Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(H) of bounded linear operators on a Hilbert space H.


Let {Tn} be a sequence of linear operators on the Hilbert space H. Consider the statement that Tn converges to some operator T in H. This could have several different meanings:

All of these notions make sense and are useful for a Banach space in place of the Hilbert space H.

List of topologies on B(H)

Diagram of relations among topologies on the space B(H) of bounded operators

There are many topologies that can be defined on B(H) besides the ones used above. These topologies 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. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.) The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.

The Banach space B(H) has a (unique) predual B(H)*, consisting of the trace class operators, whose dual is B(H). The seminorm pw(x) for w positive in the predual is defined to be (w, x*x)1/2.

If B is a vector space of linear maps on the vector space A, then σ(A, B) is defined to be the weakest topology on A such that all elements of B are continuous.

Relations between the topologies

The continuous linear functionals on B(H) for the weak, strong, and strong* (operator) topologies are the same, and are the finite linear combinations of the linear functionals (xh1, h2) for h1, h2 in H. The continuous linear functionals on B(H) for the ultraweak, ultrastrong, ultrastrong* and Arens-Mackey topologies are the same, and are the elements of the predual B(H)*.

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 B(H), the weak (operator) 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 (norm) bounded subset of B(H). 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 K of B(H), the conditions that K be closed in the ultrastrong*, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that for all r > 0, K has closed intersection with the closed ball of radius r in the strong*, strong, or weak (operator) topologies.

The norm topology is metrizable and the others are not; in fact they fail to be first-countable. However, when H is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

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 B(H) into a Banach space, but it is too strong for many purposes; for example, B(H) 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 B(H) 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 B(H) 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.

See also


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