Double commutants of $C_{\cdot 0}$ contractions. II
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- by Mitsuru Uchiyama PDF
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Abstract:
In [13], it was shown that if T is a $C{._0}$ contraction with finite defect indices $\infty > {\delta _{{T^\ast }}} > {\delta _T}$, then $\{ T\} '' = \{ \phi (T):\phi \in {H^\infty }\}$. In this note we shall extend this result to $\infty \geqslant {\delta _{{T^\ast }}} > {\delta _T}$ and show that $\{ T\} ''$ and ${H^\infty }$ is isometric isomorphic, and moreover such an operator is reflexive.References
- H. Bercovici, C. Foiaş, and B. Sz.-Nagy, Compléments à l’étude des opérateurs de classe $C_{O}$. III, Acta Sci. Math. (Szeged) 37 (1975), no. 3-4, 313–322 (French). MR 394251
- Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
- Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Eric A. Nordgren, On quasi-equivalence of matrices over $H^{\infty }$, Acta Sci. Math. (Szeged) 34 (1973), 301–310. MR 318929
- Béla Sz.-Nagy, Diagonalization of matrices over $H^{\infty }$, Acta Sci. Math. (Szeged) 38 (1976), no. 3-4, 223–238. MR 435907
- Béla Sz.-Nagy and Ciprian Foias, Jordan model for contractions of class $C._{O}$, Acta Sci. Math. (Szeged) 36 (1974), 305–322. MR 372651
- Béla Sz.-Nagy and Ciprian Foiaş, Commutants and bicommutants of operators of class $C_{0}$, Acta Sci. Math. (Szeged) 38 (1976), no. 3-4, 311–315. MR 435908
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
- D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. MR 192365, DOI 10.2140/pjm.1966.17.511
- Mitsuru Uchiyama, Hyperinvariant subspaces for contractions of class $C_{\cdot 0}$, Hokkaido Math. J. 6 (1977), no. 2, 260–272. MR 482300, DOI 10.14492/hokmj/1381758530
- Mitsuru Uchiyama, Double commutants of $C_{\cdot 0}$ contractions, Proc. Amer. Math. Soc. 69 (1978), no. 2, 283–288. MR 482301, DOI 10.1090/S0002-9939-1978-0482301-2
- Pei Yuan Wu, Commutants of $C_{0}(N)$ contractions, Acta Sci. Math. (Szeged) 38 (1976), no. 1-2, 193–202. MR 412862
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 74 (1979), 271-277
- MSC: Primary 47A45
- DOI: https://doi.org/10.1090/S0002-9939-1979-0524299-5
- MathSciNet review: 524299