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That is, if is a function on the real line and is a self-adjoint operator, we wish to define the operator (). The spectral theorem shows that if T {\displaystyle T} is represented as the operator of multiplication by h {\displaystyle h} , then f ( T ) {\displaystyle f(T)} is the operator of multiplication by the composition f ∘ h ...
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]
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).
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 ...
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: = ¯
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).
This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When is defined according to this formula, it is called the formal adjoint of T. A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian. [1]