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I was triying to make an interpretation of the all-one matrices in Hilbert spaces. It has to be defined in the infinite-dimensional setting, I don't know if there is something like this...

fina
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    Choose a Hilbert basis $e_i$. An analog of the matrix you describe must map the vector $\sum_i x_i\ e_i$ to $\sum_i (\sum_j x_j) e_i$, ie sum up all the components and then map it to the vector where every component has this sum as its value. The square norm of such a vector is then $\sum_i (\sum_j x_j)^2$ which is $\infty$ if $\sum_j x_j\neq0$. Hence a generalisaiton of htis form is not sensible. – s.harp Dec 06 '20 at 14:14
  • That's what I thought, then it should be defined in a subspace of $H$? @s.harp – fina Dec 06 '20 at 14:16
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    By the argument you can only define it on the subspace of those vectors whose components sum to $0$ and on this subspace its equal to the zero operator. – s.harp Dec 06 '20 at 14:17

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