# Almost Dunford-Pettis operators

Recall that an operator $$T$$ from a Banach space $$E$$ to a Banach space $$F$$ is called completely continuous (also called Dunford-Pettis) if $$\|Tx_{n}\|\rightarrow 0$$ for every weakly null sequence $$(x_{n})_{n}$$ in $$E$$.

In 1994, W. Wnuk introduced the concept of almost Dunford-Pettis operators. Let $$X$$ be a Banach lattice and let $$X_{+}=\{x\in X:x\geq 0\}$$ be the positive cone. An operator $$T$$ from a Banach lattice $$X$$ to a Banach space $$E$$ is called almost Dunford-Pettis if $$\|Tx_{n}\|\rightarrow 0$$ for every weakly null disjoint sequence $$(x_{n})_{n}$$ in $$X$$. He also remarked that $$T$$ is almost Dunford-Pettis if and only if $$\|Tx_{n}\|\rightarrow 0$$ for every weakly null disjoint sequence $$(x_{n})_{n}$$ in $$X_{+}$$.

It is natural to introduce the concept of positive completely continuous operators. We say that an operator $$T$$ from a Banach lattice $$X$$ to a Banach space $$E$$ is positive completely continuous if $$\|Tx_{n}\|\rightarrow 0$$ for every weakly null sequence $$(x_{n})_{n}$$ in $$X_{+}$$. Clearly, it follows from Wnuk's remark that every positive completely continuous operator is almost Dunford-Pettis. Does the converse hold?

Question. Do positive completely continuous operators and almost Dunford-Pettis operators coincide?

• I'm not sure why this received a downvote. – Jochen Glueck Apr 10 at 6:06