Operator topologies
Operator topologies are studied in the mathematical branch of functional analysis. There are various topologies on the space of continuous linear operators on a Hilbert space.
These topologies are especially for infinite-dimensional Hilbert spaces are of great interest, since they coincide for finite dimensional Hilbert spaces with the norm topology, and there are therefore unnecessary. Therefore, whether in a infinite-dimensional Hilbert space always follows and denote the algebra of continuous linear operators on.
- 3.1 seminorms
- 3.2 Features
- 4.1 seminorms
- 4.2 Properties
- 5.1 seminorms
- 5.2 Properties
- 6.1 seminorms
- 6.2 properties
- 7.1 seminorms
- 7.2 Characteristics
- 8.1 seminorms
- 8.2 properties
Norm topology
The operator norm, which each operator returns the value
Maps defined on a norm topology. It makes it a Banach algebra, with the adjunction as involution to a C *-algebra, even von Neumann algebra.
In addition to this standard topology a number of other so-called operator topologies is used to investigate and operator algebras contained therein. These are in each case by locally convex topologies are described below by defining the half- family standards. A network of operators converges in this topology if and only against one, if for all.
Weak topology
Seminorms
Like any Banach space also carries a weak topology. This is by the system of seminorms
Given. This is the dual space of.
Note
Since one can describe the steady, linear functionals on not good and since the unit ball is not compact in the weak topology with respect to due to lack of reflexivity, this topology plays only a minor role. Many authors therefore my weak topology and the weak operator topology presented below.
Strong operator topology
Seminorms
The strong operator topology (SOT ) is the topology of pointwise norm convergence, it is determined by the semi-norms
Generated.
Properties
Multiplication is not SOT -continuous. The multiplication is SOT -continuous if the left factor remains limited. In particular, the multiplication is SOT sequentially continuous, for every SOT- convergent sequence is bounded by the theorem of Banach - Steinhaus.
The involution is SOT -continuous. For example, if the unilateral shift operator, so is respect SOT, but the sequence does not converge in SOT tends to 0, but the restriction of the involution on the set of all normal operators is SOT -continuous.
The completed standard balls are SOT -complete, is almost complete and SOT- SOT- sequentially complete. The unit sphere is not SOT- compact ( infinite dimensional, as assumed in this article).
Weak operator topology
Seminorms
Weak operator topology (WOT = weak operator topology ), the topology of the point-wise weak convergence, that is, it is the semi-norms
Defined.
Properties
The multiplication is not WOT continuously, whereas the single-sided multiplications, that is, the pictures and for solid WOT continuous.
The involution is WOT -continuous.
The most important feature is the WOT - compactness of the unit sphere, and thus each ball of finite radius. Is separable, then the balls are also metrizable.
The WOT - continuous linear functionals on exactly are the functionals of the form for finite and. These are precisely the SOT- continuous functionals, which is why it follows from the separation theorem that the WOT statements and SOT- match statements of convex sets.
Strong * operator topology
Seminorms
According to the above involution is continuous with respect to and with respect to the WOT standard topology, but not to the intermediate SOT. This shortcoming can be addressed by the transition to the strong * operator topology SOT *. Considering to the topology represented by the semi-norms
Is generated.
Properties
And is designated by 0, the ball radius in such multiplication is the limited
* SOT -continuous. By construction, the involution * is SOT -continuous.
Furthermore, a linear functional on if and SOT is * - continuous if it is WOT -continuous.
Ultra Weak Topology
Seminorms
The ultra-weak topology, called by some authors - weak topology is the weak -* topology of the duality, the space of trace class operators and the duality is known to be given by. The topology is defined by the seminorms
Generated.
Properties
Since it is a weak -* topology, the unit sphere by the theorem of Banach - Alaoglu is ultra- weakly compact. It coincides on each bounded set with the WOT, but on strictly finer than WOT.
As with the WOT involution and the one-sided multiplications ultra weak are continuous.
For bounded functionals on the following statements are equivalent
- Is ultra- weakly continuous
- The restriction of the unit sphere is WOT -continuous
- There is a trace class operator with all
- Is a bounded and monotonically growing network of self-adjoint operators with supremum, then.
Ultra strong topology
Seminorms
The here -to-define ultra- strong topology, which is also known under the name - strong topology is, the ultra-weak topology in an analogous relationship as they SOT to WOT. The defining seminorms are
Properties
As with the SOT, the multiplication is not ultra- strongly continuous, but it is ultra - strongly continuous if the left factor remains limited. The involution is not ultra strong continuous.
The ultra- strong topology coincides on each bounded set with the SOT, but on strictly finer than SOT.
The ultra - strong continuous linear functionals on are consistent with the ultra - weak continuous linear functionals, in particular convex sets have matching ultra- strong and ultra-weak financial statements.
Ultra Strong * topology
Seminorms
In analogy to the SOT * it is ultra- strong * topology defined by the following system of semi-norms:
Properties
The ultra- strong * topology coincides on bounded sets with the SOT -* topology, the limited multiplication is ultra- strong * -continuous, as by definition, is the involution ultra- strong * -continuous. Furthermore, linear functional is continuous on * if and ultra strong if it is continuous ultra weak.
Agree on bounded sets the SOT * and the ultra- strong * topology with the Mackey topology coincide, the latter is the finest locally convex topology, the same continuous linear functionals as that of the ultra-weak topology.