Operator topologies

Last updated April 16, 2024

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra $B(X)$ of bounded linear operators on a Banach space $X$ .

Introduction

Let $(T_{n})_{n\in \mathbb {N} }$ be a sequence of linear operators on the Banach space $X$ . Consider the statement that $(T_{n})_{n\in \mathbb {N} }$ converges to some operator $T$ on $X$ . This could have several different meanings:

If $\|T_{n}-T\|\to 0$ , that is, the operator norm of $T_{n}-T$ (the supremum of $\|T_{n}x-Tx\|_{X}$ , where $x$ ranges over the unit ball in $X$ ) converges to 0, we say that $T_{n}\to T$ in the uniform operator topology .
If $T_{n}x\to Tx$ for all $x\in X$ , then we say $T_{n}\to T$ in the strong operator topology .
Finally, suppose that for all $x \in X$ we have $T_{n}x\to Tx$ in the weak topology of $X$ . This means that $F(T_{n}x)\to F(Tx)$ for all continuous linear functionals $F$ on $X$ . In this case we say that $T_{n}\to T$ in the weak operator topology .

List of topologies on B(H)

Diagram of relations among topologies on the space B(X) of bounded operators Optop.svg — Diagram of relations among topologies on the space $B(X)$ of bounded operators

There are many topologies that can be defined on $B(X)$ besides the ones used above; most are at first only defined when $X = H$ is a Hilbert space, even though in many cases there are appropriate generalisations. The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms.

In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.) The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.

If $H$ is a Hilbert space, the Hilbert space $B(X)$ has a (unique) predual $B(H)_{*}$ , consisting of the trace class operators, whose dual is $B(X)$ . The seminorm $p w (x)$ for w positive in the predual is defined to be $B(w, x * x) 1/2$ .

If $B$ is a vector space of linear maps on the vector space $A$ , then $σ(A, B)$ is defined to be the weakest topology on $A$ such that all elements of $B$ are continuous.

The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on $B(H)$ . It is stronger than all the other topologies below.
The weak (Banach space) topology is $σ(B(H), B(H) *)$ , in other words the weakest topology such that all elements of the dual $B(H) *$ are continuous. It is the weak topology on the Banach space $B(H)$ . It is stronger than the ultraweak and weak operator topologies. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on $B(H)$ such that the dual is $B(H) *$ , and is also the uniform convergence topology on $Bσ(B(H) *$ , $B(H)$ -compact convex subsets of $B(H) *$ . It is stronger than all topologies below.
The σ-strong-^* topology or ultrastrong-^* topology is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms $p w (x)$ and $p w (x *)$ for positive elements $w$ of $B(H) *$ . It is stronger than all topologies below.
The σ-strong topology or ultrastrong topology or strongest topology or strongest operator topology is defined by the family of seminorms $p w (x)$ for positive elements $w$ of $B(H) *$ . It is stronger than all the topologies below other than the strong^* topology. Warning: in spite of the name "strongest topology", it is weaker than the norm topology.)
The σ-weak topology or ultraweak topology or weak-^* operator topology or weak-* topology or weak topology or $σ(B(H), B(H) *$ ) topology is defined by the family of seminorms |(w, x)| for elements w of $B(H) *$ . It is stronger than the weak operator topology. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
The strong-^* operator topology or strong-^* topology is defined by the seminorms ||x(h)|| and ||x^*(h)|| for $h \in H$ . It is stronger than the strong and weak operator topologies.
The strong operator topology (SOT) or strong topology is defined by the seminorms ||x(h)|| for $h \in H$ . It is stronger than the weak operator topology.
The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h₁), h₂)| for $h 1, h 2 \in H$ . (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)

Relations between the topologies

The continuous linear functionals on $B(H)$ for the weak, strong, and strong^* (operator) topologies are the same, and are the finite linear combinations of the linear functionals (xh₁, h₂) for $h 1, h 2 \in H$ . The continuous linear functionals on $B(H)$ for the ultraweak, ultrastrong, ultrastrong^* and Arens-Mackey topologies are the same, and are the elements of the predual $B(H) *$ .

By definition, the continuous linear functionals in the norm topology are the same as those in the weak Banach space topology. This dual is a rather large space with many pathological elements.

