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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.
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint ... Operator Algebras. Theory of C*-Algebras and von Neumann ...
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 ().
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: = ¯
Self-adjoint operators [ edit ] If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator ), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
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 ...
A bounded linear functional on a C*-algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signed measures .
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]