Operator theory

Builds on Linear map Normed vector space Continuous linear map Banach space Hilbert space

Definition

Let $X$ and $Y$ be normed spaces, and let $D(T)$ be a linear subspace of $X$ . An operator from $X$ to $Y$ is a linear map $T \colon D(T) \to Y$ . Operator theory is the branch of mathematics that studies operators. Its main objects are the continuous linear maps between Banach spaces and the operators on a Hilbert space. Its central questions concern invertibility, the spectrum, adjoints, invariant subspaces and the classification of operators.

Remarks

Example: the right shift $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ on the space $\ell^2$ of square-summable sequences is an operator. It preserves norms, so it is bounded with norm $1$ . It is injective but not surjective, since every vector in its image has first coordinate $0$ .

Non-example: the map $x \mapsto \norm{x}^2 x$ on $\ell^2$ is not an operator in this sense, since it is not linear. Such maps belong to nonlinear functional analysis.

On a finite-dimensional space, operator theory reduces to the linear algebra of matrices.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Banach space concept
Continuous linear map concept
Hilbert space concept
Linear map concept
Normed vector space concept
Operator theory definition

Tag	`QEKVW55VV7`
URI	`m2://QEKVW55VV7`
Kind	definition
Writing	W0 stub
MSC	47A05, 47B01, 47B02
Defines	Operator theory
Uses	Linear map, Normed vector space, Continuous linear map, Banach space, Hilbert space
Changed	2 November 2026

Operator theory

Definition

Remarks

Reading path

Further reading

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