\def\papertitle{On the strongly mid p-summing operators and application}
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{University of M'sila}

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\begin{abstract}
In this paper, we introduce and study the new ideal of strongly mid $p$-summing linear operators between Banach spaces. We prove, an operator is strongly  mid $p$-summing if and only if its adjoint is absolutely mid $p^*$-summing. This result led us to prove, $X$ is $p$-Dunford-Pettis property $(1<p<\infty)$ if and only if, so is $X^*$.
\end{abstract}

\section{Notation and preliminaries}\label{section1}
The notation used in the paper is in general standard. The letters $X,X_1,\cdots,X_m, \ Y$ ($m$ be in $\mathbb{N}$ ) shall denote Banach spaces over  $\mathbb{K}$ (real or complex scalars field). We will denote by $ \mathcal{L}(X_{1},\cdots,X_{m};Y)$ the Banach space of all bounded multilinear operators from $X_{1}\times \cdots\times X_{m}$ into $Y$ equipped with the operator norm. $\mathcal{L}(X;Y) $ the Banach space of all bounded linear operators $T:X\rightarrow Y$ endowed with the usual sup norm. The closed unit ball of $X$  is denoted by $B_X$ and its topological dual by $X^*=\mathcal{L}(X;\mathbb{K})$.

For a Banach space $X$, and $1\leq p \leq \infty$, let $p^*$ be its conjugate that is $1/p+1/p^*=1$. Let us define the  sequences spaces we shall work with: (see \cite{DJT,mid1}).\\
$\ell_{p}(X)=$ the Banach space of absolutely $p$-summable sequences with the norm ($1\leq p < \infty$) $$\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{p}:=\left( \sum_{j=1}^{\infty}\parallel x_{j}\parallel^{p}\right) ^{1/p}.$$
$\ell_{p}^{w}(X)=$ the Banach space of weakly $p$-summable sequences with the norm ($1\leq p < \infty$) $$\parallel(x_{j})_{j=1}^{\infty}\parallel_{w, p}:= \sup _{x^* \in B_{X} *}\left(\sum_{j=1}^{\infty}\left|x^*\left(x_{j}\right)\right|^{p}\right) ^{1/p}.$$
$\ell_{p}^{mid}(X)$ the Banach space  of mid $p$-summable sequences with the norm ($1\leq p < \infty$) $$\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{mid, p}:=\sup _{\left(x_{n}^{*}\right)_{n=1}^{\infty} \in B_{\ell_p^w\left(X^{*}\right)}} \left( \sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left|x_{n}^{*}\left(x_{j}\right)\right|^p\right) ^{1/p}.$$
When $p=\infty$ we have $\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{\infty}=\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{w,\infty}=\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{mid,\infty}=\sup_j \Vert x_j\Vert$. The relationships between the various sequence spaces are given by \cite{mid1}
$$\ell_{p}(X)\subsetneq  \ell_{p}^{mid}(X)\subsetneq  \ell_{p}^{w}(X), $$with $$\parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{p}\leq \parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{w,p} \leq \parallel\left(x_{j}\right)_{j=1}^{\infty}\parallel_{mid,p}.$$

Recall that, for $1\leq p <\infty$, we say that a continuous linear operator $T:X\rightarrow Y$ is absolutely $p$-summing if $$(T(x_j))_{j=1}^\infty \in \ell_p(Y)   \   whenever \ (x_j)_{j=1}^\infty \in \ell^w_p(Y).$$ The class of absolutely $p$-summing operator from $X$ to $Y$ will be represented by $\Pi_p(X;Y)$. An equivalent formulation asserts that $T$ is absolutely $p$-summing if  there is a positive constant $C$ such that $$\Vert(T(x_j))_{j=1}^\infty \Vert_p \leq C \Vert(x_j)_{j=1}^\infty\Vert_{w,p}$$ for all $(x_j)_{j=1}^\infty\in \ell_p^w(X)$. The infimum of all $C>0$ that satisfy the above inequality define a norm, denoted by $ \pi_{p}(T)$ .

