This is an exercise from Dirk Werner's "Funktionalanalysis" (Aufgabe II.5.4): Let $(\Omega,\Sigma, \mu)$ be some $\sigma$-finite measure space. Denote by $L^p_{\mathbb{R}}(\mu)$ and $L^p_{\mathbb{C}}(\mu)$ with $p\in[1,\infty]$ the space of $L^p$-functions taking real or complex values respectively. Let $$U:L^p_{\mathbb{R}}(\mu)\to L^q_{\mathbb{R}}(\mu) $$ be a linear and bounded operator and define $$ U_{\mathbb{C}}(f+ig)=Uf+iUg.$$ Then $U_{\mathbb{C}}$ defines a linear and bounded operator between $L^p_{\mathbb{C}}(\mu)$ and $L^q_{\mathbb{C}}(\mu)$ and we have $$\|U_{\mathbb{C}}\| \leq 2\|U\|,$$ which gives a constant $2$ for the real version of the Riesz-Thorin interpolation inequality. The question is then to give an explicit example such that $\|U_{\mathbb{C}}\|>\|U\|$ is valid, where the norm here is the usual operator norm between the $L^p$ and $L^q$ spaces. I have no idea how to begin with this problem, since there is no hint for this exercise. Also it seems that the examples given in this book are irrelevant to this problem. The problem here is that if one wants to keep that $\|f+ig\|=1$, then one needs something like $\|f\|=\|g\|=1/\sqrt{2}$, but then the value of $Uf+iUg$ will remain invariant when compare it to the single term $Uf$.
Any idea and hint is appreciate!