Representation theorems for compact operators

Author:
Daniel J. Randtke

Journal:
Proc. Amer. Math. Soc. **37** (1973), 481-485

MSC:
Primary 47B05; Secondary 46B15

DOI:
https://doi.org/10.1090/S0002-9939-1973-0310685-X

MathSciNet review:
0310685

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Abstract: It is shown that (the Banach space of zero-convergent sequences) is the only Banach space with basis that satisfies the following property: For every compact operator from into a Banach space *E*, there is a sequence in and an unconditionally summable sequence in *E* such that for each in . This result is then used to show that a linear operator from a locally convex space *E* into a Fréchet space *F* has a representation of the form , where is a sequence in is an equicontinuous sequence in the topological dual of *E* and is an unconditionally summable sequence in *F*, if and only if *T* can be ``compactly factored'' through .

**[1]**C. Bessaga and A. Pełczyński,*On bases and unconditional convergence of series in Banach spaces*, Studia Math.**17**(1958), 151-164. MR**22**#5872. MR**0115069 (22:5872)****[2]**J. B. Conway,*The strict topology and compactness in the space of measures*. II, Trans. Amer. Math. Soc.**126**(1967), 474-486. MR**34**#6503. MR**0206685 (34:6503)****[3]**M. M. Day,*Normed linear spaces*, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Heft 21, Academic Press, New York; Springer-Verlag, Berlin, 1962. MR**26**#2847.**[4]**S. Goldberg,*Unbounded linear operators*:*Theory and applications*, McGraw-Hill, New York, 1966. MR**34**#580. MR**0200692 (34:580)****[5]**A. Pietsch,*Nukleare lokalkonvexe Räume*, 2nd ed., Akademie-Verlag, Berlin, 1969. MR**0181888 (31:6114)****[6]**D. Randtke,*Characterizations of precompact maps, Schwartz spaces and nuclear spaces*, Trans. Amer. Math. Soc.**165**(1972), 87-101. MR**0305009 (46:4139)****[7]**I. Singer,*Bases in Banach spaces*. I, Springer-Verlag, New York, 1970. MR**0298399 (45:7451)****[8]**T. Terzioglu,*A characterization of compact linear mappings*, Arch. Math.**22**(1971), 76-78. MR**0291865 (45:954)****[9]**-,*On compact and infinite-nuclear mappings*, Bull. Math. Soc. Sci. Math. R.S. Roumanie**14**(**62**) (1970), 93-99. MR**0336414 (49:1189)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0310685-X

Keywords:
Normed space,
Banach space,
Fréchet space,
compact linear operator,
precompact linear operator,
unconditionally summable sequence,
weakly unconditionally summable sequence,
equicontinuous sequence,
normalized basis,
associated sequence of coefficient forms

Article copyright:
© Copyright 1973
American Mathematical Society