Operator topologies
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space.
Introduction
Let be a sequence of linear operators on the Banach space. Consider the statement that converges to some operator on.This could have several different meanings:
- If, that is, the operator norm of (the supremum of, where ranges over the unit ball in converges to 0, we say that in the uniform operator topology.
- If for all, then we say in the strong operator topology.
- Finally, suppose that for all we have in the weak topology of. This means that for all linear functionals on. In this case we say that in the weak operator topology.
List of topologies on B(''H'')
The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms.
In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak.
The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.
If is a Hilbert space, the Hilbert space has a predual,
consisting of the trace class operators, whose dual is.
The seminorm for w positive in the predual is defined to be
If is a vector space of linear maps on the vector space, then is defined to be the weakest topology on such that all elements of are continuous.
- The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on. It is stronger than all the other topologies below.
- The weak topology is, in other words the weakest topology such that all elements of the dual are continuous. It is the weak topology on the Banach space. It is stronger than the ultraweak and weak operator topologies.
- The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on such that the dual is, and is also the uniform convergence topology on, -compact convex subsets of. It is stronger than all topologies below.
- The σ-strong* topology or ultrastrong* topology is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms and for positive elements of. It is stronger than all topologies below.
- The σ-strong topology or ultrastrong topology or strongest topology or strongest operator topology is defined by the family of seminorms for positive elements of. It is stronger than all the topologies below other than the strong* topology. Warning: in spite of the name "strongest topology", it is weaker than the norm topology.)
- The σ-weak topology or ultraweak topology or weak* operator topology or weak * topology or weak topology or ) topology is defined by the family of seminorms || for elements w of. It is stronger than the weak operator topology.
- The strong* operator topology or strong* topology is defined by the seminorms ||x|| and ||x*|| for. It is stronger than the strong and weak operator topologies.
- The strong operator topology or strong topology is defined by the seminorms ||x|| for. It is stronger than the weak operator topology.
- The weak operator topology or weak topology is defined by the seminorms || for.
Relations between the topologies
for.
The continuous linear functionals on for the ultraweak, ultrastrong, ultrastrong* and Arens-Mackey topologies are the same, and are the elements of the predual.
By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology.
This dual is a rather large space with many pathological elements.
On norm bounded sets of, the weak and ultraweak topologies coincide. This can be seen via, for instance, the Banach–Alaoglu theorem.
For essentially the same reason, the ultrastrong
topology is the same as the strong topology on any bounded subset of.
Same is true for the Arens-Mackey topology, the ultrastrong*, and the strong* topology.
In locally convex spaces, closure of convex sets can be characterized by the continuous linear functionals. Therefore, for a convex subset of, the conditions that be closed in the ultrastrong*, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that
for all, has closed intersection with the closed ball of radius in the strong*, strong, or weak topologies.
The norm topology is metrizable and the others are not; in fact they fail to be first-countable.
However, when is separable, all the topologies above are metrizable when restricted to the unit ball.
Which topology should I use?
The most commonly used topologies are the norm, strong, and weak operator topologies.The weak operator topology is useful for compactness arguments, because the unit ball is compact by the Banach–Alaoglu theorem.
The norm topology is fundamental because it makes into a Banach space, but it is too strong for many purposes; for example, is not separable in this topology.
The strong operator topology could be the most commonly used.
The ultraweak and ultrastrong topologies are better-behaved than the weak and strong operator topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed.
For example, the dual space of in the weak or strong operator topology is too small to have much analytic content.
The adjoint map is not continuous in the strong operator and ultrastrong topologies, while the strong* and ultrastrong* topologies are modifications so that the adjoint becomes continuous. They are not used very often.
The Arens–Mackey topology and the weak Banach space topology are relatively rarely used.
To summarize, the three essential topologies on are the norm, ultrastrong, and ultraweak topologies.
The weak and strong operator topologies are widely used as convenient approximations to the ultraweak and ultrastrong topologies. The other topologies are relatively obscure.