Let mathscr{H}_1 and mathscr{H}_2 be Hilbert C^* -modules. In this paper we give some necessary and sufficient conditions for the positivity of a block matrix on the Hilbert C^* -module mathscr{H}_1 oplus mathscr{H}_2 If mathscr{H}_1, J_1) and ( mathscr{H}_2, J_2) are two Krein C^* -modules, we study the { bf tilde{J}}$-positivity of 2 times 2 block matrix begin{eqnarray*} left( begin{array}{cc} A & X X^{ sharp}& B end{array} right) end{eqnarray*} on the Krein C^* -module ( mathscr{H}_1 oplus mathscr{H}_2,{ bf tilde{J}}=J_1 oplus J_2) , where $X^{\\\\sharp}=J_2X^*J_1$ is the $(J_2,J_1)$-adjoint of the operator $X$. We prove that if $A$ is $J_1$-selfadjoint and $B$ is $J_2$-selfadjoint and $A$ is invertible, then the operator $\\\\left(\\\\begin{array}{cc}A & X\\\\\\\\ X^{\\\\sharp} & B\\\\end{array}\\\\right)$ is ${\\\\bf\\\\tilde{J}}$-positive if and only if $A\\\\geq^{J_1}0$, $B\\\\geq^{J_2}0$ and $X^{\\\\sharp}A^{-1}X\\\\leq^{J_2}B$. We also present more equivalent conditions for the ${\\\\bf\\\\tilde{J}}$-positivity of this operator.

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