On norm bounded sets of $B(H)$ , the weak (operator) and ultraweak topologies coincide. This can be seen via, for instance, the Banach–Alaoglu theorem. For essentially the same reason, the ultrastrong topology is the same as the strong topology on any (norm) bounded subset of $B(H)$ . Same is true for the Arens-Mackey topology, the ultrastrong^*, and the strong^* topology.

In locally convex spaces, closure of convex sets can be characterized by the continuous linear functionals. Therefore, for a convex subset $K$ of $B(H)$ , the conditions that $K$ be closed in the ultrastrong^*, ultrastrong, and ultraweak topologies are all equivalent and are also equivalent to the conditions that for all $r > 0$ , $K$ has closed intersection with the closed ball of radius $r$ in the strong^*, strong, or weak (operator) topologies.

The norm topology is metrizable and the others are not; in fact they fail to be first-countable. However, when $H$ is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).

Topology to use

The most commonly used topologies are the norm, strong, and weak operator topologies. The weak operator topology is useful for compactness arguments, because the unit ball is compact by the Banach–Alaoglu theorem. The norm topology is fundamental because it makes $B(H)$ into a Banach space, but it is too strong for many purposes; for example, $B(H)$ is not separable in this topology. The strong operator topology could be the most commonly used.

The ultraweak and ultrastrong topologies are better-behaved than the weak and strong operator topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed. For example, the dual space of $B(H)$ in the weak or strong operator topology is too small to have much analytic content.

The adjoint map is not continuous in the strong operator and ultrastrong topologies, while the strong* and ultrastrong* topologies are modifications so that the adjoint becomes continuous. They are not used very often.

The Arens–Mackey topology and the weak Banach space topology are relatively rarely used.

To summarize, the three essential topologies on $B(H)$ are the norm, ultrastrong, and ultraweak topologies. The weak and strong operator topologies are widely used as convenient approximations to the ultraweak and ultrastrong topologies. The other topologies are relatively obscure.

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.

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 a generalization of the intuitive notion of "length" in the physical world. If $is a vector space over, where is a field equal to or to, then a norm on is a map, typically denoted by, satisfying the following four axioms:$

Non-negativity: for every $, .$
Positive definiteness: for every $, if and only if is the zero vector.$
Absolute homogeneity: for every $and,$
Triangle inequality: for every $and,$

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 topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from $into its bidual is a homeomorphism. A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this evaluation map is an isometric isomorphism and the normed space is a Banach space. Those space for which the canonical evaluation map is surjective are called semi-reflexive spaces.$

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space $, such that the functional sending an operator to the complex number is continuous for any vectors and in the Hilbert space.$

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 $, as x varies in H .$

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis, a branch of mathematics, a compact operator is a linear operator $, where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of . Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces.$

In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.

In functional analysis, the ultrastrong topology, or σ-strong topology, or strongest topology on the set B(H) of bounded operators on a Hilbert space is the topology defined by the family of seminorms

In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual is bijective. If this map is also an isomorphism of TVSs then it is called reflexive.

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Shultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

In the theory of C*-algebras, the universal representation of a C*-algebra is a faithful representation which is the direct sum of the GNS representations corresponding to the states of the C*-algebra. The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. The close relationship between an arbitrary representation of a C*-algebra and its universal representation can be exploited to obtain several criteria for determining whether a linear functional on the algebra is ultraweakly continuous. The method of using the properties of the universal representation as a tool to prove results about the C*-algebra and its representations is commonly referred to as universal representation techniques in the literature.

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk $is bounded: in this case, the auxiliary normed space is with norm$

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

This is a glossary for the terminology in a mathematical field of functional analysis.

References

Functional analysis, by Reed and Simon, ISBN 0-12-585050-6
Theory of Operator Algebras I, by M. Takesaki (especially chapter II.2) ISBN 3-540-42248-X

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v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets /set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

v t e Duality and spaces of linear maps
Basic concepts	Dual space Dual system Dual topology Duality Operator topologies Polar set Polar topology Topologies on spaces of linear maps
Topologies	Norm topology Dual norm Ultraweak/Weak-* Weak polar operator in Hilbert spaces Mackey Strong dual polar topology operator Ultrastrong
Main results	Banach–Alaoglu Mackey–Arens
Maps	Transpose of a linear map
Subsets	Saturated family Total set
Other concepts	Biorthogonal system

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category