For example, the classification of normal operators in terms of their spectra falls into this category.
The spectral theorem is any of a number of results about linear operators or about matrices.
[1] In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces.
In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find.
In more abstract language, the spectral theorem is a statement about commutative C*-algebras.
See also spectral theory for a historical perspective.
The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
A normal operator on a complex Hilbert space
[2] Normal operators are important because the spectral theorem holds for them.
Today, the class of normal operators is well understood.
Examples of normal operators are The spectral theorem extends to a more general class of matrices.
be an operator on a finite-dimensional inner product space.
is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have
is normal if and only if there exists a unitary matrix
Unlike the Hermitian case, the entries of
The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator.
[3] The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P. The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues.
The existence of a polar decomposition is a consequence of Douglas' lemma: Lemma — If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then there exists a contraction C such that A = CB.
The operator C can be defined by C(Bh) = Ah, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B).
where (A*A)1/2 is the unique positive square root of A*A given by the usual functional calculus.
Notice that an analogous argument can be used to show A = P'U' , where P' is positive and U' a partial isometry.
When H is finite dimensional, U can be extended to a unitary operator; this is not true in general (see example above).
By property of the continuous functional calculus, |A| is in the C*-algebra generated by A.
A similar but weaker statement holds for the partial isometry: the polar part U is in the von Neumann algebra generated by A.
For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit disk with unimodular boundary values almost everywhere on the circle.
Beurling interpreted the unilateral shift as multiplication by the independent variable on the Hardy space.
[4] The success in studying multiplication operators, and more generally Toeplitz operators (which are multiplication, followed by projection onto the Hardy space) has inspired the study of similar questions on other spaces, such as the Bergman space.
A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map * : A → A.
For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure: