On weakly compact operators on Banach lattices

Authors:
C. D. Aliprantis and O. Burkinshaw

Journal:
Proc. Amer. Math. Soc. **83** (1981), 573-578

MSC:
Primary 47B55; Secondary 46B30

DOI:
https://doi.org/10.1090/S0002-9939-1981-0627695-X

MathSciNet review:
627695

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a Banach lattice and an order bounded weakly compact operator . The purpose of this note is to study the weak compactness of operators that are related with in some order sense. The main results are the following.

(1) If is a positive weakly compact operator and an operator satisfies , then is weakly compact. (Examples show that need not be weakly compact.)

(2) If and are as in (1) and either is an orthomorphism or has an order continuous norm, then is weakly compact.

(3) If is an abstract -space and is weakly compact, then the modulus is weakly compact.

**[1]**C. D. Aliprantis and O. Burkinshaw,*Locally solid Riesz spaces*, Academic Press, New York, 1978. MR**0493242 (58:12271)****[2]**O. Burkinshaw,*Weak compactness in the order dual of a vector lattice*, Trans. Amer. Math. Soc.**187**(1974), 183-201. MR**0394098 (52:14904)****[3]**N. Dunford and J. T. Schwartz,*Linear operators*. I, Wiley (Interscience), New York, 1958.**[4]**W. A. J. Luxemburg and A. C. Zaanen,*Riesz spaces*. I, North-Holland, Amsterdam, 1971.**[5]**H. H. Schaefer,*Banach lattices and positive operators*, Springer-Verlag, Berlin and New York and Heidelberg, 1974. MR**0423039 (54:11023)****[6]**A. R. Schep,*Compactness properties of an operator which imply that is is an integral operator*, Trans. Amer. Math. Soc.**265**(1981), 111-119. MR**607110 (82i:47088)****[7]**A. W. Wickstead,*Representation and duality of multiplication operators on Archimedean Riesz spaces*, Compositio Math.**35**(1977), 225-238. MR**0454728 (56:12976)****[8]**-,*Extremal structure of cones of operators*, Quart. J. Math. Oxford Ser. (2)**32**(1981), 239-253. MR**615198 (82i:47069)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47B55,
46B30

Retrieve articles in all journals with MSC: 47B55, 46B30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0627695-X

Keywords:
Banach lattice,
positive operator,
weakly compact operator

Article copyright:
© Copyright 1981
American Mathematical Society