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fa.practical evaluation – Is it workable to categorise non-closed subspaces of Hilbert’s house? Answer

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fa.practical evaluation – Is it workable to categorise non-closed subspaces of Hilbert’s house?

Let $H$ breathe Hilbert’s house.

Motivated by my earlier query about wildly discontinuous linear functionals, which can breathe interpreted as an try to classify dense hyperplanes in $H$, let me now refer straight to the purpose:

Questions.

  1. Are there any important variations amongst dense hyperplanes in $H$?

  2. If $L$ and $M$ are two dense hyperplanes in $H$, is there a unitary operator mapping $L$ to $M$?

  3. Assuming the respond to (2) is traverse, what number of orbits are there for the
    unaffected motion of the unitary group $mathscr U(H)$ on the clique of dense hyperplanes?


Speaking about common (not essentially closed or dense) subspaces of $H$, there are some things one could say in that
respect.

For instance, not all such areas could breathe described because the meander of a bounded operator and, specifically, no
dense hyperplane qualifies. This is as a result of, if the meander of such an operator has finite co-dimension, it should breathe
closed (this follows simply from the Closed Graph Theorem).

The meander of a compact operator doesn’t acquire any innumerable dimensional closed subspace, so that’s one other property one might employ
to categorise subspaces.

More Questions.

  1. Is there a needful and adequate situation, expressed in topological/analytical phrases, characterizing the meander of
    a bounded (resp. compact) operator amongst all subspaces of $H$?

  2. How many unitary equivalence lessons of non-closed subspaces of $H$ are there? How many of those could breathe described in
    topological/analytical phrases?

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