I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf and get stuck at the last step. We construct the complex function \begin{equation*} F(z)=\int g_zTf_z=\sum_{i,j}|a_i|^{\frac{p}{p(z)}}\frac{a_i}{|a_i|}|b_j|^{\frac{q}{q(z)}}\frac{b_j}{|b_j|}\int_{B_j}T\chi_{A_i} \end{equation*} and finally we have \begin{equation*} \sup_{y\in\mathbb{R}}F(iy)=\sup_{y\in\mathbb{R}}\int g_{yi}Tf_{yi}\leq \|T\|_{L_{p_0}(\mathbb{C}^n)\rightarrow L_{q_0}(\mathbb{C}^n)}\|g_{yi}\|_{L_{q_0}}\|f_{yi}\|_{L_{p_0}} \end{equation*} However, if we come back to the theorem, what we actually want to show is \begin{equation} \sup_{y\in\mathbb{R}}F(iy)\leq N_0\|g_{yi}\|_{L_{q_0}} \|f_{yi}\|_{L_{p_0}}=\|T\|_{L_{p_0}(\mathbb{R}^n)\rightarrow L_{q_0}(\mathbb{R}^n)}\|g_{yi}\|_{L_{q_0}} \|f_{yi}\|_{L_{p_0}} \end{equation} I understand that $\|g_{yi}\|_{L_{q_0}}=\|g_0\|_{L_{q_0}}$ and $\|f_{yi}\|_{L_{p_0}}=\|f_0\|_{L_{p_0}}$, but as we know the complex norm is never smaller than real norm. So we need to show \begin{equation*} \|T\|_{L_{p_0}(\mathbb{C}^n)\rightarrow L_{q_0}(\mathbb{C}^n)}=\|T\|_{L_{p_0}(\mathbb{R}^n)\rightarrow L_{q_0}(\mathbb{R}^n)} \end{equation*} which is not obvious. Moreover, if we have to add a constant $C$ such that \begin{equation*} \|T\|_{L_{p_0}(\mathbb{C}^n)\rightarrow L_{q_0}(\mathbb{C}^n)}\leq C\|T\|_{L_{p_0}(\mathbb{R}^n)\rightarrow L_{q_0}(\mathbb{R}^n)} \end{equation*} Then the statement of this theorem must be modified accordingly. I tried to simplify it into matrix case and regard the linear map $T$ as an $n\times n$ matrix, then $T:\mathbb{C}^n\rightarrow\mathbb{C}^n$ can be looked as another $2n\times 2n$ matrix $T\otimes I_2$, however, the $L_p$ norm in $\mathbb{R}^{2n}$ is different from that in $\mathbb{C}^n$ when $p\neq 2$.

In conclusion, my questions are:

(1) Is the statement of Riesz-Thorin theorem in the above website true for real linear map $T$? If it is true, how to prove it?

(2) In fact I mainly care about how to prove it when $T$ is a symmetric matrix, so is there an easier way to prove the matrix version of this theorem?

(3) If possible, I hope some one can provide me with some reference about Riesz-Thorin theorem and some related topics so that I can learn more. (matrix version is enough but I would be glad to learn the results in more general version)

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