Self-adjoint extensions of symmetric operators
In Sections 5.4 and 5.5 we have already learned that certain symmetric operators (the semi-bounded and continuously invertible ones) possess self-adjoint extensions. The question of whether all (or which) symmetric operators have self-adjoint extensions could not be answered there. The key to our studies was the fact that λ — T was continuously invertible for some λ ∈ ℝ; however, this is not always the case. In this chapter we develop the von Neumann extension theory, which completely answers this question. Moreover, we shall prove certain theorems about the spectra of all self-adjoint extensions of a symmetric operator.
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