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)



A question about the proof of Riesz-Thorin interpolation theorem
Concerning your Question (3): A nice proof of the Riesz--Thorin theorem is given in Section 3.5 in [Lax's book](https://bookstore.ams.org/ulect-58/). A chapter devoted to interpolation of linear operators, including different versions of the Riesz--Thorin theorem, with further references therein, is contained in [Mashreghi's book](https://books.google.com/books/about/Representation_Theorems_in_Hardy_Spaces.html?id=PvM4VJlKcnkC) as Chapter 8. 


As for your Question (1), as Jochen Glueck commented, you probably meant $L\_p(\mathbb R^n,\mathbb R)$ and $L\_p(\mathbb R^n,\mathbb C)$ instead of $L\_p(\mathbb R^n)$ and $L\_p(\mathbb C^n)$, respectively. If so, the answer to your Question 1 is negative. Indeed, let $\ell\_{2,p}$ denote the space $\mathbb R^2$ with the $\ell\_p$ norm. 
Since $\ell\_{2,p}$ can be isometrically embedded into $L\_p(\mathbb R^n,\mathbb R)$, we may restrict the consideration to $2\times2$ matrices $T$ with the corresponding operator norms $\|T\|\_{p,q}$, $\|T\|\_{p\_0,q\_0}$, $\|T\|\_{p\_1,q\_1}$. 


Then we find that the inequality 
$$\|T\|\_{p,q}\le\|T\|\_{p\_0,q\_0}^{1-t}\|T\|\_{p\_1,q\_1}^t \tag{1}
$$ 
fails to hold if
$$T=\left(\begin{array}{cc}
 0.448531 & 0.812143 \\
 -0.772457 & 0.469272 \\
\end{array}
\right)
$$
and 
$$(p\_0, q\_0, p\_1, q\_1, t)=(3.99136, 1.32751, 8.82177, 1, 0.854335); 
$$
then the ratio of the left-hand side of (1) to its right-hand side is about $1.00041>1$. 


