# A generalization of $b$-weakly compact operators

## TL;DR

This paper clarifies the distinction between $b$-weakly compact and $KB$-operators, providing new insights into their properties and relationships within normed vector lattices.

## Contribution

It resolves an open problem by proving that $b$-weakly compact and $KB$-operators are different classes of operators.

## Key findings

- $KB$-operators have specific convergence properties.
- $b$-weakly compact operators are a distinct class.
- Relationships between $KB$-operators and $b$-weakly compact operators are characterized.

## Abstract

A. Bahramnezhad and K. Haghnejad Azar introduced the classes of $KB$-operators and $WKB$-operators, and they studied some of theirs properties. In the present paper, we give answer for an open problem from that paper, which two classifications of operators, $b$-weakly compact operators and $KB$-operators are different. A continuous operator $T$ from a normed vectoe lattice $E$ into a normed space $X$ is said to be $KB$-operator (respectively, $WKB$-operator) if $\{Tx_n\}_n$ has a norm (respectively, weak) convergent subsequence in $X$ for every positive increasing sequence $\{x_n\}_n$ in the closed unit ball $B_E$ of $E$. We investigate some other properties of $KB$-operators and its relationships with $b-$weakly compact operators.