Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator

Jiang, Chunlan; Sun, Shunhua; Wang, Zongyao

doi:10.1090/S0002-9947-97-01754-6

Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator
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by Chunlan Jiang, Shunhua Sun and Zongyao Wang

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

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

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Abstract:

It is shown that given an essentially normal operator $T$ with connected spectrum, there exists a compact operator $K$ such that $T+K$ is strongly irreducible.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
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
L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C^{\ast }$-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 58–128. MR 0380478
John B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR 634507
M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Bull. Amer. Math. Soc. 83 (1977), no. 1, 131–133. MR 500206, DOI 10.1090/S0002-9904-1977-14215-8
M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187–261. MR 501368, DOI 10.1007/BF02545748
Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
C. K. Fong and C. L. Jiang, Approximation by Jordan type operators, Houston J. Math. 19 (1993), no. 1, 51–62. MR 1218079
C. K. Fong and Chun Lan Jiang, On irreducible operators, Northeast. Math. J. 8 (1992), no. 4, 385–390. MR 1210193
Frank Gilfeather, Strong reducibility of operators, Indiana Univ. Math. J. 22 (1972/73), 393–397. MR 303322, DOI 10.1512/iumj.1972.22.22032
P. R. Halmos, Irreducible operators, Michigan Math. J. 15 (1968), 215–223. MR 231233, DOI 10.1307/mmj/1028999975
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
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, DOI 10.1307/mmj/1029004135
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, DOI 10.1007/978-3-0348-5475-7_{9}
Z. J. Jiang, A lecture on operator theory, The report in the seminar of functional analysis in Jilin Univ., Chang Chun, 1979.
Z. J. Jiang and S. L. Sun, On completely irreducible operators, Acta Scientiarum Naturalium Universitatis Jilinensis 4 (1992), 20–29.
Chun Lan Jiang, Strongly irreducible operators and Cowen-Douglas operators, Northeast. Math. J. 7 (1991), no. 1, 1–3. MR 1099733
Chun Lan Jiang, Similarity, reducibility and approximation of the Cowen-Douglas operators, J. Operator Theory 32 (1994), no. 1, 77–89. MR 1332444
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.
C. L. Jiang and Z. Y. Wang, A class of strongly irreducible operators with nice property, J. Operator Theory (to appear).