Previous |  Up |  Next


MSC: 46H05, 47A10, 47A53
Full entry | Fulltext not available (moving wall 24 months)      Feedback
trace; index; Fredholm element
We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index.
[1] Aupetit, B.: A Primer on Spectral Theory. Universitext Springer, New York (1991). MR 1083349 | Zbl 0715.46023
[2] Aupetit, B., Mouton, H. du T.: Trace and determinant in Banach algebras. Stud. Math. 121 (1996), 115-136. MR 1418394 | Zbl 0872.46028
[3] Bonsall, F. F., Duncan, J.: Complete Normed Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete 80 Springer, New York (1973). MR 0423029 | Zbl 0271.46039
[4] Grobler, J. J., Raubenheimer, H.: The index for Fredholm elements in a Banach algebra via a trace. Stud. Math. 187 (2008), 281-297. DOI 10.4064/sm187-3-5 | MR 2417458
[5] Kordula, V., Müller, V.: On the axiomatic theory of spectrum. Stud. Math. 119 (1996), 109-128. MR 1391471
[6] Kraljević, H., Suljagić, S., Veselić, K.: Index in semisimple Banach algebras. Glas. Mat., III. Ser. 17 (1982), 73-95. MR 0682475
[7] Mbekhta, M., Müller, V.: On the axiomatic theory of spectrum. II. Stud. Math. 119 (1996), 129-147. MR 1391472
[8] Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications 139 Birkhäuser, Basel (2007). MR 2355630
[9] Müller, V.: Axiomatic theory of spectrum. III: Semiregularities. Stud. Math. 142 (2000), 159-169. MR 1792602
[10] Puhl, J.: The trace of finite and nuclear elements in Banach algebras. Czech. Math. J. 28 (1978), 656-676. MR 0506439 | Zbl 0394.46041
Partner of
EuDML logo