Let $(\mathcal{I}(X;Y),\alpha(\cdot))$ be an operator ideal, we put $$\mathcal{I}^d(X;Y)=\{ T\in \mathcal{L}(X;Y): T^*\in \mathcal{I}(Y^*;X^*) \}.$$ For an operator $T\in \mathcal{I}^d(X;Y)$, we put $\alpha^d(T)=\alpha(T^*)$. With these notations $(\mathcal{I}^d(X;Y),\alpha^d(\cdot))$ is also a Banach operator ideal and is called the dual ideal of $(\mathcal{I}(X;Y),\alpha(\cdot))$ (see \cite[Section 4]{Pie80}).

Let us recall now the definition of strongly $p$-summing linear operators. An operator $T\in \mathcal{L}(X;Y)$ is strongly $p$-summing if there is a constant $C$ such that  $$\Vert ( \langle T(x_j),y_j^*\rangle  ) _{j=1}^{\infty }\Vert _{1}\leq C\Vert
( x_{j}) _{j=1}^{\infty }\Vert _{p}\Vert (y_{j}^{\ast }) _{j=1}^{\infty }\Vert _{w,p^*}$$ for all $(x_j)_{j=1}^\infty\in \ell_p(X)$ and $(y^*_j)_{j=1}^\infty\in \ell_{p^*}^w(Y^*)$. The space $\mathcal{D}_p(X;Y)$ of all strongly $p$-summing linear operators from $X$ into $Y$ which is a Banach space with the norm $d_p(T)$  the infimum of all $C>0$ that satisfy the above inequality. According to \cite{Cohen,ab} we obtain $\mathcal{D}_p^d(X;Y)=\Pi_{p^*}(X;Y)$ and $\Pi_{p}^d(X;Y)=\mathcal{D}_{p^*}(X;Y)$.

Our results are presented as follows. In first section \ref{section1}, we recall important results and definitions to be used later.
In section \ref{section2}, we introduce and investigate a new ideal of strongly mid $p$-summing operators. We present a characterization given by a summability property. We also prove the related dual result: an operator $T\in \mathcal{L}(X;Y)$ is strongly mid $p$-summing ($T\in \mathcal{D}_p^{mid}(X;Y)$) if and only if its adjoint $T^*$ is absolutely mid $p^*$-summing ($T^*\in \Pi_{p^*}^{mid}(Y^*;X^*)$). We study Banach space $X$ for which $id_X\in \Pi_p^{mid}$(Theorem   \ref{theorema1}), $id_X\in \mathcal{D}_p^{mid}$(Theorem  \ref{theorema2}), $\mathcal{D}_{p}(\ell_p;X)=\mathcal{L}(\ell_p;X)$ (Theorem \ref{theorema4}). We also prove, $X$ has the $p$-Dunford-Pettis property if and only if $X^*$ has the $p$-Dunford-Pettis property.
\section{Strongly mid summing linear operators}\label{section2}
Botelho, Campos and Santos \cite{mid1} introduced the concept of absolutely mid $p$-summing operators. For $1\leq p<\infty$, an operator $T\in \mathcal{L}(X;Y)$ is absolutely mid $p$-summing if $$(T(x_j))_{j=1}^\infty\in \ell_p^{mid}(Y) \  whenever  \ (x_j)_{j=1}^\infty\in \ell_p(X).$$ By $\Pi_p^{mid}(X;Y)$, we denote the space of absolutely mid $p$-summing operators.

