Operator topology

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In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the algebra L(H) of bounded linear operators on a Hilbert space H.

Contents

[edit] Introduction

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:

  • If \|T_n - T\| \to 0, that is, the supremum of Tnx - T x converges to 0, where x ranges over the unit ball in H, we say that T_n \to T in the uniform operator topology.
  • If T_n x \to Tx for all x in H, then we say T_n \to T in the strong operator topology.
  • Finally, suppose T_n x \to Tx in the weak topology of H. This means that F(T_n x) \to F(T x) for all linear functionals F on H. In this case we say that T_n \to T in the weak operator topology.

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

[edit] List of topologies on L(H)

There are many topologies that can be defined on L(H) besides the ones used above. These topologies are all locally convex, which implies that they are defined by a family of seminorms.

The Banach space L(H) has a (unique) predual L(H)*, consisting of the trace class operators, whose dual is L(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.

  • The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on L(H). It is stronger than all the other topologies below.
  • The weak (Banach space) topology is σ(L(H), L(H)*), in other words the weakest topology such that all elements of the dual L(H)* are continuous. It is the weak topology on the Banach space L(H). It is stronger than the ultraweak and weak operator topologies. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
  • The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on L(H) such that the dual is L(H)*, and is also the uniform convergence topology on σ(L(H)*, L(H))-compact convex subsets of L(H)*. 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 pw(x) and pw(x*) for positive elements w of L(H)*. 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 pw(x) for positive elements w of L(H)*. 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 σ(L(H), L(H)*) topology is defined by the family of seminorms |(w, x)| for elements w of L(H)*. It is stronger than the weak operator topology. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
  • The strong* operator topology or strong* topology is defined by the seminorms ||x(h)|| and ||x*(h)|| for h in H. It is stronger than the strong and weak operator topologies.
  • The strong operator topology (SOT) or strong topology is defined by the seminorms ||x(h)|| for h in H. It is stronger than the weak operator topology.
  • The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h1), h2)| for h1 and h2 in H. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)

[edit] Relations between the topologies

The continuous linear functionals on L(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 L(H) for the ultraweak, ultrastrong, ultrastrong* and Arens-Mackey topologies are the same, and are the elements of the predual L(H)*. The continuous linear functions in the norm topology form a rather large space with many pathological elements.

On any (norm) bounded subset of L(H), the Arens-Mackey topology, the ultrastrong*, and the strong* topology are the same. On any (norm) bounded subset of L(H) the ultrastrong topology is the same as the strong topology. On any (norm) bounded subset of L(H) the ultraweak topology is the same as the weak (operator) topology.

For a convex subset K of L(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 x, K has closed intersection with the closed ball of radius x in the strong*, strong, or weak (operator) topologies.

The closed unit ball of L(H) is compact in the weak (operator) and ultraweak topologies.

The norm topology is metrizable and the others are not. However, when H is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

[edit] Which topology should I use?

The most commonly used topologies are the norm, strong, and weak topologies. The weak topology is useful for compactness arguments as the unit ball is compact. The norm topology makes L(H) into a Banach space, but is too strong for many purposes (for example, L(H) is not separable in this topology). The strong topology is a sort of general purpose topology and probably the most commonly used.

The ultraweak and ultrastrong topologies are better in some ways than the weak and strong 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 L(H) in the weak or strong topologies is usually too small.

The adjoint map is not continuous in the strong and ultrastrong topologies, and 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 very rarely used.

To summarize, the three essential topologies are the norm, ultrastrong, and ultraweak topologies, and it is rarely necessary to use any other topology. The weak and strong topologies are widely used as cheap approximations to the ultraweak and ultrastrong topologies, and the remaining topologies are of little practical importance.

[edit] See also

[edit] References

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