A relation between norm and spectral radius for some matrix operators on Banach spaces $\ell^{p}$

Question

Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote further the spectral radius of $A^{T}A$ as $$ r_{p}(A^{T}A):=\sup\{|\lambda| \mid \lambda\in\sigma(A^{T}A)\}, $$ where $\sigma(A^{T}A)$ is the spectrum of $A^{T}A$ regarded as an operator on $\ell^{p}$.

It is well-known that, if $p=2$, then $r_{2}(A^{T}A)=\|A\|_{\ell^{2}\to\ell^{2}}^{2}$ since $A^{T}A$ is self-adjoint and $\|A^{T}A\|_{\ell^{2}\to\ell^{2}}=\|A\|_{\ell^{2}\to\ell^{2}}^{2}$.

Is there an $\ell^{p}$-variant of the equality for general $p\geq1$? That is, can be the spectral radius $r_{p}(A^{T}A)$ related to the norm $\|A\|_{\ell^{p}\to\ell^{p}}$ or perhaps to $\|A\|_{\ell^{p}\to\ell^{p}}$ and $\|A^{T}\|_{\ell^{q}\to\ell^{q}}$, where $q$ is the dual conjugate to $p$ ($1/q+1/p=1$)?

A modification of the claim or imposing additional conditions on $A$ is possible. I do not have a clear picture about the relation. I just would like to know if there are certain results of this kind. Thank you.

Answer 1

If $A,A^T=\ell^p\rightarrow\ell^p$, then the adjoints $A^T,A$ map $\ell^{p'}$ into itself. By interpolation (Riesz-Thorin), they map $\ell^2$ into itself. It will be often the case that the spectrum of $A^TA$ is not sensitive to the exponent $p$, in which case one will have $r_p(A^TA)=r_2(A^TA)=\|A\|_2\ne\|A\|_p$.

Answer 2

In general, the spectral radius of a bounded operator is bounded by its norm, i.e. $$ r_p(A^T A) \leq \Vert A^T A\Vert_{\ell^p\to\ell^p},$$ therefore you have the inequality $$ r_p(A^T A) \leq \Vert A^T \Vert_{\ell^p\to\ell^p}\ \Vert A \Vert_{\ell^p\to\ell^p}. $$