 We introduce the concept of strongly mid $p$-summing linear operators as a characterization of the conjugates of absolutely mid $p^*$-summing linear operators. This idea seems to have appeared for the first time in \cite{Cohen} and \cite{ab}.
\begin{definition}Let $1<p\leq \infty$. A mapping $T\in \mathcal{L}(X;Y)$ is strongly mid $p$-summing if there exist a constant $C>0$ such that 
\begin{equation}\label{defs2.1}
\Vert ( \langle T( x_{j}) ,y_{j}^*\rangle ) _{j=1}^{k}\Vert _{1}\leq C\Vert (x_{j}) _{j=1}^{k}\Vert _{p}\Vert ( y_{j}^*) _{j=1}^{k}\Vert _{mid,p^*}
\end{equation}
for any $( x_{j}) _{j=1}^{k}\subset X$ and $( y_{j}^{\ast })_{j=1}^{k}\subset Y^{\ast }.$ $\mathcal{D}_p^{mid}(X;Y)$ denotes the space of all  strongly mid $p$-summing operators from $X$ to $Y$. The least $C$  for which (\ref{defs2.1}) holds will be written $d_p^{mid}(T)$. From the definition it is clear that $\mathcal{D}_p(X;Y)\subset \mathcal{D}_p^{mid}(X;Y)$ and $d_p^{mid}(\cdot)\leq d_p(\cdot)$.
\end{definition}
For $T\in \mathcal{L}(X;Y)$. The induced map $\varphi_T:X\times Y^*\rightarrow \mathbb{K}$ given by $\varphi_T(x,y^*)=\langle T(x),y^*\rangle$ for all $x\in X$ and $y^*\in Y^*$ is continuous bilinear. Using  the abstract approach of \cite{transformation} and Proposition 1.9 in \cite{mid1}, we can see that the next proposition is immediate consequences of  \cite[Proposition 2.4]{transformation}.
\begin{proposition}Let $T\in \mathcal{L}(X;Y)$, the following assertions are equivalent.
\begin{enumerate}
\item $T\in \mathcal{D}_p^{mid}(X;Y)$.
\item $(\varphi_T (x_j,y^*_j))_{j=1}^{\infty } \in \ell_1$ whenever $(x_j)_{j=1}^{\infty }\in \ell_{ p}(X)$ and $(y^*_j)_{j=1}^{\infty }\in \ell_{ p^*}^{mid}(Y^*)$.
\item The induced map $\widehat{\varphi_T }:\ell _{p}(X) \times \ell	_{p^*}^{mid}(Y^*)    \rightarrow \ell_1 $ given by $\widehat{\varphi_T} ((x_{j}) _{j=1}^{\infty } ,(y^*_j)_{j=1}^{\infty })=(\varphi_T (x_j,y^*_j))_{j=1}^{\infty } $ is a well-define and continuous bilinear operator.
\item There exist a constant $C>0$ such that 
\begin{equation}\label{equ500}
\Vert ( \varphi_T(x_j,y_j^*) ) _{j=1}^{\infty }\Vert _{1}\leq C\Vert
( x_{j}) _{j=1}^{\infty }\Vert _{p}\Vert (y_{j}^{\ast }) _{j=1}^{\infty }\Vert _{mid,p^*}.
\end{equation}
for any $( x_{j}) _{j=1}^{\infty }\in \ell _{p}( X)$ and  $(y_{j}^{\ast }) _{j=1}^{\infty }\in \ell _{p^*}^{mid}( Y^*) .$
\item There exist a constant $C>0$ such that 
\begin{equation}\label{equ501}
\Vert ( \varphi_T(x_j,y_j^*) ) _{j=1}^{k }\Vert _{1}\leq C\Vert
( x_{j}) _{j=1}^{k }\Vert _{p}\Vert (y_{j}^{\ast }) _{j=1}^{k }\Vert _{mid,p^*}.
\end{equation}
for any $( x_{j}) _{j=1}^{k}\subset X$ and $( y_{j}^{\ast })_{j=1}^{k}\subset Y^{\ast }.$
\end{enumerate}
Moreover, $d_{p}^{mid}(T)=\Vert \widehat{\varphi_T } \Vert=\inf \{ C: \ (\ref{equ500}) \ holds \}=\inf \{ C: \ (\ref{equ501}) \ holds \} $.
\end{proposition}
%\begin{remark} Let $T\in \mathcal{L}(X;Y)$. Then $T\in \mathcal{D}^{mid}(X;Y)$ if and only if $\varphi_T$ satisfy Proposition.\end{remark}
\begin{theorem}$(\mathcal{D}_{p}^{mid}(X;Y),d_{p}^{mid}(\cdot ))$ is a Banach operator ideal.
\end{theorem}
\begin{proof}
Using the abstract framework, notation and language \cite{transformation}, we find that a linear operator $T$ is  strongly $p$-summing if and only if $\varphi_T$ is $(\ell_{p}(.)\ell_{p^*}^{mid}(.);\ell_1)$-summing. Since $1/p+1/p^*=1$  we obtain $$\Vert (\lambda_j^1.\lambda_j^2)_{j=1}^{\infty}\Vert_1\leq \Vert (\lambda_j^1)_{j=1}^{\infty}\Vert_{\ell_p(\mathbb{K})} \Vert (\lambda_j^2)_{j=1}^{\infty}\Vert_{\ell_{p^*}(\mathbb{K})}=\Vert (\lambda_j^1)_{j=1}^{\infty}\Vert_{\ell_p(\mathbb{K})} \Vert (\lambda_j^2)_{j=1}^{\infty}\Vert_{\ell_{p^*}^{mid}(\mathbb{K})}.$$Therefor,
