Upper and lower Fredholm spectra. I

Buoni, John J.; Harte, Robin; Wickstead, Tony

doi:10.1090/S0002-9939-1977-0454676-0

Upper and lower Fredholm spectra. I
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by John J. Buoni, Robin Harte and Tony Wickstead PDF

Proc. Amer. Math. Soc. 66 (1977), 309-314 Request permission

Abstract:

Joint upper and lower Fredholm spectra are defined for n-tuples of bounded linear operators, and the upper Fredholm spectrum is represented both as the simultaneous eigenvalues and as the simultaneous approximate eigenvalues of an n-tuple of operators obtained by a Berberian-Quigley construction.

References

Sterling K. Berberian, Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962), 111–114. MR 133690, DOI 10.1090/S0002-9939-1962-0133690-8
S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 9, Marcel Dekker, Inc., New York, 1974. MR 0415345
M. D. Choi and Chandler Davis, The spectral mapping theorem for joint approximate point spectrum, Bull. Amer. Math. Soc. 80 (1974), 317–321. MR 333780, DOI 10.1090/S0002-9904-1974-13481-6
A. T. Dash, Joint spectrum in the Calkin algebra, Bull. Amer. Math. Soc. 81 (1975), no. 6, 1083–1085. MR 390800, DOI 10.1090/S0002-9904-1975-13925-5
Chandler Davis and Peter Rosenthal, Solving linear operator equations, Canadian J. Math. 26 (1974), 1384–1389. MR 355649, DOI 10.4153/CJM-1974-132-6
P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179–192. MR 322534
Bernhard Gramsch and David Lay, Spectral mapping theorems for essential spectra, Math. Ann. 192 (1971), 17–32. MR 291846, DOI 10.1007/BF02052728
R. E. Harte, Spectral mapping theorems, Proc. Roy. Irish Acad. Sect. A 72 (1972), 89–107. MR 326394
Arnold Lebow and Martin Schechter, Semigroups of operators and measures of noncompactness, J. Functional Analysis 7 (1971), 1–26. MR 0273422, DOI 10.1016/0022-1236(71)90041-3
Heinrich P. Lotz, Über das Spektrum positiver Operatoren, Math. Z. 108 (1968), 15–32 (German). MR 240648, DOI 10.1007/BF01110453
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Z. Słodkowski and W. Żelazko, On joint spectra of commuting families of operators, Studia Math. 50 (1974), 127–148. MR 346555, DOI 10.4064/sm-50-2-127-148
Vernon Williams, Closed Fredholm and semi-Fredholm operators, essential spectra and perturbations, J. Functional Analysis 20 (1975), no. 1, 1–25. MR 0394276, DOI 10.1016/0022-1236(75)90050-6