Acta mathematica scientia, Series B >
ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS
Received date: 2007-09-06
Revised date: 2008-08-11
Online published: 2010-07-20
Supported by
This research was in part supported by a grant (No. 86-GR-SC-27) from Shiraz University Research Council
By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r(A) such that its spectrum is the closed disc {z:|z| ≤ r(A)} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.
M. Faghih Ahmadi , K. Hedayatian . ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS[J]. Acta mathematica scientia, Series B, 2010 , 30(4) : 1100 -1104 . DOI: 10.1016/S0252-9602(10)60107-5
[1] Bercovici H, Foias C, Pearcy C. Dual algebras with applications to invariant subspaces and dilation theory. Amer Math Soc, 1985
[2] Conway John B. A course in operator theory. Amer Math Soc, Vol. 21, 2000
[3] Conway John B, Dudziak J J. Von Neumann operators are reflexive. J Reine Angew Math, 1990, 408: 34--56
[4] Hadwin D, Nordgren E A. Reflexivity and direct sums. Acta Sci Math, 1991, 55: 181--197
[5] Halmos P R. A Hilbert space problem book. 2nd ed. New-York and Berlin: Springer-Verlag, 1982
[6] Hedayatian K. A non-quasinilpotent operator which commutes with a quasinilpotent shift. Int J Math, 2006, 17(5): 633--639
[7] Herrero D A, Lambert A. On strictly cyclic algebras, P-algebras and reflexive operators. Trans Amer Math Soc, 1973, 185: 229--235
[8] Radjavi H and Rosenthal P. Invariant subspaces. New York, Heidelberg and Berlin: Springer-Verlag, 1973
[9] Shields A. Weighted shift operators and analytic functions theory. Topics in Operator Theory, Math Surveys
Monographs, Vol 13. Amer Math Soc, Providence, RI, 1974: 49--128
[10] Stessin M, Zhu K. Reducing subspaces of weighted shift operators. Proc Amer Math Soc, 2002, 130: 2631--2639
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