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Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}. $$

By the Riesz-Thorin theorem, the $p$-norm of $A$, as a function of $p$, is log-convex, meaning that the function $[1,\infty]\ni p\mapsto \log(\|A\|_p)$ is convex.

Question: Is the p-norm of a matrix strictly log-convex (unless it is constant)?

The proof of the Riesz-Thorin theorem uses holomorphic functions, so it seems plausible that if $\log(\|A\|_p)$ is constant on some non-trivial interval, then $\log(\|A\|_p)$ has to be globally constant.

Edit (following Terry Tao's answer):

Can $\|A\|_p$ be constant on some non-trivial interval, yet not be globally constant?

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  • $\begingroup$ Why not just prove it directly? $\phi\left(\tfrac{p+q}{2}\right) < \sqrt{\phi(p)\phi(q)}$, where $\phi(p)=\|A\|_p$. $\endgroup$ – Suvrit May 15 '14 at 16:40
  • $\begingroup$ Do you have a proof in mind? The inequality is not true in general, for example for the identity matrix. $\endgroup$ – Hannes Thiel May 15 '14 at 16:46
  • $\begingroup$ What I meant was trying to just provide sufficient conditions for the direct inequality, by using closedness of log-cvxity under addition and multiplication. Clearly, it does not hold in general :-) $\endgroup$ – Suvrit May 15 '14 at 16:49
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The answer is "no".

First, a trivial counterexample: let $A_n$ be the $n \times n$ matrix with all $1$s on the first row and zeroes elsewhere, then $\frac{1}{p} \mapsto \|A_n\|_p = n^{1/p}$ is log-convex but non-constant and not strictly log-convex.

One can deal with this example by replacing "constant" with "log-linear". However, if we consider the $(n+m) \times (n+m)$ matrix

$$ B := \begin{pmatrix} a A_n & 0\\ 0 & b A_m \end{pmatrix} $$ for some real parameters $a,b > 0$ and distinct natural numbers $n,m > 0$, then $\frac{1}{p} \mapsto \|B\|_p = \max( a n^{1/p}, b m^{1/p})$, which is log-convex but not log-linear and not strictly log-convex.

It seems to me that a similar construction allows any log-convex function to arise as $\frac{1}{p} \mapsto \|A\|_p$, at least if one allows A to be infinite-dimensional.

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  • $\begingroup$ Thanks, I realize that my question was imprecise. I will edit it. What I want to know is: Can $\|A\|_p$ be constant on some non-trivial interval, yet not globally constant? $\endgroup$ – Hannes Thiel May 15 '14 at 17:02
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    $\begingroup$ Yes: take $m=1$ or $n=1$ in the above example. $\endgroup$ – Terry Tao May 15 '14 at 19:39

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