# Which entrywise $p,q$-norms are comparable to the $\ell^2$ induced operator norm?

Consider an infinite-dimensional matrix $$[A_{ij}\mid i,j\in\mathbb N]$$, which corresponds to a linear operator $$A:c_{00}\to\mathbb R^\mathbb N$$ from finite sequences to arbitrary sequences:

$$x\mapsto y=Ax$$

$$y_i=\sum_jA_{ij}x_j$$

We can assume that $$A$$ is symmetric ($$A_{ij}=A_{ji}$$) and positive-semidefinite ($$\sum_ix_iy_i\geq0$$).

We define the usual $$\ell^p$$ norms on sequences and matrices, and the $$\ell^2$$ induced operator norm, with values in the extended interval $$[0,+\infty]$$:

$$\lVert x\rVert_p=\left(\sum_i|x_i|^p\right)^{1/p},\qquad\lVert x\rVert_\infty=\sup_i|x_i|$$

$$\lVert A\rVert_{p,q}=\left(\sum_j\left(\sum_i|A_{ij}|^p\right)^{q/p}\right)^{1/q}$$

$$\lVert A\rVert_{\text{op}}=\sup_{0\neq x\in c_{00}}\frac{\lVert Ax\rVert_2}{\lVert x\rVert_2}$$

For which values of $$(p,q)$$ does $$\lVert A\rVert_{\text{op}}<\infty$$ imply $$\lVert A\rVert_{p,q}<\infty$$? (These are necessary for $$A$$ to be bounded.)

$$N=\{(p,q)\in[1,+\infty]^2\mid\forall A,\lVert A\rVert_{\text{op}}<\infty\implies\lVert A\rVert_{p,q}<\infty\}$$

For which values of $$(p,q)$$ does $$\lVert A\rVert_{p,q}<\infty$$ imply $$\lVert A\rVert_{\text{op}}<\infty$$? (These are sufficient for $$A$$ to be bounded.)

$$S=\{(p,q)\in[1,+\infty]^2\mid\forall A,\lVert A\rVert_{p,q}<\infty\implies\lVert A\rVert_{\text{op}}<\infty\}$$

From my previous question, it is expected that $$N\cap S=\{\}$$. It is known that $$N$$ contains $$(2,\infty)$$, and that $$S$$ contains $$(2,2)$$ and $$(1,\infty)$$ and no other $$(p,\infty)$$. Clearly if $$(p,q)\in N$$, then $$(\geq\!p,\geq\!q)\in N$$; and if $$(p,q)\in S$$, then $$(\leq\!p,\leq\!q)\in S$$.