The spectral theorem is series of results in functional analysis that explore conditions for operators in Hilbert spaces to be diagonalizable (in some appropriate sense). These results can also describe how the diagonalization takes , mainly by analyzing how the operator acts in the underlying Hilbert space.
Roughly speaking, the spectral theorems state that normal operators (or self-adjoint operators) are diagonalizable and can be expressed as a sum or, more generally, as an integral of projections. More specifically, a normal (or self-adjoint) operator is unitarily equivalent to a multiplication operator in some -space (http://planetmath.org/L2SpacesAreHilbertSpaces) and we can to it a spectral measure (or a resolution of identity) whose integration gives is .
There is a wide of spectral theorems, each one with its own , applicable to many classes of normal and self-adjoint operators.
We explore here two to motivate the spectral theorem. The first is by recalling the finite-dimensional case, corresponding to the well known result in linear algebra, the spectral theorem for Hermitian matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices) (or the self-adjoint analog). The second motivation comes from the -algebra (http://planetmath.org/CAlgebra) theory, by regarding a normal (or self-adjoint) operator as a continuous function.
2.0.1 Finite-dimensional case
Suppose is a self-adjoint operator in a finite-dimensional Hilbert space . An important fact about self-adjoint operators (not just in finite-dimensional spaces) is the following:
Proof: Let and . Then , where the last term is zero because is invariant by , i.e. . But this proves that .
Let be an eigenvector of and the subspace generated by it. For self-adjoint transformations, Fact 1 above says that is also invariant by . Thus, by restriction (http://planetmath.org/RestrictionOfAFunction), we have a self-adjoint operator and we could again find an eigenvector and repeat the same . Thus, we are decomposing as a direct sum of orthogonal one-dimensional subspaces , and the operator can be expressed as a sum
where each is the eigenvalue associated with the eigenvector and each is the orthogonal projection onto the subspace .
This is exactly the process of diagonalization of a self-adjoint matrix.
For normal operators it is more subtle as Fact 1 is no longer true. The idea to overpass this is that eigenvectors of normal operators are always orthogonal to each other (see this entry (http://planetmath.org/EigenvaluesOfNormalOperators)).
Suppose is a self-adjoint operator in some Hilbert space . The closed *-algebra generated by and the identity operator is a commutative -algebra (http://planetmath.org/CAlgebra), which we denote by . Hence, the Gelfand-Naimark theorem (http://planetmath.org/GelfandTransform) and the continuous functional calculus provide an isomorphism
where stands for the spectrum of and is the -algebra of continuous functions .
Recall that the spectrum of a self-adjoint operator is a always a compact subset of . Thus, we can think of as a continuous function defined in a subset of .
It is a well known fact from measure theory that every continuous function can be approximated by linear combinations of characteristic functions. With some additional effort it can be seen that each continuous function is in fact a (vector valued) integral of characteristic functions
where is the vector measure of characteristic functions .
We now notice that characteristic functions in are not continuous in general. Hence, they may not have a correspondent in the , the -algebra generated by and the identity. The fact is that they do have a correspondent in the von Neumann algebra generated by . Informally, this is the same as saying that characteristic functions belong to rather then .
The correspondent operators in the von Neumann algebra generated by must be projections (since characteristic functions are projections in ), and similarly, can be approximated by linear combinations of projections and can, in fact, be expressed as an integral of projections:
where is a resolution of identity for (or a projection valued measure, when is a normal operator).
3 Spectral Theorems
Here we list a series of spectral theorems, applicable to different classes of normal or self-adjoint operators.
3.0.1 Normal Operators
spectral theorem for Hermitean matrices (http://planetmath.org/SpectralTheoremForHermitianMatrices)
This is the finite-dimensional case. It is often referred to as ”the spectral theorem”, especially in linear algebra.
spectral theorem for bounded (http://planetmath.org/OperatorNorm) normal operators in separable Hilbert spaces
spectral theorem for bounded normal operators in inseparable Hilbert spaces
spectral theorem for compact (http://planetmath.org/CompactOperator) normal operators
spectral theorem for unitary operators
spectral theorem for unbounded (http://planetmath.org/OperatorNorm) normal operators
3.0.2 Self-adjoint Operators
spectral theorem for self-adjoint matrices
spectral theorem for bounded self-adjoint operators in separable Hilbert spaces
spectral theorem for bounded self-adjoint operators in inseparable Hilbert spaces
spectral theorem for self-adjoint operators
spectral theorem for unbounded self-adjoint operators
|Date of creation||2013-03-22 18:04:30|
|Last modified on||2013-03-22 18:04:30|
|Last modified by||asteroid (17536)|