The spectral and Fredholm theory of extensions of bounded linear operators

Author:
Bruce A. Barnes

Journal:
Proc. Amer. Math. Soc. **105** (1989), 941-949

MSC:
Primary 47A20; Secondary 47A10, 47A53, 47B38

MathSciNet review:
955454

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Abstract: Assume is a bounded linear operator on some Banach space , and that has a bounded extension on another space. In general almost nothing can be said concerning the relationship between the spectral and Fredholm properties of and . However, assuming the special condition that the range of lies in , it is shown that these properties are essentially the same for and .

**[1]**B. Barnes, G. Murphy, S. Smyth, and T. West,*Riesz and Fredholm theory in Banach algebras*, Pitman, Boston, 1982.**[2]**Bruce A. Barnes,*Interpolation of spectrum of bounded operators on Lebesgue spaces*, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 359–378. MR**1065835**, 10.1216/rmjm/1181073112**[3]**-,*Operators symmetric with respect to a pre-innerproduct*(preprint).**[4]**D. W. Boyd,*The spectrum of the Cesàro operator*, Acta Sci. Math. (Szeged)**29**(1968), 31–34. MR**0239441****[5]**S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood,*Calkin algebras and algebras of operators on Banach spaces*, Marcel Dekker, Inc., New York, 1974. Lecture Notes in Pure and Applied Mathematics, Vol. 9. MR**0415345****[6]**N. Dunford and J. Schwartz,*Linear operators, Part I*, Interscience, New York, 1964.**[7]**Sandy Grabiner,*Spectral consequences of the existence of intertwining operators*, Comment. Math. Prace Mat.**22**(1980/81), no. 2, 227–238. MR**641436****[8]**E. Hewitt and K. Ross,*Abstract harmonic analysis I*, Springer-Verlag, Berlin, 1963.**[9]**Vasile I. Istrăţescu,*Introduction to linear operator theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 65, Marcel Dekker, Inc., New York, 1981. MR**608969****[10]**K. Jörgens,*Linear integral operators*, Pitman, Boston, 1982.**[11]**Peter D. Lax,*Symmetrizable linear transformations*, Comm. Pure Appl. Math.**7**(1954), 633–647. MR**0068116****[12]**Joseph I. Nieto,*On the essential spectrum of symmetrizable operators*, Math. Ann.**178**(1968), 145–153. MR**0233221****[13]**Edgar Lee Stout,*The theory of uniform algebras*, Bogden & Quigley, Inc., Tarrytown-on-Hudson, N. Y., 1971. MR**0423083**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0955454-4

Keywords:
Extension,
spectral theory,
Fredholm theory

Article copyright:
© Copyright 1989
American Mathematical Society