Abstract
An operator on a complex, separable, infinite dimensional Hilbert space is strongly irreducible if it does not commute with any nontrivial idempotent. This article answers the following questions of D. A. Herrero: (i) Given an operatorT with connected spectrum, can we find a strongly irreducible operatorL such that they have same spectral picture? (ii) When we use a sequence of irreducible operators to approximateT, can the approximation be the “most economic”? i.e., does there exist a strongly irreducible operatorL such thatT ∈S(L) − (the closure of the similarity orbit ofL)? It is shown that the answer for the two questions is yes.
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References
R. A. Adams,Sobolev spaces, Academic Press, New York-San Francisco-London, 1975.
J. Agler, E. Franks and D. A. Herrero. Spectral pictures of operators quasisimilar to the unilateral shift, J. Reine und Angewandte Mathematik,422(1991), 1–20.
F. A. Al-Musallam, An upper estimate for the distance to the essentiallyG 1 operators, Ph. D. dissertation, Arizona State University (1989).
J. H. Anderson, On normal derivations, Proc. Amer. Math. Soc.38 (1973), 135–140
C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, Approximation of Hilbert space operators. II, Research Notes in Math.,102, (London-Boston-Melbourne: Pitman Books, Ltd., 1984).
L. G. Brown, P. G. Douglas and P. F. Fillmore, Unitary equivalence modulo the compact operators and extensions of C*-algebras, Proceedings of a conference on operator theory, Lect. Notes in Math., Vol. 345, Springer-Verlag, (1973) 58–128.
C. Bosch, C. Hernandez, E. de Oteyza and C. M. Pearcy, Spectral pictures of functions of operators, J. Operator Theory8 (1982), 391–400.
J. B. Conway, Subnormal operators, Research Notes in Math., Vol. 51. London-Boston-Melbourne: Pitman Books Ltd., 1981).
M. Cowen and R. G. Douglas, Complex geometry and operator theory, Bull. Amer. Math. Soc.83 (1977), 131–133.
, Complex geometry and operator theory, Acta Math.141 (1978), 187–261.
K. R. Davidson and D. A. Herrero, The Jordan form of a bitriangular operator, J. Funct. Anal.94 (1990), 27–73.
R. G. Douglas, Banach algebras techniques in operator theory, (New York and London: Academic Press, 1972).
C. K. Fong and C. L. Jiang, Approximation by Jordan type operators, Houston J. Math.,19 (1993) 51–62.
, On irreducible operators, Northeastern Math., J.8(4) (1992), 385–390.
F. Gilfeather, Strongly reducibility of operators, Indiana Univ. Math. J.22 (1972), 393–397.
P. R. Halmos,A Hilbert space problem book. (Princeton, New Jersey: D. Van Nostrand, 1967).
D. A. Herrero, Approximation of Hilbert space operators, Vol. 1 2nd ed., Pitman Res. Notes Math. 224, Longman Sci. Tech., Harlow, Essex, 1990.
, A unicellular universal quasinilpotent weighted shift, Proc. Amer. Math. Soc.110 (1990), 649–652.
D. A. Herrero and C. L. Jiang, Limits of Strongly irreducible operators, and the Riesz decomposition Theorem, Michigan Math. J.37 (1990), 283–291.
D. A. Herrero, T. Taylor and Z. Y. Wang, Variation of the point spectrum under compact perturbations, Topics in operator Theory, Constantin Apostol Memorial Issue, OT: Advances and Applications, Vol. 32. (Basel-Boston-Stuttgart: Birkhauser-Verlag, 1988), 113–158.
Z. J. Jiang, A lecture on operator theory, the report in the seminar of functional analyses of Jilin University, Changchun, 1979.
Z. J. Jiang and S. L. Sun, On completely irreducible operators, Acta Scientiarum Naturalium Universitatis Jilinensis4 (1992) 20–29.
C. L. Jiang, Strongly irreducible operators and Cowen-Douglas Operators, Northeastern Math. J.7(1) (1991), 1–3.
C. L. Jiang, Similarity, reducibility and approximation of Cowen-Douglas operators. (to appear).
C. L. Jiang and Z. Y. Wang, A class of strongly irreducible operators with nice property. (to appear).
L. Marcoux, The closure of the (U+K)-orbit of shift-like operators, Indiana University Math. J.41 (4) (1992).
I. D. Berg, K. R. Davidson, Almost commuting matrices and a quantitative version of Brown-Douglas-Fillmore theorem, Acta Mathematica,166(1991), 121–161.
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This project was partially supported by National Natural Science Foundation of China.
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Jiang, CI., Wang, Zy. The spectral picture and the closure of the similarity orbit of strongly irreducible operators. Integr equ oper theory 24, 81–105 (1996). https://doi.org/10.1007/BF01195486
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DOI: https://doi.org/10.1007/BF01195486