Operator theory (nonfiction)

From Gnomon Chronicles

In functional analysis, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study depends heavily on the topology of function spaces.

If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory.

Single operator theory

Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category.

Spectrum of operators

Main article: Spectral theorem

The spectral theorem is any of a number of results about linear operators or about matrices. 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.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.

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.

Normal operators

Main article: Normal operator

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

  • unitary operators:
  • Hermitian operators
  • anti-selfadjoint operators
  • positive operators
  • normal matrices

Polar decomposition

Main article: Polar decomposition (nonfiction)

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.

The existence of a polar decomposition is a consequence of Douglas' lemma:

Connection with complex analysis

Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the study of the operator is intimately linked to questions in function theory. 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.

Operator algebras

The theory of operator algebras brings algebras of operators such as C*-algebras to the fore.

C*-algebras

Main article: C*-algebra

A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map * : A → A.

See also:

Retrieved from "https://gnomonchronicles.com/w/index.php?title=Operator_theory_(nonfiction)&oldid=228430"