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

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.

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 $$r_p(A^T A) \leq \Vert A^T \Vert_{\ell^p\to\ell^p}\ \Vert A \Vert_{\ell^p\to\ell^p}.$$
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 $$r_p(A^TA)=r_2(A^TA)=\|A\|_2\ne\|A\|_p$$.