Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The oscillation of an operator on $ L\sp{p}$


Authors: George R. Barnes and Robert Whitley
Journal: Trans. Amer. Math. Soc. 211 (1975), 339-351
MSC: Primary 47B37
DOI: https://doi.org/10.1090/S0002-9947-1975-0405158-6
MathSciNet review: 0405158
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and discuss the oscillation of an operator T mapping $ {L^p}(S,\Sigma ,\mu )$ into a Banach space. We establish results relating the oscillation, a ``local norm", to the norm of the operator. Also using the oscillation we define a generalization of the Fredholm operators T with index $ \kappa (T) < \infty $ and a corresponding perturbation class which contains the compact operators.


References [Enhancements On Off] (What's this?)

  • [1] D. W. Boyd, The spectrum of the Cesàro operator, Acta Sci. Math. (Szeged) 29 (1968), 31-34. MR 39 #798. MR 0239441 (39:798)
  • [2] E. W. Cheney and K. H. Price, Minimal projections, Approximation Theory (Proc. Sympos., Lancaster, 1969), Academic Press, London, 1970, pp. 261-289. MR 42 #751. MR 0265842 (42:751)
  • [3] C. Foiaş and I. Singer, Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions, Math. Z. 87 (1965), 434-450. MR 31 #5093. MR 0180863 (31:5093)
  • [4] S. Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill, New York, 1966. MR 34 #580. MR 0200692 (34:580)
  • [5] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, New York, 1967.
  • [6] M. G. Krein, M. A. Krasnosel'skiĭ and D. P. Milman, On the defect numbers of linear operators in a Banach space and on some geometrical questions, Sb. Tr. Inst. Mat. Akad. Nauk Ukr. SSR, No. 11, 1948, pp. 97-112. (Russian)
  • [7] G. M. Leibowitz, The Cesàro operators and their generalizations: examples in infinite-dimensional linear analysis, Amer. Math. Monthly 80 (1973), 654-661. MR 48 #931. MR 0322569 (48:931)
  • [8] Danuta Przeworska-Rolewicz and Stefan Rolewicz, Equations in linear spaces, PWN, Warsaw, 1968.
  • [9] B. E. Rhoades, Spectra of some Hausdorff operators, Acta Sci. Math (Szeged) 32 (1971), 91-100. MR 46 #4238. MR 0305108 (46:4238)
  • [10] R. Whitley, Oscillation of an operator, Trans. Amer. Math. Soc. 165 (1972), 65-73. MR 45 #4173. MR 0295105 (45:4173)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37

Retrieve articles in all journals with MSC: 47B37


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0405158-6
Keywords: Diffuse operator, $ {c_0}$ operator, compact operator, concentrated operator, semi-Fredholm operator, Fredholm operator, perturbation theory
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society