It is true if $M$ is a diagonal matrix. Let $M=\left(\begin{array}{cc} m_{1} & 00\\ 0 & m_{2}m_{2}\\ 0 & 0\end{array}\right)$. Assume that $\left|m_{1}\right|\ge\left|m_{2}\right|$, then for any $x:=\left(\begin{array}{c} x_{1}x_{1}\\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$ with $\left\Vert x\right\Vert _{3}=1$, $\left\Vert Mx\right\Vert {3}=\left(\left|m{1}x_{1}\right|^{3}+\left|m_{2}x_{2}\right|^{3}\right)^{1/3}\geq\left(\left|m_{2}x_{1}\right|^{3}+\left|m_{2}x_{3}\right|^{3}\right)^{1/3}=\left|m_{2}\right|$. _{3}=\left(\left|m_{1}x_{1}\right|^{3}+\left|m_{2}x_{2}\right|^{3}\right)^{1/3}\geq\left(\left|m_{2}x_{1}\right|^{3}+\left|m_{2}x_{3}\right|^{3}\right)^{1/3}=\left|m_{2}\right|$. Take$V$to be the space spanned by$\left(\begin{array}{c} 00\\ 11\\ 0\end{array}\right)$and$\left(\begin{array}{c} 00\\ 00\\ 1\end{array}\right)$, then $\left\Vert M^{T}y\right\Vert {3/2}\leq\left|m{2}\right|$_{3/2}\leq\left|m_{2}\right|$ for all $y\in V$ with $\left\Vert y\right\Vert {3/2}=1$. _{3/2}=1$. Likewise for the case $\left|m{1}\right|\le\left|m_{2}\right|$.\left|m_{1}\right|\le\left|m_{2}\right|$.
It is true if $M$ is a diagonal matrix. Let $M=\left(\begin{array}{cc} m_{1} & 0\ 0 & m_{2}\ 0 & 0\end{array}\right)$. Assume that $\left|m_{1}\right|\ge\left|m_{2}\right|$, then for any $x:=\left(\begin{array}{c} x_{1}\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$ with $\left\Vert x\right\Vert _{3}=1$, $\left\Vert Mx\right\Vert {3}=\left(\left|m{1}x_{1}\right|^{3}+\left|m_{2}x_{2}\right|^{3}\right)^{1/3}\geq\left(\left|m_{2}x_{1}\right|^{3}+\left|m_{2}x_{3}\right|^{3}\right)^{1/3}=\left|m_{2}\right|$. Take $V$ to be the space spanned by $\left(\begin{array}{c} 0\ 1\ 0\end{array}\right)$ and $\left(\begin{array}{c} 0\ 0\ 1\end{array}\right)$, then $\left\Vert M^{T}y\right\Vert {3/2}\leq\left|m{2}\right|$ for all $y\in V$ with $\left\Vert y\right\Vert {3/2}=1$. Likewise for the case $\left|m{1}\right|\le\left|m_{2}\right|$.