Operator topology

Operator topologies are examined in the mathematical sub-area of functional analysis. There are different topologies in the space of continuous, linear operators on a Hilbert space .

These topologies are of great interest especially for infinite-dimensional Hilbert spaces, since they coincide with the standard topology for finite-dimensional Hilbert spaces and are therefore dispensable there. Therefore, in the following it is always an infinite-dimensional Hilbert space and denote the algebra of continuous linear operators on . ${\displaystyle H}$ ${\displaystyle L(H)}$ ${\displaystyle H}$

An overview of the operator topologies, arrows point from finer to coarser topologies

Standard topology

The operator norm that gives each operator the value ${\displaystyle A\in L(H)}$

${\displaystyle \|A\|:=\sup _{x\in H,\,\|x\|\leq 1}\|Ax\|}$

assigns, defines a standard topology . It turns into a Banach algebra , with the adjunction as involution to a C * algebra , even a Von Neumann algebra . ${\displaystyle L(H)}$ ${\displaystyle L(H)}$

In addition to this standard topology, a number of other so-called operator topologies are used to investigate or contain operator algebras. These are locally convex topologies , which are described below by a defining family of semi-norms . A network of operators in this topology converges to one if and only if for all . ${\displaystyle L(H)}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle A\in L(H)}$ ${\displaystyle p(A-A_{i}){\xrightarrow[{i\in I}]{}}0}$ ${\displaystyle p\in {\mathcal {P}}}$

Weak topology

Semi-norms

Like every Banach space , it also has a weak topology . This is through the system of semi-norms ${\displaystyle L(H)}$

${\displaystyle p(A):=|f(A)|,\quad f\in L(H)'}$

given. Here is the dual space of . ${\displaystyle L(H)'}$ ${\displaystyle L(H)}$

annotation

Since the continuous, linear functionals cannot be described well (in general) and since the unit sphere is not compact in terms of the weak topology due to a lack of reflexivity , this topology only plays a subordinate role. Many authors therefore also mean the weak operator topology presented below by weak topology . ${\displaystyle L(H)}$ ${\displaystyle L(H)}$

Strong operator topology

Semi-norms

The strong operator topology (SOT = strong operator topology ) is the topology of point-by-point norm convergence, it is created by the semi-norms

${\displaystyle p_{x}(A):=\|Ax\|,\quad x\in H}$

generated.

properties

The multiplication is not SOT continuous. The multiplication becomes SOT-continuous if the left factor remains constrained. In particular, the multiplication SOT- is continuous , because every SOT-convergent sequence is bounded by the Banach-Steinhaus theorem . ${\displaystyle L(H)\times L(H)\rightarrow L(H),\,(A,B)\mapsto AB}$

The involution is not SOT continuous. If, for example, is the unilateral shift operator , then with respect to SOT, but the sequence does not converge in SOT to 0. But the restriction of the involution to the set of all normal operators is SOT-continuous. ${\displaystyle L(H)\rightarrow L(H),\,A\mapsto A^{*}}$ ${\displaystyle S}$ ${\displaystyle (S^{n})^{*}{\xrightarrow[{n\to \infty }]{}}0}$ ${\displaystyle (S^{n})_{n\in \mathbb {N} }}$

The closed standard spheres are SOT- complete , SOT- is quasi-complete and SOT- sequence complete . The unit sphere is not SOT-compact ( infinite-dimensional, as assumed in this article). ${\displaystyle L(H)}$ ${\displaystyle H}$

Weak operator topology

Semi-norms

The weak operator topology (WOT = weak operator topology ) is the topology of point-wise weak convergence, that is, it is through the semi-norms

${\displaystyle p_{x,y}(A):=|\langle Ax,y\rangle |,\quad x,y\in H}$

Are defined.

properties

The multiplication is not WOT-continuous, whereas the one-sided multiplications, i.e. the images and for fixed , are WOT-continuous. ${\displaystyle L(H)\times L(H)\rightarrow L(H),\,(A,B)\mapsto AB}$ ${\displaystyle A\mapsto AB}$ ${\displaystyle A\mapsto BA}$ ${\displaystyle B\in L(H)}$

The involution is WOT-continuous. ${\displaystyle L(H)\rightarrow L(H),\,A\mapsto A^{*}}$

Probably the most important property is the WOT-compactness of the unit sphere and thus every sphere with a finite radius. If separable , the balls can also be metrised . ${\displaystyle H}$

The WOT-continuous linear functionals on are exactly the functionals of the form for finite and . These are exactly the SOT-continuous functionals, which is why the separation theorem shows that the WOT-terminations and SOT-terminations of convex sets match. ${\displaystyle L(H)}$ ${\displaystyle \textstyle L(H)\rightarrow \mathbb {C} ,\,A\mapsto \sum _{i=1}^{n}\langle Ax_{i},y_{i}\rangle }$ ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\in H}$

Strong * operator topology

Semi-norms

According to the above, the involution is continuous with regard to WOT and with regard to the standard topology, but not for the SOT in between. This deficiency can be countered by switching to the strong * operator topology SOT *. To do this, one considers the topology that is defined by the semi-norms

${\displaystyle p_{x}(A):=\|Ax\|+\|A^{*}x\|,\quad x\in H}$

is produced.

properties

If and denotes the sphere around 0 with radius in , then is the restricted multiplication ${\displaystyle 0<r<\infty }$ ${\displaystyle L(H)_{r}}$ ${\displaystyle r}$ ${\displaystyle L(H)}$

${\displaystyle L(H)_{r}\times L(H)_{r}\rightarrow L(H),\quad (A,B)\mapsto AB}$

SOT * -continuous. According to the construction, the involution SOT * is also continuous.

