# A question about the proof of Riesz-Thorin interpolation theorem

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 $$$$\sup_{y\in\mathbb{R}}F(iy)\leq N_0\|g_{yi}\|_{L_{q_0}} \|f_{yi}\|_{L_{p_0}}=\|T\|_{L_{p_0}(\mathbb{R}^n)\rightarrow L_{q_0}(\mathbb{R}^n)}\|g_{yi}\|_{L_{q_0}} \|f_{yi}\|_{L_{p_0}}$$$$ I understand that $$\|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)

• I am not finding there any statement that the real and complex versions of the norm are the same. Apr 4, 2019 at 12:39
• @losifPinelis Me too, so I am a little doubt about this proof... or are there easier way to prove this theorem for the case $T$ is a symmetric matrix? Apr 4, 2019 at 12:49
• If you don't see such a statement there (in the lecture notes?), then what is your question? Apr 4, 2019 at 15:03
• @IosifPinelis Sorry I didn't state it clearly. I have changed my statement and listed my questions above. Apr 5, 2019 at 5:33
• The formulation of the question seems to indicate somekind of misunderstanding: the space $\mathbb{R}^n$ in the linked notes is the underlying measure space of $L_p$. I can't see why the measure space $\mathbb{R^n}$ should be changed to $\mathbb{C}^n$ anywhere in the argument. Apr 5, 2019 at 15:54

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 $$1.00041>1$$.
• I'm not sure I follow your answer to Question (1): If $T$ is an operator on a real-valued $L^p$-space, then the operator norm of $T$ coincides with the operator norm of the canonical extension of $T$ to the complex-valued $L^p$-space. Thus, the Riesz-Thorin theorem is certainly also true on real-valued $L^p$-spaces. Apr 5, 2019 at 17:49
• @JochenGlueck : "If $T$ is an operator on a real-valued $L^p$-space, then the operator norm of $T$ coincides with the operator norm of the canonical extension of $T$ to the complex-valued $L^p$-space." Why would that be so? In fact, for the operator (given by matrix) $T$ in my answer, its "real" norm is about $0.903636$ of its "complex" norm -- this is how that $T$ was found. Also, you are welcome to check the calculations. Apr 5, 2019 at 21:13
• @JochenGlueck Thanks for your reference which help me understand what I originally want to know(when $T$ is a symmetric matrix and $p$ equals to $q$). To sum up your two links, they tell us the real norm and complex norm are coincide when $p=q$, and $\|T\|_{\mathbb{C}}\leq 2\|T\|_{\mathbb{R}}$ in general. And this result does not contradict with the numerical example introduced above. Apr 6, 2019 at 13:08