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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.
Download QR code; Print/export ... an element of a *-algebra is called self-adjoint if it is the same as its ... Operator Algebras. Theory of C*-Algebras and von ...
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.
M. G. Krein has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator T. If T, S are non-negative self-adjoint operators, write if, and only if, (/) (/)
A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define B ≥ A {\displaystyle B\geq A} if the following hold: A {\displaystyle A} and B {\displaystyle B} are self-adjoint
(For general, non-self-adjoint operators on Banach spaces, by definition, a complex number is in the discrete spectrum if it is a normal eigenvalue; or, equivalently, if it is an isolated point of the spectrum and the rank of the corresponding Riesz projector is finite.)
Download QR code; Print/export Download as PDF; ... von Neumann's theorem is a result in the operator theory of linear operators on Hilbert ... and it is self-adjoint.