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.

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the norm topology</em>}}⟺\parallel T_{\alpha }-T\parallel ⟶0$$

0.2 2. Strong Operator Topology

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

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the strong operator topology</em>}}⟺\parallel (T_{\alpha }-T)x\parallel ⟶0\mathit{\hspace{1em}},\forall x\in X$$

0.3 3. Weak Operator Topology

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

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the weak operator topology</em>}}⟺\parallel f((T_{\alpha }-T)x)\parallel ⟶0\mathit{\hspace{1em}},\forall x\in X,\forall f\in X^{*}$$

$$

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

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the weak operator topology</em>}}⟺|⟨(T_{\alpha }-T)x,y⟩|⟶0\mathit{\hspace{1em}},\forall x,y\in X$$

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 $\parallel \cdot {\parallel }_S,S\in K(X)$, defined by ${\parallel T\parallel }_S:=\parallel TS\parallel$. That means

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the <xmath xmlns="http://dlmf.nist.gov/LaTeXML"> <xmtok name="sigma" role="UNKNOWN">σ</xmtok> </xmath>-strong operator topology</em>}}⟺\parallel (T_{\alpha }-T)S\parallel ⟶0\mathit{\hspace{1em}},\forall S\in K(X)$$

$$

Equivalently, $T_{\alpha }⟶T⟺T_{\alpha }S⟶TS$ 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

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the <xmath xmlns="http://dlmf.nist.gov/LaTeXML"> <xmtok name="sigma" role="UNKNOWN">σ</xmtok> </xmath>-weak operator topology</em>}}⟺|Tr[(T_{\alpha }-T)S]|⟶0\mathit{\hspace{1em}},\forall S\in B{(X)}_{*}$$

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 $\parallel \cdot {\parallel }_x,x\in X$ defined by ${\parallel T\parallel }_x:=\parallel Tx\parallel +\parallel T^{*}x\parallel$. That means

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the strong-* operator topology</em>}}⟺\parallel (T_{\alpha }-T)x\parallel +\parallel (T_{\alpha }^{*}-T^{*})x\parallel ⟶0\mathit{\hspace{1em}},\forall x\in X$$

Equivalently, $T_{\alpha }⟶T$ if and only if $T_{\alpha }x⟶Tx$ and $T_{\alpha }^{*}x⟶T^{*}x$, 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 $\parallel \cdot {\parallel }_S,S\in K(X)$ defined by ${\parallel T\parallel }_S:=\parallel TS\parallel +\parallel T^{*}S\parallel$. That means

$$T_{\alpha }⟶T\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">in the <xmath xmlns="http://dlmf.nist.gov/LaTeXML"> <xmtok name="sigma" role="UNKNOWN">σ</xmtok> </xmath>-strong-* operator topology</em>}}⟺\parallel (T_{\alpha }-T)S\parallel +\parallel (T_{\alpha }^{*}-T^{*})S\parallel ⟶0\mathit{\hspace{1em}},\forall S\in K(X)$$

$$

Equivalently, $T_{\alpha }⟶T$ if and only if $T_{\alpha }S⟶TS$ and $T_{\alpha }^{*}S⟶T^{*}S$ 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 $𝒰,𝒱$ the notation $𝒰\to 𝒱$ means $𝒰$ is weaker than $𝒱$:

Title	operator topologies
Canonical name	OperatorTopologies
Date of creation	2013-03-22 17:22:04
Last modified on	2013-03-22 17:22:04
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	18
Author	asteroid (17536)
Entry type	Definition
Classification	msc 54E99
Classification	msc 47L05
Classification	msc 46A32
Related topic	OperatorNorm
Defines	strong operator topology
Defines	weak operator topology
Defines	$\sigma$-weak operator topology
Defines	$\sigma$-strong operator topology
Defines	strong-* operator topology
Defines	$\sigma$-strong-* operator topology
Defines	ultra-strong operator topology
Defines	ultra-weak operator topology
Defines	ultra-stro