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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.
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.
3 languages. Deutsch; Français ... an element of a *-algebra is called self-adjoint if it is the same as its adjoint ... Self-adjoint matrix; Self-adjoint operator ...
Example.Multiplication by a non-negative function on an L 2 space is a non-negative self-adjoint operator.. Example.Let U be an open set in R n.On L 2 (U) we consider differential operators of the form
An operator is called essentially self-adjoint if its closure is self-adjoint. [40] An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension. [24] A symmetric operator may have more than one self-adjoint extension, and even a continuum of them. [26] A densely defined, symmetric operator T is ...
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 ().
A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define if the following hold: and are self-adjoint; It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces. [2]
A compact operator K on H is symmetrizable if there is a bounded self-adjoint operator S on H such that S is positive with trivial kernel, i.e. (Sx,x) > 0 for all non-zero x, and SK is self-adjoint: =. In many applications S is also compact. The operator S defines a new inner product on H