Essentially Normal Operator + Compact

Operator = Strongly Irreducible Operator

Authors:
Chunlan Jiang, Shunhua Sun and Zongyao Wang

Journal:
Trans. Amer. Math. Soc. **349** (1997), 217-233

MSC (1991):
Primary \, 47A10, 47A55, 47A58

DOI:
https://doi.org/10.1090/S0002-9947-97-01754-6

MathSciNet review:
1376549

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that given an essentially normal operator with connected spectrum, there exists a compact operator such that is strongly irreducible.

**1.**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****2.**Constantin Apostol, Lawrence A. Fialkow, Domingo A. Herrero, and Dan Voiculescu,*Approximation of Hilbert space operators. Vol. II*, Research Notes in Mathematics, vol. 102, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**735080****3.**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of 𝐶*-algebras*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR**0380478****4.**John B. Conway,*Subnormal operators*, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR**634507****5.**M. J. Cowen and R. G. Douglas,*Complex geometry and operator theory*, Bull. Amer. Math. Soc.**83**(1977), no. 1, 131–133. MR**500206**, https://doi.org/10.1090/S0002-9904-1977-14215-8**6.**M. J. Cowen and R. G. Douglas,*Complex geometry and operator theory*, Acta Math.**141**(1978), no. 3-4, 187–261. MR**501368**, https://doi.org/10.1007/BF02545748**7.**Ronald G. Douglas,*Banach algebra techniques in operator theory*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR**0361893****8.**C. K. Fong and C. L. Jiang,*Approximation by Jordan type operators*, Houston J. Math.**19**(1993), no. 1, 51–62. MR**1218079****9.**C. K. Fong and Chun Lan Jiang,*On irreducible operators*, Northeast. Math. J.**8**(1992), no. 4, 385–390. MR**1210193****10.**Frank Gilfeather,*Strong reducibility of operators*, Indiana Univ. Math. J.**22**(1972/73), 393–397. MR**303322**, https://doi.org/10.1512/iumj.1972.22.22032**11.**P. R. Halmos,*Irreducible operators*, Michigan Math. J.**15**(1968), 215–223. MR**231233****12.**Domingo A. Herrero,*Approximation of Hilbert space operators. Vol. 1*, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR**1088255****13.**Domingo A. Herrero and Chun Lan Jiang,*Limits of strongly irreducible operators, and the Riesz decomposition theorem*, Michigan Math. J.**37**(1990), no. 2, 283–291. MR**1058401**, https://doi.org/10.1307/mmj/1029004135**14.**Domingo A. Herrero, Thomas J. Taylor, and Zong Y. Wang,*Variation of the point spectrum under compact perturbations*, Topics in operator theory, Oper. Theory Adv. Appl., vol. 32, Birkhäuser, Basel, 1988, pp. 113–158. MR**951959**, https://doi.org/10.1007/978-3-0348-5475-7_9**15.**Z. J. Jiang,*A lecture on operator theory*, The report in the seminar of functional analysis in Jilin Univ., Chang Chun, 1979.**16.**Z. J. Jiang and S. L. Sun,*On completely irreducible operators*, Acta Scientiarum Naturalium Universitatis Jilinensis**4**(1992), 20-29.**17.**Chun Lan Jiang,*Strongly irreducible operators and Cowen-Douglas operators*, Northeast. Math. J.**7**(1991), no. 1, 1–3. MR**1099733****18.**Chun Lan Jiang,*Similarity, reducibility and approximation of the Cowen-Douglas operators*, J. Operator Theory**32**(1994), no. 1, 77–89. MR**1332444****19.**C. L. Jiang and Z. Y. Wang,*The spectral pictures of completely irreducible operators and decomposition theorem of Hilbert space operators*, Northeastern Math. J.**10**(2) (1994), 145-148.**20.**C. L. Jiang and Z. Y. Wang,*A class of strongly irreducible operators with nice property*, J. Operator Theory (to appear).

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Additional Information

**Chunlan Jiang**

Affiliation:
Department of Mathematics, Jilin University, Changchun 130023, P.R. of China

**Shunhua Sun**

Affiliation:
Department of Mathematics, Sichuan University, Chengdu 610064, P.R. of China

**Zongyao Wang**

Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, P.R. of China

DOI:
https://doi.org/10.1090/S0002-9947-97-01754-6

Keywords:
\ Essentially normal operator,
strongly irreducible operator,
Sobolev space,
Cowen-Douglas operator,
subnormal operator,
Rosenblum operator

Received by editor(s):
May 25, 1995

Additional Notes:
The research supported by National Natural Science Foundation of China.

Article copyright:
© Copyright 1997
American Mathematical Society