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.


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.


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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