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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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), no. 2, 641–657. MR**645335**, https://doi.org/10.1090/S0002-9947-1982-0645335-6**[2]**James A. Deddens,*Another description of nest algebras*, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 77–86. MR**526534****[3]**James A. Deddens and Tin Kin Wong,*The commutant of analytic Toeplitz operators*, Trans. Amer. Math. Soc.**184**(1973), 261–273. MR**0324467**, https://doi.org/10.1090/S0002-9947-1973-0324467-0**[4]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[5]**Alan Lambert,*Unitary equivalence and reducibility of invertibly weighted shifts*, Bull. Austral. Math. Soc.**5**(1971), 157–173. MR**0295128**, https://doi.org/10.1017/S000497270004702X**[6]**A. L. Lambert and T. R. Turner,*The double commutant of invertibly weighted shifts*, Duke Math. J.**39**(1972), 385–389. MR**0310683****[7]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR**0361899**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47B37

Retrieve articles in all journals with MSC: 47B37

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