Essential self - adjointness

1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a prototypical natural unbounded operator. It is undeniably not continuous in the L 2 topology: on L 2 [0, 1] the norm of f (x) = x n is 1/ √ 2n + 1, and the second derivative of x n is n(n − 1)x n−2 , so operator norm d 2 dx 2 on L 2 [0, 1] ≥ sup n≥1 n(n − 1) · 1 √ 2n−3 1 √ 2n+1 = +∞ That is, d 2 dx 2 is not a L 2-bounded operator on polynomials on [0, 1], so has no bounded extension [1] to L 2 [0, 1]. Nevertheless, the geometric structure of Hilbert spaces is extremely useful, especially the simple duality and adjoint phenomena. This motivates reconsideration of unbounded, not-everywhere-defined, but densely defined operators on Hilbert spaces. [2] Let V be a Hilbert space, with hermitian inner product , and corresponding norm | · |. Let T be an unbounded linear map T : D T → V defined on a dense subspace D T of V. We say that T is on V , even though its domain may be strictly smaller. We are interested in symmetric operators, meaning that T v, w = v, T w (for all v, w ∈ D T) For unbounded operators, specification of the domain is critical. In this notation, in terms of the adjoint [3] T * of T ,    T symmetric ⇐⇒ T ⊂ T * T self-adjoint ⇐⇒ T = T * (for densely-defined T) [1] Whether or not the Axiom of Choice is used to artificially extend d 2 dx 2 to L 2 [0, 1], that extension is not continuous, because the restriction to polynomials is already not continuous. The unboundedness/non-continuity is inescapable. [2] Alternatively, one might allow more complicated topologies than that of a single Hilbert space, as did Friedrichs, Sobolev, Schwartz, and Grothendieck. In fact, a combination of approaches seems optimal. [3] For an unbounded operator T , symmetric or not, to have a well-defined adjoint T * requires the domain D …