providence, R. I., 1974. MR**0361899 (50:14341)**

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Additional Information

DOI:
https://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