Similarity of parts to the whole for certain multiplication operators
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- by Paul S. Bourdon PDF
- Proc. Amer. Math. Soc. 99 (1987), 563-567 Request permission
Abstract:
We show that the Bergman shift $B$, multiplication by $z$ on the Bergman space ${A^2}$, is similar to its part $B\left | {_N} \right .$ if and only if $N = \varphi {A^2}$, where $\varphi$ is a finite product of interpolating Blaschke products. In addition, we show that $B$ is not unitarily equivalent to any of its parts. For the analytic Toeplitz operator ${T_f}$ on ${H^2}$, we obtain that ${T_f}$ is similar to each of its parts if and only if ${T_f}$ is unitarily equivalent to each of its parts if and only if $f$ is a weak-star generator of ${H^\infty }$.References
- Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255. MR 27954, DOI 10.1007/BF02395019
- John B. Conway, On quasisimilarity for subnormal operators, Illinois J. Math. 24 (1980), no. 4, 689–702. MR 586807
- P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on $H^{p}$ spaces with $0<p<1$, J. Reine Angew. Math. 238 (1969), 32–60. MR 259579
- Charles Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), no. 1, 201–213. MR 427650
- Daniel H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11. MR 602889
- G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), no. 4, 595–611. MR 542947, DOI 10.1512/iumj.1979.28.28042
- D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. MR 192365
- Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
- Hsiao Lan Wang and Joseph G. Stampfli, Uniform operators, Trans. Amer. Math. Soc. 289 (1985), no. 1, 163–169. MR 779057, DOI 10.1090/S0002-9947-1985-0779057-0
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 563-567
- MSC: Primary 47B38; Secondary 47B35
- DOI: https://doi.org/10.1090/S0002-9939-1987-0875398-4
- MathSciNet review: 875398