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Not every $ d$-symmetric operator is GCR


Author: C. Ray Rosentrater
Journal: Proc. Amer. Math. Soc. 81 (1981), 443-446
MSC: Primary 47B47; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1981-0597659-3
MathSciNet review: 597659
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Abstract: Let $ T$ be an element of $ \mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on the Hilbert space $ \mathcal{H}$. The derivation induced by $ T$ is the map $ {\delta _T}(X) = TX - XT$ from $ \mathcal{B}(\mathcal{H})$ into itself. $ T$ is $ d$-symmetric if the norm closure of the range of $ {\delta _T}$, $ \mathcal{R}{({\delta _T})^\_}$, is closed under taking adjoints. This paper answers the question of whether every $ d$-symmetric operator is GCR by giving an example of an NGCR weighted shift that is also $ d$-symmetric.


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  • [An] J. Anderson, On normal derivations, Proc. Amer. Math. Soc. 38 (1973), 135-140. MR 0312313 (47:875)
  • [ABDW] J. Anderson, J. Bunce, J. Deddens and J. P. Williams, $ {C^*}$-algebras and derivation ranges, Acta Sci. Math. 40 (1978), 211-227. MR 515202 (80i:47047)
  • [A] W. Arveson, An invitation to $ {C^*}$-algebras, Springer-Verlag, New York, Heidelberg and Berlin, 1976. MR 0512360 (58:23621)
  • [BD] J. Bunce and J. Deddens, $ {C^*}$-algebras generated by weighted shifts, Indiana Univ. Math. J. 23 (1973), 257-271. MR 0341108 (49:5858)
  • [JW] B. E. Johnson and J. P. Williams, The range of a normal derivation, Pacific J. Math. 58 (1975), 105-122. MR 0380490 (52:1390)
  • [O] D. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1-25. MR 0385632 (52:6492)
  • [S] A. 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 (50:14341)
  • [St] J. G. Stampfli, Derivations on $ \mathcal{B}(\mathcal{H})$: The range, Illinois J. Math. 17 (1973), 518-524. MR 0318914 (47:7460)
  • [W1] J. P. Williams, On the range of a derivation, Pacific J. Math. 38 (1971), 273-279. MR 0308809 (46:7923)
  • [W2] -, On the range of a derivation. II, Proc. Roy. Irish Acad. Sect. A 74 (1974), 299-310. MR 0358370 (50:10836)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0597659-3
Article copyright: © Copyright 1981 American Mathematical Society

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