Strong operator topology
Encyclopedia
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 operator
s on a Hilbert space
(or, more generally, on a Banach space
) such that the evaluation map sending an operator T to the real number
is continuous for each vector x in the Hilbert space.
The SOT is stronger than the weak operator topology
and weaker than the norm topology
.
The SOT lacks some of the nicer properties that the weak operator topology
has, but being stronger, things are sometimes easier to prove in this topology. It is more natural too, since it is simply the topology of pointwise convergence for an operator.
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 functional
s on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the WOT
. Because of this, the closure of a convex set
of operators in the WOT is the same as the closure of that set in the SOT.
It should also be noted that the above language translates into convergence properties of Hilbert space operators. One especially observes that for a complex Hilbert space, by way of the polarization identity, one easily verifies that Strong Operator convergence implies Weak Operator convergence.
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...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the strong operator topology, often abbreviated SOT, is the weakest locally convex 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...
(or, more generally, on a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
) such that the evaluation map sending an operator T to the real number
is continuous for each vector x in the Hilbert space.The SOT is stronger than the weak operator topology
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
and weaker than the norm topology
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
.
The SOT lacks some of the nicer properties that the weak operator topology
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
has, but being stronger, things are sometimes easier to prove in this topology. It is more natural too, since it is simply the topology of pointwise convergence for an operator.
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus
Continuous functional calculus
In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,Theorem...
.
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 SOT are precisely those continuous in the WOT
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
. Because of this, 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 should also be noted that the above language translates into convergence properties of Hilbert space operators. One especially observes that for a complex Hilbert space, by way of the polarization identity, one easily verifies that Strong Operator convergence implies Weak Operator convergence.
See also
- Strongly continuous semigroup
- Topologies on the set of operators on a Hilbert space