In
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the weak operator topology, often abbreviated WOT, is the weakest
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
on the set of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on a
Hilbert space , such that the
functional sending an operator
to the complex number
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
for any vectors
and
in the Hilbert space.
Explicitly, for an operator
there is
base of neighborhoods of the following type: choose a finite number of vectors
, continuous functionals
, and positive real constants
indexed by the same finite set
. An operator
lies in the neighborhood if and only if
for all
.
Equivalently, a
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
of bounded operators converges to
in WOT if for all
and
, the net
converges to
.
Relationship with other topologies on ''B''(''H'')
The WOT is the weakest among all common
topologies on , the bounded operators on a Hilbert space
.
Strong operator topology
The
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
, or SOT, on
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
and consider the sequence
of unilateral shifts. An application of Cauchy-Schwarz shows that
in WOT. But clearly
does not converge to
in SOT.
The
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s on the set of bounded operators on a Hilbert space that are continuous in the
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
are precisely those that are continuous in the WOT (actually, the WOT is the weakest operator topology that leaves continuous all strongly continuous linear functionals on the set
of bounded operators on the Hilbert space ''H''). Because of this fact, the closure of a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
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
converges to
in SOT if and only if
in WOT.
Weak-star operator topology
The predual of ''B''(''H'') is the
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
operators C
1(''H''), and it generates the w*-topology on ''B''(''H''), called the
weak-star operator topology
In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set ''B''(''H'') of bounded operators on a Hilbert space is the weak-* topology ob ...
or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in ''B''(''H'').
A net ⊂ ''B''(''H'') converges to ''T'' in WOT if and only Tr(''T
αF'') converges to Tr(''TF'') for all
finite-rank operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional.
Finite-rank operators on a Hilbert space A canonical form
Finite-rank operators are ...
''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
:
So converges to ''T'' in WOT means
:
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
:
where the series
converges. Suppose
and
in WOT. For every trace-class ''S'',
:
by invoking, for instance, the
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
.
Therefore every norm-bounded set is compact in WOT, by the
Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.
A common ...
.
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
be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both ''T
n'' and ''T*
n'' converges to 0 in WOT. But ''T*
nT
n'' is the identity operator for all
. (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 ''T
i'' → ''T'' in WOT, then ''ST
i'' → ''ST'' and ''T
iS'' → ''TS'' in WOT.
SOT and WOT on ''B(X,Y)'' when ''X'' and ''Y'' are normed spaces
We can extend the definitions of SOT and WOT to the more general setting where ''X'' and ''Y'' are
normed spaces
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
and
is the space of bounded linear operators of the form
. In this case, each pair
and
defines a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
on
via the rule
. The resulting family of seminorms generates the weak operator topology on
. Equivalently, the WOT on
is formed by taking for
basic open neighborhoods those sets of the form
:
where
is a finite set,
is also a finite set, and
. The space
is a locally convex topological vector space when endowed with the WOT.
The strong operator topology on
is generated by the family of seminorms
via the rules
. Thus, a topological base for the SOT is given by open neighborhoods of the form
:
where as before
is a finite set, and
Relationships between different topologies on ''B(X,Y)''
The different terminology for the various topologies on
can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is
. The
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on a normed space
is the coarsest topology that makes the linear functionals in
continuous; when we take
in place of
, the weak topology can be very different than the weak operator topology. And while the WOT is formally weaker than the SOT, the SOT is weaker than the operator norm topology.
In general, the following inclusions hold:
:
and these inclusions may or may not be strict depending on the choices of
and
.
The WOT on
is a formally weaker topology than the SOT, but they nevertheless share some important properties. For example,
:
Consequently, if
is convex then
:
in other words, SOT-closure and WOT-closure coincide for convex sets.
See also
*
*
*
{{Duality and spaces of linear maps
Topological vector spaces
Topology of function spaces