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I was reading Kato's book on Perturbations of Linear Operators and have the following questions:

  1. If we have a self-adjoint operator, what kinds of perturbations (other than relatively bounded ones) will result in self-adjoint operators?

  2. If we have a negative-semidefinite operator, what kinds of perturbations will result in negative-semidefinite operators?

I realize my question is kind of vague. I do not have a specific aim in mind, I am just trying to gain a better understanding of what perturbations can do to unbounded operators.

Thanks in advance.

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It depends in which sense you understand "sum". Usually Little can be said.

If you think about generalized sums, then there is a lot where you do not need relative boundedness (i.e., inclusions of the domains).

As you are reading Kato, you can find a lot about the "form sum", which is a far reaching generalization. It has a Connection to the Lie-Trotter product formula, see

T. Kato, "Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups" I. Gohberg (ed.) M. Kac (ed.) , Topics in functional analysis , Acad. Press (1978) pp. 185–195

meaning that using the Lie-Trotter product formula a lot can be said about the sum of two negative semidefinite Operators.

Based on this or on the construction of the Friedrichs Extension, a lot has been done in the context of schrödinger Operators, see for example the recent paper by Rostyslav O. Hryniv, Yaroslav V. Mykytyuk.

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You might take a look at Reed and Simon, "Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness", sections X.2 to X.6.

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