nLab operator topology

Idea

By an operator topology one means a topology (the structure of a topological space) on the set of continuous linear operators between topological vector spaces.

In other words, given a sequence of linear operators $(T_n)_{n \in \mathbb{N}}$, an operator topology determines whether this converges to a given limit operator $T \;\colon\; V \longrightarrow W$.

There are three key examples:

1. the strong operator topology requires that for each vector $v \in V$ the sequence of image vectors $T_n(v)$ converges to $T(v)$ in $W$,

2. the weak operator topology requires less: namely that for every $v \in V$ and every linear functional $\omega$ on $W$, the sequence of numbers $\omega\big(T_n(v)\big)$ converges to $\omega\big(T(v)\big)$,

3. the norm topology requires more (assuming now that $V$ and $W$ are equipped with norms $\vert-\vert$), namely that the supremum over all unit norm vectors, $\vert v \vert = 1$, of the norm $\vert T_n(v) - T(v)\vert$ converges to zero.

Definition

Let

• $k$ denote the ground field, assumed normed,

• $V$, $W$ be topological vector space,

• $L(V,W) = Hom_{TVS}(V,W)$ denote the set of continuous linear operators between them.

Definition

The weak operator topology on $L(V,W)$ is given by the basis of open neighborhoods of zero given by subsets of the form

$$U(v,f) \;=\; \Big\{ T\in L(V,W) \,\Big\vert\, \omega\big(T(v)\big) \vert \lt 1 \Big\} \mathrlap{\,,}$$

where $v\in V$ and $\omega \in W^* = Hom_{TVS}(W,k)$.

Definition

The strong operator topology on $L(V,W)$ is given by the basis of open neighborhoods of zero is given by subsets of the form

$$N(v,U) \;=\; \Big\{ T \in L(V,W) \,\Big|\, T(v) \in U \Big\} \,,$$

where $v\in V$ and $U$ is a neighborhood of zero in $W$.

Definition

Assuming that $V,W$ are normed vector spaces with norms $p_V$, $p_W$ then the uniform operator topology or norm topology on $L(V,W)$ is induced by the norm given by the formula

$$p(T) \;=\; sup_{v\neq 0} \frac{p_W(T v)}{p_V (v)} \mathrlap{\,.}$$

Examples

The unitary group

The reason that in the definition of a unitary representation, the strong operator topology on $\mathcal{U}(\mathcal{H})$ is used and not the norm topology, is that only few group homomorphisms turn out to be continuous in the norm topology.

For instance, let

• $G$ be a compact Lie group,

• $L^2(G)$ denote the Hilbert space of square integrable measurable functions with respect to its Haar measure,

then the (right) regular representation of $G$ on $L^2(G)$, defined as

$$\begin{array}{ccc} G &\longrightarrow& \mathcal{U}\big(L^2(G)\big) \\ g &\mapsto& \big( U(-) \mapsto U( - \cdot g) \big) \mathrlap{\,,} \end{array}$$

is generally not continuous in the norm topology, but is always continuous in the strong operator topology.

(The exception is of course when $G$ is discrete, hence finite.)

Related entries

• norm topology

• U(ℋ)

References

• A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

See also:

• Wikipedia: Operator topologies