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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 .
Each positive element of a C*-algebra is self-adjoint. [3] ... is the direct sum of two real linear subspaces, ... Operator Algebras. Theory of C*-Algebras and von ...
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.
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 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 more general continuous functional calculus can be defined for any self-adjoint (or even normal, in the complex case) bounded linear operator on a Hilbert space. The compact case described here is a particularly simple instance of this functional calculus.
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).
That is, if is a compact self-adjoint operator on , then the essential spectra of and that of + coincide, i.e. () = (+). This explains why it is called the essential spectrum : Weyl (1910) originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions.