# Operator topology

A topology on the space $L( E, F )$ of continuous linear mappings from one topological vector space $E$ into another topological vector space $F$, converting the space $L( E, F )$ into a topological vector space. Let $F$ be a locally convex space and let $\mathfrak S$ be a family of bounded subsets of $E$ such that the linear hull of the union of the sets of this family is dense in $E$. Let $\mathfrak B$ be a basis of neighbourhoods of zero in $F$. The family

$$M( S, V) = \{ {f } : {f \in L( E, F ), f( S) \subset V } \} ,$$

where $S$ runs through $\mathfrak S$ and $V$ through $\mathfrak B$, is a basis of neighbourhoods of zero for a unique topology that is invariant with respect to translation, which is an operator topology and which converts the space $L( E, F )$ into a locally convex space; this topology is called the $\mathfrak S$- topology on $L( E, F )$.

Examples. I) Let $E, F$ be locally convex spaces. 1) Let $\mathfrak S$ be the family of all finite subsets in $E$; the corresponding $\mathfrak S$- topology (on $L( E, F )$) is called the topology of simple (or pointwise) convergence. 2) Let $\mathfrak S$ be the family of all convex balanced compact subsets of $E$; the corresponding topology is called the topology of convex balanced compact convergence. 3) Let $\mathfrak S$ be the family of all pre-compact subsets of $E$; the corresponding $\mathfrak S$- topology is called the topology of pre-compact convergence. 4) Let $\mathfrak S$ be the family of all bounded subsets; the corresponding topology is called the topology of bounded convergence.

II) If $E, F$ are Banach spaces considered simultaneously in the weak or strong (norm) topology, then the corresponding spaces $L( E, F )$ coincide algebraically; the corresponding topologies of simple convergence are called the weak or strong operator topologies on $L( E, F )$. The strong operator topology majorizes the weak operator topology; both are compatible with the duality between $L( E, F )$ and the space of functionals on $L( E, F )$ of the form $f( A) = \sum \phi _ {i} ( A \xi _ {i} )$, where $\xi _ {i} \in E$, $\phi _ {i} \in F ^ { * }$, $A \in L( E, F )$.

III) Let $E, F$ be Hilbert spaces and let $\widetilde{E} , \widetilde{F}$ be countable direct sums of the Hilbert spaces $E _ {n} , F _ {n}$, respectively, where $E _ {n} = E$, $F _ {n} = F$ for all integer $n$; let $\psi$ be the imbedding of the space $L( E, F )$ into $L( \widetilde{E} , \widetilde{F} )$ defined by the condition that for any operator $A \in L( E, F )$ the restriction of the operator $\psi ( A)$ to the subspace $E _ {n}$ maps $E _ {n}$ into $F _ {n}$ and coincides on $E _ {n}$ with the operator $A$. Then the complete pre-image in $L( E, F )$ of the weak (strong) operator topology on $L( \widetilde{E} , \widetilde{F} )$ is called the ultra-weak (correspondingly, ultra-strong) operator topology on $L( E, F )$. The ultra-weak (ultra-strong) topology majorizes the weak (strong) operator topology. A symmetric subalgebra $\mathfrak A$ of the algebra $L( E)$ of all bounded linear operators on a Hilbert space $E$, containing the identity operator, coincides with the set of all operators from $L( E)$ that commute with each operator from $L( E)$ that commutes with all operators from $\mathfrak A$, if and only if $\mathfrak A$ is closed in the weak (or strong, or ultra-weak, or ultra-strong) operator topology, i.e. is a von Neumann algebra.

How to Cite This Entry:
Operator topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_topology&oldid=48050
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article