Search results
Results from the WOW.Com Content Network
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.
For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator A on a complex Hilbert space H is an conjugate-linear operator A ∗ : H → H with the property:
In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. [1]: 91 This adjoint is sometimes called a reflector, or localization. [2] Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint.
Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras . One may also define a conjugation for quaternions and split-quaternions : the conjugate of a + b i + c j + d k {\textstyle a+bi+cj+dk} is a − b i − c j − d ...
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.
Proof that a common eigenbasis implies commutation. Let {| } be a set of orthonormal states (i.e., | =,) that form a complete eigenbasis for each of the two compatible observables and represented by the self-adjoint operators ^ and ^ with corresponding (real-valued) eigenvalues {} and {}, respectively.
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 ().
For a Hilbert space, T* normally denotes the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H), the Borel functional calculus gives additional ways to break up the spectrum naturally.