# Dunford Pettis property

I am trying the following problem.

Let $$\mu$$ be a probability measure. Show that an operator $$T : L_1(\mu) \rightarrow X$$ is Dunford Pettis if and only if $$T|_{L_2(\mu)}$$ is compact.

I was trying to use the definition. But I cannot come up with anything.Could you give me some hint?

• The Dunford-Pettis property is usually a property of a Banach space, not of a single operator. What do you mean by "Dunford Pettis property of $T$"? – uniquesolution Apr 11 at 5:28
• One way is easy: any $L^{2}$ bounded sequence is uniformly integrable, hence weakly relatively compact. – Kavi Rama Murthy Apr 11 at 5:33
• This may work: Suppose $(f_n)\subset L_1$ converges weakly to $0$. We need to show $Tf_n$ converges in norm to $0$ (see Prop 5.4.2 in Albiac/Kalton). Towards this end, fix $M>0$ and set $g_n=f_n\cdot\chi_{A_n}$ where $A_n=\{ x : |f_n (x)|<M\}$. Then $(g_n)\subset L_2$ and, being a bounded sequence, its image under $T$ tends to $0$ in norm. Since $(f_n)$ is uniformly integrable, $\sup_n \int_{A_n^c } |f_n|$ can be made small (by choosing $M$ large). It seems this will imply the result. – David Mitra Apr 11 at 17:43