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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 ...
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 ...
That is, if K is a compact self-adjoint operator on X, then the essential spectra of T and that of + coincide. 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.
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).
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 ().
In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator.It is named after the mathematician Kurt Friedrichs.