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It is stronger 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, h 2 ∈ 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.)
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the study of the operator is intimately linked to questions in function theory. For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit ...
One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak , ultrastrong , and *-ultrastrong topologies.
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus. The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT).
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