# operator topologies

Let $X$ be a normed vector space and $B(X)$ the space of bounded operators in $X$. There are several interesting topologies that can be given to $B(X)$. In what follows, $T_{\alpha}$ denotes a net in $B(X)$ and $T$ denotes an element of $B(X)$.

Note: On 4, 5, 6 and 7, $X$ must be a Hilbert space.

## 0.1 1. Norm Topology

This is the topology induced by the usual operator norm.

 $T_{\alpha}\longrightarrow T\text{\emph{in the norm topology}}\;\;% \Longleftrightarrow\;\|T_{\alpha}-T\|\longrightarrow 0$

## 0.2 2. Strong Operator Topology

This is the topology generated by the family of semi-norms $\|\cdot\|_{x}\;,x\in X$ defined by $\|T\|_{x}:=\|Tx\|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the strong operator topology}}\;\;% \Longleftrightarrow\;\|(T_{\alpha}-T)x\|\longrightarrow 0\quad,\forall x\in X$

## 0.3 3. Weak Operator Topology

This is the topology generated by the family of semi-norms $\|\cdot\|_{f,x}\;$, where $x\in X$ and $f$ is a linear functional of $X$ (written $f\in X^{*}$, the dual vector space of $X$), defined by $\|T\|_{f,x}:=|f(Tx)|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the weak operator topology}}\;\;% \Longleftrightarrow\;\|f((T_{\alpha}-T)x)\|\longrightarrow 0\quad,\forall x\in X% ,\;\forall f\in X^{*}$

$\,$

In case $X$ is an Hilbert space with inner product $\langle\cdot,\cdot\rangle$, we have that

 $T_{\alpha}\longrightarrow T\text{\emph{in the weak operator topology}}\;\;% \Longleftrightarrow\;|\langle(T_{\alpha}-T)x,y\rangle|\longrightarrow 0\quad,% \forall x,y\in X$

## 0.4 4. $\sigma$-Strong Operator Topology

In this topology $X$ must be a Hilbert space. Let $K(X)$ denote the space of compact operators on $X$.

The $\sigma$-strong operator topology is the topology generated by the family of semi-norms $\|\cdot\|_{S}\;,S\in K(X)$, defined by $\|T\|_{S}:=\|TS\|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the \sigma-strong operator % topology}}\;\;\Longleftrightarrow\;\|(T_{\alpha}-T)S\|\longrightarrow 0\quad,% \forall S\in K(X)$

$\,$

Equivalently, $T_{\alpha}\longrightarrow T\;\;\Longleftrightarrow\;T_{\alpha}S\longrightarrow TS$ in norm for every $S\in K(X)$.

This topology is also called the ultra-strong operator topology.

## 0.5 5. $\sigma$-Weak Operator Topology

In this topology $X$ must be a Hilbert space. Let $B(X)_{*}$ denote the space of trace-class operators on $X$ and $Tr(S)$ the trace of an operator $S\in B(X)_{*}$.

The $\sigma$-weak operator topology is the topology generated by the family of semi-norms $\{\omega_{S}:S\in B(X)_{*}\}$ defined by $\omega_{S}(T):=|Tr(TS)|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the \sigma-weak operator topology}% }\;\;\Longleftrightarrow\;|Tr[(T_{\alpha}-T)S]|\longrightarrow 0\quad,\forall S% \in B(X)_{*}$

This topology is also called the ultra-weak operator topology.

## 0.6 6. Strong-* Operator Topology

In this topology $X$ must be a Hilbert space. In the following $T^{*}$ denotes the adjoint operator of $T$.

The strong-* operator topology is the topology generated by the family of semi-norms $\|\cdot\|_{x}\;,x\in X$ defined by $\|T\|_{x}:=\|Tx\|+\|T^{*}x\|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the strong-* operator topology}}\;\;% \Longleftrightarrow\;\|(T_{\alpha}-T)x\|+\|(T_{\alpha}^{*}-T^{*})x\|% \longrightarrow 0\quad,\forall x\in X$

Equivalently, $T_{\alpha}\longrightarrow T$ if and only if $T_{\alpha}x\longrightarrow Tx$ and $T_{\alpha}^{*}x\longrightarrow T^{*}x$, for every $x\in X$.

## 0.7 7. $\sigma$-Strong-* Operator Topology

In this topology $X$ must be a Hilbert space. Let $K(X)$ denote the space of compact operators on $X$. In the following $T^{*}$ denotes the adjoint operator of $T$.

The $\sigma$-strong-* operator topology is the topology generated by the family of semi-norms $\|\cdot\|_{S}\;,S\in K(X)$ defined by $\|T\|_{S}:=\|TS\|+\|T^{*}S\|$. That means

 $T_{\alpha}\longrightarrow T\text{\emph{in the \sigma-strong-* operator % topology}}\;\;\Longleftrightarrow\;\|(T_{\alpha}-T)S\|+\|(T_{\alpha}^{*}-T^{*}% )S\|\longrightarrow 0\quad,\forall S\in K(X)$

$\,$

Equivalently, $T_{\alpha}\longrightarrow T$ if and only if $T_{\alpha}S\longrightarrow TS$ and $T_{\alpha}^{*}S\longrightarrow T^{*}S$ in norm, for every $S\in K(X)$.

This topology is also called ultra-strong-* operator topology.

## 0.8 Comparison of Operator Topologies

• The norm topology is the strongest of the topologies defined above.

• The weak operator topology is weaker than the strong operator topology, which is weaker than the norm topology.

• In Hilbert spaces we can summarize the relations of the above topologies in the following diagram. Given two topologies $\mathcal{U},\mathcal{V}$ the notation $\mathcal{U}\rightarrow\mathcal{V}$ means $\mathcal{U}$ is weaker than $\mathcal{V}$:

 $\xymatrix{weak\ar[r]\ar[d]&strong\ar[r]\ar[d]&\emph{strong-*}\ar[d]\\ \emph{\sigma-weak}\ar[r]&\emph{\sigma-strong}\ar[r]&\emph{\sigma-strong-% *}\ar[r]&Norm}$
Title operator topologies OperatorTopologies 2013-03-22 17:22:04 2013-03-22 17:22:04 asteroid (17536) asteroid (17536) 18 asteroid (17536) Definition msc 54E99 msc 47L05 msc 46A32 OperatorNorm strong operator topology weak operator topology $\sigma$-weak operator topology $\sigma$-strong operator topology strong-* operator topology $\sigma$-strong-* operator topology ultra-strong operator topology ultra-weak operator topology ultra-stro