The oscillation of an operator
HTML articles powered by AMS MathViewer
- by Robert Whitley PDF
- Trans. Amer. Math. Soc. 165 (1972), 65-73 Request permission
Erratum: Trans. Amer. Math. Soc. 172 (1972), 507.
Abstract:
Foiaş and Singer introduced the oscillation of a bounded linear operator mapping $C(S)$ into a Banach space. Using this concept we define a generalization of the Fredholm operators T with $\mathcal {K}(T) < \infty$ and a corresponding perturbation class which contains the weakly compact operators. We show that a bounded linear operator on c is a conservative summability matrix which sums every bounded sequence if and only if it has zero oscillation at infinity.References
- S. Banach, Théorie des opérations linéaires, Chelsea, New York, 1955. MR 17, 175.
- E. W. Cheney and K. H. Price, Minimal projections, Approximation Theory (Proc. Sympos., Lancaster, 1969) Academic Press, London, 1970, pp. 261–289. MR 0265842
- I. K. Daugavet, A property of completely continuous operators in the space $C$, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 157–158 (Russian). MR 0157225 M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 21, Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Ciprian Foiaş and Ivan Singer, Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions, Math. Z. 87 (1965), 434–450. MR 180863, DOI 10.1007/BF01111723
- Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692
- Richard H. Herman, Generalizations of weakly compact operators, Trans. Amer. Math. Soc. 132 (1968), 377–386. MR 223929, DOI 10.1090/S0002-9947-1968-0223929-2
- M. A. Krasnosel′skiĭ, A class of linear operators in a space of abstract continuous functions, Mat. Zametki 2 (1967), 599–604 (Russian). MR 222683
- Elton Lacey and R. J. Whitley, Conditions under which all the bounded linear maps are compact, Math. Ann. 158 (1965), 1–5. MR 173159, DOI 10.1007/BF01370391
- A. Pełczyński, On simultaneous extension of continuous functions. A generalization of theorems of Rudin-Carleson and Bishop, Studia Math. 24 (1964), 285–304. MR 174996, DOI 10.4064/sm-24-3-285-304
- A. Pełczyński, Some linear topological properties of separable function algebras, Proc. Amer. Math. Soc. 18 (1967), 652–660. MR 213883, DOI 10.1090/S0002-9939-1967-0213883-6
- A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in $C(S)$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 31–36. MR 177300
- Edward O. Thorp and Robert J. Whitley, Operator representation theorems, Illinois J. Math. 9 (1965), 595–601. MR 181900
- Albert Wilansky, Functional analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0170186
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 165 (1972), 65-73
- MSC: Primary 47A99; Secondary 40J05
- DOI: https://doi.org/10.1090/S0002-9947-1972-0295105-X
- MathSciNet review: 0295105