PlanetMath: operator topologies

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operator topologies

(Definition)

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

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

1. Norm Topology

This is the topology induced by the usual operator norm.

$\displaystyle T_{\alpha} \longrightarrow T$ in the norm topology $\displaystyle \;\; \Longleftrightarrow \; \Vert T_{\alpha} - T \Vert \longrightarrow 0$

2. Strong Operator Topology

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

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

3. Weak Operator Topology

This is the topology generated by the family of semi-norms $\Vert \cdot \Vert _{f,x}\;$ , where $x \in X$ and

is a linear functional of

(written $f\in X^*$ , the dual vector space of

), defined by $\Vert T \Vert _{f,x} := \vert f(Tx)\vert$ . That means

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

$\,$

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

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

4. $\sigma$ -Strong Operator Topology

In this topology

must be a Hilbert space. Let

denote the space of compact operators on

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

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

$\,$

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.

5. $\sigma$ -Weak Operator Topology

In this topology

must be a Hilbert space. Let

denote the space of trace-class operators on

and

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 $\omega_{S}(T) := \vert Tr(TS)\vert$ . That means

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

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

6. Strong-* Operator Topology

In this topology

must be a Hilbert space. In the following

denotes the adjoint operator of

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

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

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

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

In this topology

must be a Hilbert space. Let

denote the space of compact operators on

. In the following

denotes the adjoint operator of

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

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

$\,$

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.

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 $\mathcal{V}$ :
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ weak \ar[r] \ar[d] & strong \ar[... ...ph{$\sigma$-strong} \ar[r] & \emph{$\sigma$-strong-*} \ar[r] & Norm} } \end{xy}$

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