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:

$$x\mapsto y=Ax$$


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 $(\leq\!p,\leq\!q)\in S$.


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