Czechoslovak Mathematical Journal, first online, pp. 1-19


Generalized spectral perturbation and the boundary spectrum

Sonja Mouton

Received February 4, 2020.   Published online February 2, 2021.

Abstract:  By considering arbitrary mappings $\omega$ from a Banach algebra $A$ into the set of all nonempty, compact subsets of the complex plane such that for all $a \in A$, the set $\omega(a)$ lies between the boundary and connected hull of the exponential spectrum of $a$, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.
Keywords:  exponential spectrum; singular spectrum; boundary spectrum; boundary and hull; (strong) Riesz property; Mobius spectrum
Classification MSC:  46H10, 47A10
DOI:  10.21136/CMJ.2021.0046-20

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] B. Aupetit: Propriétés spectrales des algèbres de Banach. Lecture Notes in Mathematics 735. Springer, Berlin (1979). (In French.) DOI 10.1007/BFb0064204 | MR 0549769 | Zbl 0409.46054
[2] B. Aupetit: A Primer on Spectral Theory. Universitext. Springer, New York (1991). DOI 10.1007/978-1-4612-3048-9 | MR 1083349 | Zbl 0715.46023
[3] L. Burlando: Comparisons between different spectra of an element in a Banach algebra. Int. J. Math. Sci. 16 (1993), 819-822. DOI 10.1155/S0161171293001036 | MR 1234831 | Zbl 0808.46065
[4] J. B. Conway: A Course in Functional Analysis. Graduate Texts in Mathematics 96. Springer, New York (2010). DOI 10.1007/978-1-4757-4383-8 | MR 1070713 | Zbl 0706.46003
[5] L. Groenewald, R. E. Harte, H. Raubenheimer: Perturbation by inessential and Riesz elements. Quaest. Math. 12 (1989), 439-446. DOI 10.1080/16073606.1989.9632195 | MR 1021942 | Zbl 0705.46022
[6] L. Groenewald, H. Raubenheimer: A note on the singular and exponential spectrum in Banach algebras. Quaest. Math. 11 (1988), 399-408. DOI 10.1080/16073606.1988.9632154 | MR 0969558 | Zbl 0673.46022
[7] R. E. Harte: The exponential spectrum in Banach algebras. Proc. Am. Math. Soc. 58 (1976), 114-118. DOI 10.1090/S0002-9939-1976-0407603-5 | MR 0407603 | Zbl 0338.46043
[8] R. E. Harte: Invertibility and Singularity for Bounded Linear Operators. Pure and Applied Mathematics 109. Marcel Dekker, New York (1988). MR 0920812 | Zbl 0636.47001
[9] R. E. Harte, A. W. Wickstead: Boundaries, hulls and spectral mapping theorems. Proc. R. Ir. Acad., Sect. A 81 (1981), 201-208. MR 0654819 | Zbl 0489.46041
[10] L. Lindeboom, H. Raubenheimer: A note on the singular spectrum. Extr. Math. 13 (1998), 349-357. MR 1695568 | Zbl 1054.46510
[11] L. Lindeboom, H. Raubenheimer: Different exponential spectra in Banach algebras. Rocky Mt. J. Math. 29 (1999), 957-970. DOI 10.1216/rmjm/1181071617 | MR 1733077 | Zbl 0969.46038
[12] L. Lindeboom, H. Raubenheimer: On regularities and Fredholm theory. Czech. Math. J. 52 (2002), 565-574. DOI 10.1023/A:1021727829750 | MR 1923262 | Zbl 1010.46045
[13] H. du T. Mouton: On inessential ideals in Banach algebras. Quaest. Math. 17 (1994), 59-66. DOI 10.1080/16073606.1994.9632217 | MR 1276008 | Zbl 0818.46053
[14] H. du T. Mouton, S. Mouton, H. Raubenheimer: Ruston elements and Fredholm theory relative to arbitrary homomorphisms. Quaest. Math. 34 (2011), 341-359. DOI 10.2989/16073606.2011.622893 | MR 2844530 | Zbl 1274.46095
[15] H. du T. Mouton, H. Raubenheimer: Fredholm theory relative to two Banach algebra homomorphisms. Quaest. Math. 14 (1991), 371-382. DOI 10.1080/16073606.1991.9631656 | MR 1143042 | Zbl 0763.46035
[16] S. Mouton: On the boundary spectrum in Banach algebras. Bull. Aust. Math. Soc. 74 (2006), 239-246. DOI 10.1017/S0004972700035681 | MR 2260492 | Zbl 1113.46044
[17] S. Mouton: Mapping and continuity properties of the boundary spectrum in Banach algebras. Ill. J. Math. 53 (2009), 757-767. DOI 10.1215/ijm/1286212914 | MR 2727353 | Zbl 1210.46033
[18] V. Müller: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications 139. Birkhäuser, Basel (2007). DOI 10.1007/978-3-0348-7788-6 | MR 2355630 | Zbl 1208.47001
[19] H. Raubenheimer, A. Swartz: Radius preserving (semi)regularities in Banach algebras. Quaest. Math. 42 (2019), 811-822. DOI 10.2989/16073606.2018.1501439 | MR 3989360 | Zbl 1439.46039
[20] H. Raubenheimer, A. Swartz: Regularity-type properties of the boundary spectrum in Banach algebras. Rocky Mt. J. Math. 49 (2019), 2747-2754. DOI 10.1216/RMJ-2019-49-8-2747 | MR 4058347 | Zbl 07163196
[21] A. E. Taylor, D. C. Lay: Introduction to Functional Analysis. John Wiley & Sons, New York (1980). MR 0564653 | Zbl 0501.46003
[22] S. Č. Živkovič-Zlatanovič, R. E. Harte: Polynomially Riesz elements. Quaest. Math. 38 (2015), 573-586. DOI 10.2989/16073606.2015.1026558 | MR 3403668 | Zbl 06696038

Affiliations:   Sonja Mouton, Department of Mathematical Sciences, Stellenbosch University, P/Bag X1, Matieland 7602, Stellenbosch, South Africa, e-mail: smo@sun.ac.za


 
PDF available at: