Weak operator topology
Encyclopedia
In functional analysis
, the weak operator topology, often abbreviated WOT, is the weakest topology
on the set of bounded operator
s on a Hilbert space
H, such that the functional
sending an operator T to the complex number <Tx, y> is continuous for any vectors x and y in the Hilbert space.
Equivalently, a net Ti ⊂ B(H) of bounded operators converges to T ∈ B(H) in WOT if for all y* in H* and x in H, the net y*(Tix) converges to y*(Tx).
, or SOT, on B(H) is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is stronger than WOT. The following example shows that this inclusion is strict. Let H = ℓ 2(N) and consider the sequence {Tn} where T is the unilateral shift. An application of Cauchy-Schwarz shows that Tn → 0 in WOT. But clearly Tn does not converge to 0 in SOT.
The linear functional
s on the set of bounded operators on a Hilbert space that are continuous in the strong operator topology
are precisely those that are continuous in the WOT. Because of this fact, the closure of a convex set
of operators in the WOT is the same as the closure of that set in the SOT.
It follows from the polarization identity
that a net Tα → 0 in SOT if and only if Tα*Tα → 0 in WOT.
operators C1(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 {Tα} ⊂ B(H) converges to T in WOT if and only Tr(TαF) converges to Tr(TF) for all finite-rank operator F. Since every finite-rank operator is trace-class, this implies that WOT is weaker than the σ-weak topology. To see why the claim is true, recall that every finite-rank operator F is a finite sum F = ∑ λi uivi*. So {Tα} converges to T in WOT means Tr(TαF) = ∑ λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TF).
Extending slightly, one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form S = ∑ λi uivi*, where the series of positive numbers ∑λi converges. Suppose supα ||Tα|| = k < ∞, and Tα converges to T in WOT. For every trace-class S, Tr (TαS) = ∑λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TS), by invoking, for instance, the dominated convergence theorem
.
Therefore every norm-bounded set is compact in WOT, by the Banach–Alaoglu theorem.
Multiplication is not jointly continuous in WOT: again let T be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both Tn and T*n converges to 0 in WOT. But T*nTn is the identity operator for all n. (Because WOT coincides with the σ-weak topology on bounded sets, multiplication is not jointly continuous in the σ-weak topology.)
However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net Ti → T in WOT, then STi → ST and TiS → TS in WOT.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the weak operator topology, often abbreviated WOT, is the weakest topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
on the set of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, such that the functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
sending an operator T to the complex number <Tx, y> is continuous for any vectors x and y in the Hilbert space.
Equivalently, a net Ti ⊂ B(H) of bounded operators converges to T ∈ B(H) in WOT if for all y* in H* and x in H, the net y*(Tix) converges to y*(Tx).
Relationship with other topologies on B(H)
The WOT is the weakest among all common topologies on B(H), the bounded operators on a Hilbert space H.Strong operator topology
The strong operator topologyStrong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
, or SOT, on B(H) is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is stronger than WOT. The following example shows that this inclusion is strict. Let H = ℓ 2(N) and consider the sequence {Tn} where T is the unilateral shift. An application of Cauchy-Schwarz shows that Tn → 0 in WOT. But clearly Tn does not converge to 0 in SOT.
The linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s on the set of bounded operators on a Hilbert space that are continuous in the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
are precisely those that are continuous in the WOT. Because of this fact, the closure of a convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
of operators in the WOT is the same as the closure of that set in the SOT.
It follows from the polarization identity
Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...
that a net Tα → 0 in SOT if and only if Tα*Tα → 0 in WOT.
Weak-star operator topology
The predual of B(H) is the trace classTrace class
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
operators C1(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 {Tα} ⊂ B(H) converges to T in WOT if and only Tr(TαF) converges to Tr(TF) for all finite-rank operator F. Since every finite-rank operator is trace-class, this implies that WOT is weaker than the σ-weak topology. To see why the claim is true, recall that every finite-rank operator F is a finite sum F = ∑ λi uivi*. So {Tα} converges to T in WOT means Tr(TαF) = ∑ λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TF).
Extending slightly, one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form S = ∑ λi uivi*, where the series of positive numbers ∑λi converges. Suppose supα ||Tα|| = k < ∞, and Tα converges to T in WOT. For every trace-class S, Tr (TαS) = ∑λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TS), by invoking, for instance, the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
.
Therefore every norm-bounded set is compact in WOT, by the Banach–Alaoglu theorem.
Other properties
The adjoint operation T → T*, as an immediate consequence of its definition, is continuous in WOT.Multiplication is not jointly continuous in WOT: again let T be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both Tn and T*n converges to 0 in WOT. But T*nTn is the identity operator for all n. (Because WOT coincides with the σ-weak topology on bounded sets, multiplication is not jointly continuous in the σ-weak topology.)
However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net Ti → T in WOT, then STi → ST and TiS → TS in WOT.
See also
- Weak topologyWeak topologyIn mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
- Weak-star topologyWeak-star topologyWeak-star topology may refer to:* A topology related to the weak topology* The weak-star operator topology on the set of bounded operators on a Hilbert space* A star network topology in networking...
- Topologies on the set of operators on a Hilbert space