The oscillation of an operator on

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

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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and discuss the oscillation of an operator *T* mapping 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 and a corresponding perturbation class which contains the compact operators.

**[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)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0405158-6

Keywords:
Diffuse operator,
operator,
compact operator,
concentrated operator,
semi-Fredholm operator,
Fredholm operator,
perturbation theory

Article copyright:
© Copyright 1975
American Mathematical Society