 $ \  \ell_{p}(\mathbb{K}) \ell_{p^*}^{mid}(\mathbb{K})= \ell_{p}(\mathbb{K}) \ell_{p^*}(\mathbb{K}) \overset{1}{\hookrightarrow }\ell_{1}.$ In addition, all the sequence classes involved are linearly stable (see \cite[Proposition 1.10]{mid1}). So, form \cite[Theorem 3.6]{transformation} it follows that  $( \mathcal{D}^{mid}_{p},d^{mid}_{ p}(.))$ is a Banach operators ideal.
\end{proof}
In the next theorem, we will show that in fact the absolutely mid $p$-summing linear operators is the adjoint of  strongly mid $p$-summing linear operators that will be useful throughout our section.
\begin{theorem}\label{theequ} Let $1<p\leq \infty$, $T\in \mathcal{L}(X;Y)$. Then  $T\in \mathcal{D}_p^{mid}(X;Y)$ if and only if $T^*\in \Pi_{p^*}^{mid}(Y^*;X^*)$. Moreover, $d_p^{mid}(T)=\pi_{p^*}^{mid}(T^*)$.
\end{theorem}
\begin{proof}Assume that $T\in \mathcal{D}_p^{mid}(X;Y)$. For $(x_{j})_{j=1}^{\infty} \in  \ell _{p}( X)$ and $(y_{j}^{\ast })_{j=1}^{\infty} \in \ell _{p^*}^{mid}( Y^*)$. We have 
\begin{equation*}
\Vert ( \langle T( x_{j}),y_{j}^*\rangle ) _{j=1}^{\infty}\Vert _{1}=\Vert ( \langle T^{\ast }( y_{j}^{\ast }),x_{j}\rangle ) _{j=1}^{\infty}\Vert _{1}\leq d_p^{mid}(T) \Vert( x_{j}) _{j=1}^{\infty}\Vert _{p}\Vert ( y_{j}^{\ast}) _{j=1}^{\infty}\Vert _{mid,p^*}
\end{equation*}
by taking the supremum over the unit ball in $\ell _{p}( X) $ we obtain 
$\Vert (T^{\ast }( y_{j}^{\ast }) )_{j=1}^{\infty}\Vert _{p^{\ast }}\leq C\Vert (y_{j}^{\ast}) _{j=1}^{\infty}\Vert _{mid,p^*}
$. Therefore, $T^{\ast }\colon Y^{\ast }\rightarrow X^{\ast }$ is absolutely mid $p^*$-summing and $d_p^{mid}(T)\geq \pi_p^{mid}(T^*)$. Conversely, let $T^*\in \Pi_{p^*}^{mid}(Y^*;X^*)$. For $(x_{j})_{j=1}^{\infty} \in  \ell _{p}( X)$ and $(y_{j}^{\ast })_{j=1}^{\infty} \in \ell _{p^*}^{mid}( Y^*)$. By H\"{o}lder's inequality we have 
$$
\sum_{j=1}^{\infty}| \langle T(x_{j}),y_{j}^{\ast}\rangle | \leq \Vert (x_{j})_{j=1}^{\infty}\Vert_{p}\Vert (T^*(y_{j}^*))_{j=1}^{\infty}\Vert _{p^{\ast }} \leq \pi_p^{mid}(T^*) \Vert (x_{j})_{j}^\infty\Vert _{p}\Vert ( y_{j}^{\ast})) _{j=1}^{\infty}\Vert_{mid,p^{\ast }}.
$$Therefore $T$ is strongly mid $p$-summing and $d_p^{mid}(T)=\pi_p^{mid}(T^*)$.
\end{proof}
As a consequence, we obtain the following corollary which is a straightforward consequence of the preceding theorem and Theorem 2.8 in \cite{mid1}.
\begin{corollary}
Let $1\leq p<\infty$, $T\in \mathcal{L}(X;Y)$. If $T^*\in \mathcal{D}_{p^*}^{mid}(Y^*;X^*)$ then $T\in \Pi_p^{mid}(X;Y)$.
\end{corollary}
Recall that an operator ideal $\mathcal{I}(X;Y)$ is surjective if $T\in\mathcal{I}(X;Y) $ whenever $T\circ Q \in\mathcal{I}(X_0;Y)$, where $Q:X_0\rightarrow X$ is the quotient map.
As consequence, we obtain the following proposition which is a straightforward consequence of Theorem \ref{theequ}, and \cite[Proposition 2.8]{mid1} and \cite[Section 4]{Pie80}.
\begin{proposition}The operator ideal $(\mathcal{D}_p^{mid},d^{mid}_p)$ is surjective.
\end{proposition}
 The Dvoretzky–Rogers theorem (see \cite{DJT}) states that the Banach space $X$ is ﬁnite dimensional if, and only if, $id_X$ is $p$-summing. Now we will give a results in this context.
 