Furthermore, a linear functional is SOT * -continuous if and only if it is WOT-continuous. ${\displaystyle L(H)}$

Ultra-weak topology

Semi-norms

The ultra- weak topology, also called -weak topology by some authors , is the weak - * - topology of duality , where the space of the trace class operators is and the duality is known to be given by . The topology is determined by the semi-norms ${\displaystyle \sigma }$ ${\displaystyle N(H)'\cong L(H)}$ ${\displaystyle N(H)}$ ${\displaystyle \varphi _{T}(A):=\mathrm {Spur} (TA),\,T\in N(H),A\in L(H)}$

${\displaystyle p_{(x_{n})_{n},(y_{n})_{n}}(A):=|\sum _{n=1}^{\infty }\langle Ax_{n},y_{n}\rangle |,\quad (x_{n})_{n},(y_{n})_{n}}$Follow in with , ${\displaystyle H}$ ${\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|^{2}<\infty }$ ${\displaystyle \sum _{n=1}^{\infty }\|y_{n}\|^{2}<\infty }$

generated.

properties

Since it is a weak - * - topology, the unit sphere is extremely weakly compact according to the Banach-Alaoglu theorem . It agrees with the WOT on every limited amount, but is really finer than WOT on the other hand . ${\displaystyle L(H)}$

As with the WOT, the involution and the one-sided multiplications are ultra-weakly continuous.

The following statements are equivalent for restricted functionals on : ${\displaystyle f}$ ${\displaystyle L(H)}$

• ${\displaystyle f}$ is ultra weak steady
• The restriction of to the unit sphere is WOT-continuous ${\displaystyle f}$
• There is a trace class with all ${\displaystyle T\in N(H)}$ ${\displaystyle f(A)=\mathrm {Spur} (TA)}$ ${\displaystyle A\in L(H)}$
• If a bounded and monotonically growing network of self-adjoint operators with supremum , then is . ${\displaystyle (A_{i})_{i}}$ ${\displaystyle A\in L(H)}$ ${\displaystyle \textstyle \lim _{i\in I}f(A_{i})=f(A)}$

Ultra strong topology

Semi-norms

The ultra-strong topology to be defined here, which is also known under the name -strong topology, has an analogous relationship to the ultra-weak topology as the SOT to the WOT. The defining semi-norms are ${\displaystyle \sigma }$

${\displaystyle p_{(x_{n})_{n}}(A):=\left(\sum _{n\in \mathbb {N} }\|Ax_{n}\|^{2}\right)^{1/2},\quad (x_{n})_{n}}$Follow in with . ${\displaystyle H}$ ${\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|^{2}<\infty }$

properties

As with the SOT, the multiplication is not ultra-strong continuous, but it becomes ultra-strong continuous if the left factor remains constrained. The involution is not ultra-strong and continuous.

The ultra-strong topology matches the SOT on any constrained set, but is genuinely finer than SOT. ${\displaystyle L(H)}$

The ultra-strong-continuous linear functionals on agree with the ultra-weak-continuous linear functionals, in particular convex sets have corresponding ultra-strong and ultra-weak terminations. ${\displaystyle L(H)}$

Ultra strong * topology

Semi-norms

In analogy to the SOT *, the ultra-strong * topology is defined by the following system of semi-norms:

${\displaystyle p_{(x_{n})_{n}}(A):=\left(\sum _{n=1}^{\infty }\|Ax_{n}\|^{2}+\sum _{n=1}^{\infty }\|A^{*}x_{n}\|^{2}\right)^{1/2}\,\quad (x_{n})_{n}}$Follow in with . ${\displaystyle H}$ ${\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|^{2}<\infty }$

properties

The ultra-strong * topology agrees with the SOT * topology on limited sets, the restricted multiplication is ultra-strong * -continuous, and by definition the involution is ultra-strong * -continuous. Furthermore, a linear functional is ultra-strong * -continuous precisely when it is ultra-weakly continuous. ${\displaystyle L(H)_{r}\times L(H)_{r}\rightarrow L(H),\quad (A,B)\mapsto AB}$ ${\displaystyle L(H)}$

On limited sets, the SOT * and the ultra-strong * topology agree with the Mackey topology , the latter being the finest locally convex topology that has the same continuous linear functionals as the ultra-weak topology.

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0-1239-3301-3 , sentence 2.5.11
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0-1239-3301-3 , Section 5.1
3. ^ Ola Bratteli , Derek W. Robinson : Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , theorem 2.4.5
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 3.6.4
5. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter 3, Sections 1 to 3
6. ^ Ola Bratteli, Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1 , Springer-Verlag (1979), ISBN 0-387-09187-4 , theorem 2.4.5
7. ^ Bing-Ren Li: Introduction to Operator Algebras , World Scientific Pub Co (1992), ISBN 9-8102-0941-X , Chapter 1.11