Abstract
Our aim is to solve the problem asked by Bahramnezhad and Azar in "KB-operators on Banach lattices and their relationships with Dunford-Pettis and order weakly compact operators". We show that a continuous operators R from a Banach lattice N into a Banach space M is a b-weakly compact operator if and only if R is a KB-operator.
Highlights
In [6], Bahramnezhad and Azar defined a new classes of operators, named KB -operator and they have examined some of their properties and asked the following problem; Problem 1.1 ([6], Problem 2.27 ) Give an operator R from a Banach lattice N into a Banach space M which is a KB -operator but is not b-weakly compact
We recall the definitions of b-weakly compact operator and KB -operator
Definition 1.2 Let R be a continuous operator from a Banach lattice N into a Banach space M. (i) R is called KB -operator if R(xn) has a norm convergent subsequence in M for every positive increasing sequence of the closed unit ball BN of N. (ii) R is called b -weakly compact if R(xn) is norm convergent for every positive increasing sequence of the closed unit ball BN of N
Summary
In [6], Bahramnezhad and Azar defined a new classes of operators, named KB -operator and they have examined some of their properties and asked the following problem; Problem 1.1 ([6] , Problem 2.27 ) Give an operator R from a Banach lattice N into a Banach space M which is a KB -operator but is not b-weakly compact.We answer the question in the negative. 1. Introduction In [6], Bahramnezhad and Azar defined a new classes of operators, named KB -operator and they have examined some of their properties and asked the following problem; Problem 1.1 ([6] , Problem 2.27 ) Give an operator R from a Banach lattice N into a Banach space M which is a KB -operator but is not b-weakly compact. A lot of properties and results on b-weakly compact operators were given in [2 − −5, 7]. We recall the definitions of b-weakly compact operator and KB -operator.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