We say that a Banach space $X$ is a weak mid $p$-space (or $p$-Dunford-Pettis property see \cite{KS}) if $\ell^{mid}_p(X)=\ell_p^w(X)$  and it is a strong mid $p$-space if $\ell^{mid}_p(X)=\ell_p(X)$.

The next result is a reformulation of \cite[Theorem 2.7]{mid1}.
\begin{theorem}\label{theorema1} The following are equivalent:\\
1) $X$ is a strong mid $p$-space.\\
2) $id_{X}\in \Pi_{p}^{mid}(X;X)$. \\
3) $\mathcal{L}(X;Y)=\Pi_p^{mid}(X;Y)$ for every Banach space $Y$.\\
4) $\mathcal{L}(Y;X)=\Pi_p^{mid}(Y;X)$ for every Banach space $Y$.\\
5) $X$ is a subspace of $L_p(\mu)$ for some Borel measure $\mu$.
\end{theorem}
\begin{corollary} If $X^{**}$ is a strong mid $p$-space then $X$ is a strong mid $p$-space.
\end{corollary}
\begin{proof}
Is a direct consequence of Theorem \ref{theorema1} and \cite[Proposition 2.8]{mid1}.
\end{proof}
\begin{theorem}\label{theorema2} The following are equivalent:\\
1) $X^*$ is a strong mid $p^*$-space.\\
2) $id_{X}\in \mathcal{D}_{p}^{mid}(X;X)$.\\
3)  $\mathcal{L}(X;Y)=\mathcal{D}_p^{mid}(X;Y)$ for every Banach space $Y$.\\ 
4)  $\mathcal{L}(Y;X)=\mathcal{D}_p^{mid}(Y;X)$ for every Banach space $Y$.\\
5)  $id_{X^*}\in \Pi_{p^*}^{mid}(X^*;X^*)$.\\
6)  $\mathcal{L}(X^*;Y)=\Pi_{p^*}^{mid}(X^*;Y)$ for every Banach space $Y$.\\ 
7)  $\mathcal{L}(Y;X^*)=\Pi_{p^*}^{mid}(Y;X^*)$ for every Banach space $Y$.\\ 
8)  $X^*$ is a subspace of $L_{p^*}(\mu)$ for some Borel measure $\mu$.
\end{theorem}
\begin{proof} By Theorem \ref{theequ} and Theorem \ref{theorema1}, we have
$$\begin{tabular}{ c c c c c c c}
  &   &   &   & 7 &   &   \\ 
  &   &   &   & $\Updownarrow$ &   &   \\ 
8 & $\Leftrightarrow$ & 1 & $\Leftrightarrow$ & 5 & $\Leftrightarrow$ & 6 \\ 
  &   &   &   & $\Updownarrow$ &   &   \\ 
  &   & 3 & $\Leftrightarrow$ & 2 & $\Leftrightarrow$ & 4 \\ 
\end{tabular} $$

