OPERATOR TOPOLOGY

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

Contents

Introduction

List of topologies on ''B''(''H'')

Relations between the topologies

Which topology should I use?

See also

References

Introduction

Let {''T''_''n''} be a sequence of linear operators on the Hilbert space ''H''. Consider the statement that ''T''_''n'' converges to some operator ''T'' in ''H''.This could have several different meanings:

★ If $|T\_n\; -\; T|\; o\; 0$, that is, the supremum of ''T''_''n''''x'' - ''T'' ''x'' converges to 0, where ''x'' ranges over the unit ball in ''H'', we say that $T\_n\; o\; T$ in the 'uniform operator topology'.

★ If $T\_n\; x\; o\; Tx$ for all ''x'' in ''H'', then we say $T\_n\; o\; T$ in the 'strong operator topology'.

★ Finally, suppose $T\_n\; x\; o\; Tx$ in the weak topology of ''H''. This means that $F(T\_n\; x)\; o\; F(T\; x)$ for all linear functionals ''F'' on ''H''. In this case we say that $T\_n\; o\; T$ 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''.

List of topologies on ''B''(''H'')

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.
Because of the proliferation of adjectives similar to "strong" and "weak", we will compare the topologies using the words "fine" and "coarse". A topology is fine if it has many open sets and coarse if it is 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 fine to coarse.
The Banach space ''B''(''H'') has a (unique) predual ''B''(''H'')
_★ ,
consisting of the trace class operators, whose dual is ''B''(''H''). The seminorm ''p''_''w''(''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 coarsest 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 ''B''(''H''). It is finer than all the other topologies below.

★ The 'weak (Banach space) topology' is σ(''B''(''H''), ''B''(''H'')
^★ ), in other words the coarsest topology such that all elements of the dual ''B''(''H'')
^★ are continuous. It is the weak topology on the Banach space ''B''(''H''). It is finer 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 finest locally convex topology on ''B''(''H'') such that the dual is ''B''(''H'')
_★ , and is also the uniform convergence topology on σ(''B''(''H'')
_★ , ''B''(''H'')-compact convex subsets of ''B''(''H'')
_★ . It is finer than all topologies below.

★ The 'σ-strong
^★ topology' or 'ultrastrong
^★ topology' is the coarsest topology finer than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms ''p''_''w''(''x'') and ''p''_''w''(''x''
^★ ) for positive elements ''w'' of ''B''(''H'')
_★ . It is finer 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 ''p''_''w''(''x'') for positive elements ''w'' of ''B''(''H'')
_★ . It is finer than all the topologies below other than the strong
^★ topology. Warning: in spite of the name "strongest topology", it is coarser than the norm topology.)

★ The 'σ-weak topology' or 'ultraweak topology' or 'weak
^★ operator topology' or 'weak
★ topology' or 'weak topology' or 'σ(''B''(''H''), ''B''(''H'')
_★ ) topology' is defined by the family of seminorms |(''w'', ''x'')| for elements ''w'' of ''B''(''H'')
_★ . It is finer 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 finer 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 finer than the weak operator topology.

★ The 'weak operator topology' (WOT) or 'weak topology' is defined by the seminorms |(''x''(''h''₁), ''h''₂)| for ''h''₁ and ''h''₂ in ''H''. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)

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
(x''h''₁, ''h''₂) for ''h''₁, ''h''₂ 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'')
_★ . The continuous linear functions in the norm topology
form a rather large space with many pathological elements.
On any (norm) bounded subset of ''B''(''H''), the Arens-Mackey topology, the ultrastrong
^★ , and the strong
^★ topology are the same.
On any (norm) bounded subset of ''B''(''H'') the ultrastrong
topology is the same as the strong topology. On any (norm) bounded subset of ''B''(''H'') the ultraweak topology is the same as the weak (operator) topology.
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 ''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 ''B''(''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).

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 fine 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 very 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

★ Topology

★ Hilbert space

★ Bounded operator

References

★ ''Functional analysis'',by Reed and Simon, ISBN 0-12-585050-6

★ ''Theory of Operator Algebras I'', by M. Takesaki (especially chapter II.2) ISBN 3-540-42248-X