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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.
Linear Operators is a three-volume textbook on the theory of linear operators, written by Nelson Dunford and Jacob T. Schwartz. The three volumes are (I) General Theory; (II) Spectral Theory, Self Adjoint Operators in Hilbert Space; and (III) Spectral Operators. The first volume was published in 1958, the second in 1963, and the third in 1971.
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 ...
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.
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 ().
The operator C can be defined by C(Bh) = Ah, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B). The operator C is well-defined since A*A ≤ B*B implies Ker(B) ⊂ Ker(A). The lemma then follows. In particular, if A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) ⊂ Ker(C).
The set of self-adjoint elements is a real linear subspace of . From the previous property, it follows that A {\displaystyle {\mathcal {A}}} is the direct sum of two real linear subspaces, i.e. A = A s a ⊕ i A s a {\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}} .