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In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.
The set of self-adjoint elements is a real linear subspace of . From the ... Self-adjoint matrix; Self-adjoint operator; Notes References. Blackadar, Bruce (2006). ...
The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators, (,), satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on ().
An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of ) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all.
Many authors define a positive operator to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity.
Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint).
We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.
A is a non-negative self-adjoint operator such that T 1 =A - I extends T. T 1 is the Friedrichs extension of T. Another way to obtain this extension is as follows. Let :: be the bounded inclusion operator. The inclusion is a bounded injective with dense image.