Criteria for the property (UWE) and the a-Weyl theorem. (English. Russian original) Zbl 1522.47013
Funct. Anal. Appl. 56, No. 3, 216-224 (2022); translation from Funkts. Anal. Prilozh. 56, No. 3, 76-88 (2022).
Summary: In this paper, the property (UWE) and the a-Weyl theorem for bounded linear operators are studied in terms of the property of topological uniform descent. Sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space to have the property (UWE) and satisfy the a-Weyl theorem are established. In addition, new criteria for the fulfillment of the property (UWE) and the a-Weyl theorem for an operator function are discussed. As a consequence of the main theorem, results on the stability of the property (UWE) and the a-Weyl theorem are obtained.
MSC:
| 47A10 | Spectrum, resolvent |
| 47A11 | Local spectral properties of linear operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 47B02 | Operators on Hilbert spaces (general) |
References:
| [1] | Weyl, H. V., Über beschränkte quadratische formen, deren differenz vollstetig ist, Rend. Circ. Mat. Palermo, 27, 1, 373-392 (1909) · JFM 40.0395.01 · doi:10.1007/BF03019655 |
| [2] | Harte, R.; Lee, W. Y., Another note on Weyl’s theorem, Trans. Amer. Math. Soc., 349, 5, 2115-2124 (1997) · Zbl 0873.47001 · doi:10.1090/S0002-9947-97-01881-3 |
| [3] | Rakočevi\‘c, V., Operators obeying a-Weyl’s theorem, Rev. Roum. Math. Pures Appl., 34, 10, 915-919 (1989) · Zbl 0686.47005 |
| [4] | Rakočevi\‘c, V., On a class of operators, Mat. Vesnik, 37, 4, 423-426 (1985) · Zbl 0596.47001 |
| [5] | Li, C. G.; Zhu, S.; Feng, Y. L., Weyl’s theorem for functions of operators and approximation, Integral Equations Oper. Theory, 67, 4, 481-497 (2010) · Zbl 1229.47007 · doi:10.1007/s00020-010-1796-5 |
| [6] | An, I.; Han, Y., Weyl’s theorem for algebraically quasi-class A operators, Integral Equations Oper. Theory, 62, 1, 1-10 (2008) · Zbl 1201.47021 · doi:10.1007/s00020-008-1599-0 |
| [7] | Cao, X. H.; Guo, M. Z.; Meng, B., Weyl Spectra and Weyl’s theorem, J. Math. Anal. Appl., 288, 758-767 (2003) · Zbl 1061.47004 · doi:10.1016/j.jmaa.2003.09.026 |
| [8] | Goldberg, S., Unbounded Linear Operators (1966), New York: McGraw-Hill, New York · Zbl 0148.12501 |
| [9] | Berkani, M.; Kachad, M., New Browder and Weyl type theorems, Bull. Korean Math. Soc., 52, 2, 439-452 (2015) · Zbl 1325.47028 · doi:10.4134/BKMS.2015.52.2.439 |
| [10] | Grabiner, S., Uniform ascent and descent of bounded operators, J. Math. Soc. Japan, 34, 2, 317-337 (1982) · Zbl 0477.47013 · doi:10.2969/jmsj/03420317 |
| [11] | Radjavi, H.; Rosenthal, P., Invariant Subspaces (2003), Mineola: Dover Publications, Mineola · Zbl 1043.47001 |
| [12] | Zhu, S.; Li, C. G.; Zhou, T. T., Weyl type theorems for functions of operators, Glasg. Math. J., 54, 3, 493-505 (2012) · Zbl 1256.47014 · doi:10.1017/S0017089512000092 |
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