Timeline for When the adjoint of an unbounded operator on a Hilbert space coincides with the formal adjoint on its natural domain?
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| Jan 10, 2021 at 19:35 | comment | added | Christian Remling | @apyshkin: Finite band matrices is the same thing, you can build the same type of example with a limit circle Jacobi matrix (= tridiagonal). I'm not completely sure if I understand your other comments correctly, but one remark is that there need not be an obvious "natural domain" for given matrix elements. Of course, if we do have the domain, then everything is determined and we can (in principle, at least) find $A^*$. | |
| Jan 10, 2021 at 0:44 | comment | added | apyshkin | This example is useful, thanks. I understand that there is a freedom in defining the unbounded operator even when we fix its values on some dense subset. Accordingly, the domain of adjoint varies even for the same matrix. What I am trying to find out is whether there are some simple conditions when $A$ is defined on its natural domain and its formal adjoint $A_*$ is also defined on its natural domain. I wonder when these two are adjoint (maybe it is known for simpler cases of finite-band $A_{ij}$). Probably, the original statement of the question is not the most accurate. | |
| Jan 8, 2021 at 22:24 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Jan 8, 2021 at 15:30 | history | answered | Christian Remling | CC BY-SA 4.0 |