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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.
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.
3 languages. Deutsch; Français ... an element of a *-algebra is called self-adjoint if it is the same as its adjoint ... Operator Algebras. Theory of C*-Algebras and ...
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
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]
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 ...
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).
Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator. By contrast, the article spectral theorem qualifies this further, stating that any bounded self-adjoint operator is unitarily equivalent to a multiplication operator. So, does an operator need ...