Under what conditions is an infinite matrix a bounded linear transformation in $\ell^2$?

Question

Let $(e_n)_{n\in\mathbb N}$ be the canonical basis of $\ell^2$, and $\mathcal L(\ell^2)$ be the set of bounded linear transformations from $\ell^2$ to itself. If $A\in\mathcal L(\ell ^2)$ we can set

$$a_{ij}=\left<Ae_i, e_j\right>$$

so that $A=(a_{ij})_{i,j\in\mathbb N}$ looks just like an infinite matrix. I'm interested in determining necessary and sufficient conditions for an infinite matrix to be in $\mathcal L(\ell^2)$. It's straightforward to show that if a row or a column of $A$ is not square summable, then you can choose an element $x\in\ell^2$ so that $Ax$ doesn't converge. Also if

$$\sup_{n\in\mathbb N} \frac 1n\sum_{i,j\leq n}|a_{ij}|^2=\infty$$

for every $N\in\mathbb R$ you can choose an $n$ and $j_0\leq n$ so that $\sum_{i\leq n} |a_{ij_0}|^2>N$ so that $\lVert A e_{j_0}\rVert>N$. So my guess is that

$$A=(a_{ij})_{i,j\in\mathbb N}\in \mathcal L(\ell^2)\iff \sup_{n\in\mathbb N} \frac 1n\sum_{i,j\leq n}|a_{ij}|^2<\infty\text{ and } \sup_{j_0\in\mathbb N}\lVert(a_{ij_0})_i\rVert<\infty,\ \sup_{i_0\in\mathbb N}\lVert(a_{i_0j})_j\rVert<\infty.$$

That is, an infinite matrix is a bounded linear transformation in $\ell^2$ if and only if the $\ell^2$ norm of its rows and columns is uniformly bounded and the quantity $\frac 1n\sum_{i,j\leq n}|a_{ij}|^2$ is bounded. Is that correct?

Answer 1

Let $A_n$ denote the $n\times n$ matrix with entries $n^{-1/2}.$ Then the $\ell^2$ norms of the columns and rows are equal $1.$ The matrix $A_n$ satisfies $A_n^2=n^{1/2}A_n.$ Therefore $\|A_n\|=n^{1/2}.$ Let $A$ be the direct sum of the matrices $A_n,$ i.e. $$A=\begin{pmatrix} A_1 & 0 & 0 &\ldots \\ 0 & A_2 & 0 &\ldots \\ 0& 0 & A_3 & \ldots \\ \vdots & \vdots &\vdots & \ddots \end{pmatrix} $$ Then $A$ is unbounded, but its columns and rows have $\ell^2$ norms equal $1.$

Answer 2

(a) Necessary and sufficient conditions for the matrix $A$ to define a map from $l^2$ to $l^2$? This is a hard problem.

(b) Necessary and sufficient conditions for the matrix $A$ to define a bounded linear transformation from $l^2$ to $l^2$? This is the same as (a), according to the closed graph theorem.

There is no known simple answer to (a). In particular, I would guess the criterion proposed by the OP is wrong. If so, a counterexample could be given as an answer.