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 $𝒱$: