Search results
Results From The WOW.Com Content Network
The strong operator topology (SOT) or strong topology is defined by the seminorms ||x(h)|| for h ∈ H. 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 ...
Given a topological space (,), the cocountable extension topology on is the topology having as a subbasis the union of τ and the family of all subsets of whose complements in are countable. Cofinite topology; Double-pointed cofinite topology; Ordinal number topology
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 ...
You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made.
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).
The ultrastrong topology is stronger than the strong operator topology. One problem with the strong operator topology is that the dual of B(H) with the strong operator topology is "too small". The ultrastrong topology fixes this problem: the dual is the full predual B * (H) of all trace class operators. In general the ultrastrong topology is ...
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators , and consideration may be given to nonlinear operators .
For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular, the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual of B(H) with the