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In mathematics, a self-adjoint operator on a complex vector space V with inner product , is a linear map A (from V to itself) that is its own adjoint. That is, A x , y = x , A y {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle } for all x , y {\displaystyle x,y} ∊ V .
For example, the space H can be decomposed as the orthogonal direct sum of two T–invariant closed linear subspaces: the kernel of T, and the orthogonal complement (ker T) ⊥ of the kernel (which is equal to the closure of the range of T, for any bounded self-adjoint operator). These basic facts play an important role in the proof of the ...
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.
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 ().
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).
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}} .
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every (), , and , , where is the domain of .