3
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $$ r_p(A^T A) \leq \Vert A^T \Vert_{\ell^p\to\ell^p}\ \Vert A \Vert_{\ell^p\to\ell^p}. $$

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

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 $r_p(A^TA)=r_2(A^TA)=\|A\|_2\ne\|A\|_p$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.