\end{proof}
\begin{theorem}\label{2.10}\cite[Theorem 2.6]{mid1}\cite[Theorem 3.7]{KS} The following are equivalent:\\
1) $X$ has the $p$-Dunford-Pettis property.\\
2) $\Pi_{p}^{mid}(X;Y)=\Pi_{p}(X;Y)$ for every Banach space $Y$.\\ 
3) $\Pi_{p}^{mid}(X;\ell_{p})=\Pi_{p}(X;\ell_{p})=\mathcal{L}(X;\ell_{p})$.\\
4) $\mathcal{D}_{p^*}(\ell_{p^*};X^*)=\mathcal{L}(\ell_{p^*};X^*)$.
\end{theorem}
\begin{theorem}\label{theorema4}The following are equivalent:\\
1) $X^*$ has the $p$-Dunford-Pettis property.\\
2) $\mathcal{D}_{p^*}^{mid}(Y;X)=\mathcal{D}_{p^*}(Y;X)$ for every Banach space $Y$.\\
3) $\Pi_{p}^{mid}(X^*;Y^*)=\Pi_{p}(X^*;Y^*)$ for every Banach space $Y$.\\
4) $\Pi_{p}^{mid}(X^*;\ell_{p})=\Pi_{p}(X^*;\ell_{p})=\mathcal{L}(X^*;\ell_{p})$.\\ 
5) $\mathcal{D}_{p^*}^{mid}(\ell_{p^*};X)=\mathcal{D}_{p^*}(\ell_{p^*};X)=\mathcal{L}(\ell_{p^*};X)$.\\
6) $\mathcal{D}_{p^*}(\ell_{p^*};X)=\mathcal{L}(\ell_{p^*};X)$.
\end{theorem}
\begin{proof}
follows from Theorem \ref{theequ} and Theorem \ref{2.10}.
\end{proof}
\begin{corollary} $X^{**}$ has the $p$-Dunford-Pettis property if and only if $X$ has the $p$-Dunford-Pettis property.
\end{corollary}
\begin{corollary}\cite[Corollary 3.9]{KS} If $X^{*}$ has the $p$-Dunford-Pettis property, so does $X$.
\end{corollary}
\begin{theorem}$X^{*}$ has the $p$-Dunford-Pettis property if and only if  $X$ has the $p$-Dunford-Pettis property.
\end{theorem}
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\bibitem{mid1}G. Botelho, J.R. Campos and J. Santos, Operator ideals related to absolutely summing and Cohen strongly summing operators, Pacific J. Math. 287 (2017), 1–17.
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\bibitem{KS} A. Karn and D. Sinha, An operator summability of sequences in Banach spaces, Glasg. Math. J. 56 (2014), no. 2, 427–437.

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\bibitem{ab}Apiola, H.:  {Duality between spaces of $p$-summable sequences, $(p, q)$-summing operators and characterization of nuclearity.} Math. Ann \textbf{219}, 53–64 (1976).	
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