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Consider the n-by-n matrix that has $n^{-1/2}$ as every entry. That satisfies your condition with B=1. The image of the unit vector that has $n^{-1/2}$ as every coordinate is a vector that has 1 as every coordinate, and therefore norm $n^{1/2}$.

If you now put a whole lot of these as blocks down the diagonal, you can create an unbounded operator that satisfies your condition with B=1.

I'd say that the main general problem with the condition you suggested is that it is too tied to one particular basis. I'm not sure it's all that easy to come up with nice conditions of the kind you are looking for.

Additional remark: if you take spaces like ell_1 and ell_infinity, where the definition of the norm is much more closely tied to a particular basis, then it tends to be easier to find nice matrix conditions for boundedness.

1

Consider the n-by-n matrix that has $n^{-1/2}$ as every entry. That satisfies your condition with B=1. The image of the unit vector that has $n^{-1/2}$ as every coordinate is a vector that has 1 as every coordinate, and therefore norm $n^{1/2}$.

If you now put a whole lot of these as blocks down the diagonal, you can create an unbounded operator that satisfies your condition with B=1.

I'd say that the main general problem with the condition you suggested is that it is too tied to one particular basis. I'm not sure it's all that easy to come up with nice conditions of the kind you are looking for.