The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.

For complex numbers $a_1,\ldots,a_n$, I will use the notation

$$ \mbox{circ}(a_1,\ldots,a_n)=\left(\begin{array}{llll} a_1& a_2 & \cdots & a_{n-1} & a_n \\ a_n& a_1 & \cdots & a_{n-2} & a_{n-1} \\ \vdots & \vdots & \ddots & \vdots &\vdots\\ a_3& a_4 & \cdots & a_1 & a_2\\ a_2& a_3 & \cdots & a_{n} & a_1\\ \end{array}\right). $$ I am interested in their operator $p$-norms (not to be confused with their Schatten $p$-norms). The operator $p$-norm on $M_n$ is the norm induced by regarding each $n\times n$ matrix as an operator on $\ell^p(\{1,\ldots,n\})\cong \mathbb{C}^n$ in the usual way. Such norms are not invariant under multiplication by unitary matrices (except when $p=2$).

Ideally I would like an explicit formula for the operator $p$-norm of $\mbox{circ}(a_1,\ldots,a_n)$ in terms of $a_1,\ldots,a_n$, but this is probably too much to ask for. I've looked at the available literature and I haven't been able to find anything in this direction (although there is a lot on other norms).

Question 1:Can the $p$-norms of at least some (non-trivial) circulant matrices be computed more or less explicitly?

If they can't be computed explicitly in general, maybe one can say something about the following:

Question 2:Suppose $a$ is a circulant matrix and $\|a\|_1>\|a\|_2$. Does it follow that $\|a\|_p>\|a\|_2$ for every $p \in (1,2)$?

In reference to question 2 (which is the one I'm most interested in): it is known that if $a$ is a circulant matrix, then $\|a\|_p=\|a\|_q$ whenever $p$ and $q$ are conjugate exponents, and that $\|a\|_p\geq \|a\|_2$ for all $p$. The function $p\mapsto \|a\|_p$ is log-convex and decreasing on $[1,2]$ by Riesz-Thorin. So to answer question 2 affirmatively, one would have to argue why $\|a\|_p$ can't be constant on some neighborhood of 2 without being constant everywhere.

As a very special case, set $\omega=e^{2\pi i/n}$ and consider the matrix

$$ a=\mbox{circ}(2,1+\omega^{n-1},1+\omega^{n-2},\ldots,1+\omega)=\left(\begin{array}{llll} 2& 1+\omega^{n-1} & \cdots & 1+\omega^{2} & 1+\omega \\ 1+\omega& 2 & \cdots & 1+\omega^{3} & 1+\omega^{2} \\ \vdots & \vdots & \ddots & \vdots &\vdots\\ 1+\omega^{n-2}& 1+\omega^{n-3} & \cdots & 2 & 1+\omega^{n-1}\\ 1+\omega^{n-1}& 1+\omega^{n-2} & \cdots & 1+\omega & 2\\ \end{array}\right). $$

One checks that $\|a\|_1>\|a\|_2$. Is it true that $\|a\|_p>\|a\|_2$ for the other values of $p$?

Finally, I would like to point out that it is most likely the case that if one omits the assumption that $a$ be a circulant matrix in question 2, then the answer is in general "no".