Complex analysis and operator theory

In functional analysis, a branch of mathematics, a compact operator is a linear operator mwe-math-element>, where mwe-math-element> are normed vector spaces, with the property that mwe-math-element> maps bounded subsets of mwe-math-element> to relatively compact subsets of mwe-math-element>.
Such an operator is necessarily a bounded operator, and so continuous.
Some authors require that mwe-math-element> are Banach, but the definition can be extended to more general spaces.

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.

In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1.
This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling.
The analysis of contractions provides insight into the structure of operators, or a family of operators.
The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.

In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations.
They are named in honour of Erik Ivar Fredholm.
By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel mwe-math-element> and finite-dimensional (algebraic) cokernel mwe-math-element>, and with closed range mwe-math-element>.
The last condition is actually redundant.

In operator theory, a multiplication operator is an operator texhtml >Tf defined on some vector space of functions and whose value at a function texhtml mvar style=font-style:italic>φ is given by multiplication by a fixed function texhtml mvar style=font-style:italic>f.
That is,

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.

In mathematics, the operator norm measures the size of certain linear operators by assigning each a real number called its operator norm.
Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Informally, the operator norm mwe-math-element> of a linear map mwe-math-element> is the maximum factor by which it lengthens vectors.

In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator.
Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space.

In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function texhtml >x ↦ f(x)
to its translation texhtml >x ↦ f(x + a).
In time series analysis, the shift operator is called the lag operator.