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Topologies on operators on a Hilbert space
functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X. Let ( T n ) n ∈ N
Operator_topologies
List of concrete topologies and topological spaces
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics.
List_of_topologies
Locally convex topology on function spaces
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
Strong_operator_topology
Weak topology on function spaces
common topologies on B ( H ) {\displaystyle B(H)} , the bounded operators on a Hilbert space H {\displaystyle H} . The strong operator topology, or SOT
Weak_operator_topology
Mathematical concept
mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance
Weak_topology
compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual
Ultraweak_topology
variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology
Von Neumann bicommutant theorem
Von_Neumann_bicommutant_theorem
Measure of the "size" of linear operators
targets Operator algebra – Branch of functional analysis Operator theory – Mathematical study of linear operators Topologies on the set of operators on a
Operator_norm
Branch of functional analysis
mechanics – Formulation of quantum mechanics Topologies on the set of operators on a Hilbert space Vertex operator algebra – Algebra used in 2D conformal field
Operator_algebra
Mathematical study of linear operators
linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function
Operator_theory
set the strong operator and ultrastrong topologies are the same. The ultrastrong topology is stronger than the strong operator topology. One problem with
Ultrastrong_topology
*-algebra of bounded operators on a Hilbert space
many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in
Von_Neumann_algebra
Index of articles associated with the same name
a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on
Strong_topology
Type of topology in mathematics
referred to such topologies as Alexandrov topologies. F. G. Arenas independently proposed this name for the general version of these topologies. McCord also
Alexandrov_topology
Branch of mathematics that studies dynamical systems
converge to the projector ET in the strong operator topology of Lp if 1 ≤ p ≤ ∞, and in the weak operator topology if p = ∞. More is true if 1 < p ≤ ∞ then
Ergodic_theory
Mathematical operator
which prevents confusion with the "closure operators" studied in topology. E. H. Moore studied closure operators in his 1910 Introduction to a form of general
Closure_operator
Theory in functional analysis
space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional
Spectral theory of compact operators
Spectral_theory_of_compact_operators
Largest open subset of some given set
other topologies rather than the standard one: If X {\displaystyle X} is the real numbers R {\displaystyle \mathbb {R} } with the lower limit topology, then
Interior_(topology)
Mathematical exercise
possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships. All possible polar topologies on a dual
Comparison_of_topologies
functionals continuous for the weak or strong topologies. As a consequence the weak and strong topologies coincide on a JBW algebra. In a JBW algebra,
Jordan_operator_algebra
Closure operator
In topology, a preclosure operator or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that
Preclosure_operator
Analog of Grothendieck topology
f^{-1}({\bar {s}})} . Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets
Lawvere–Tierney_topology
density Topologies on the set of operators on a Hilbert space norm topology ultrastrong topology strong operator topology weak operator topology weak-star
List of functional analysis topics
List_of_functional_analysis_topics
Functional analysis concept
the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm. As such, results from
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Generalization of the exponential function
some Banach space X {\textstyle X} that is continuous in the strong operator topology. A strongly continuous semigroup on a Banach space X {\displaystyle
C0-semigroup
Mathematical space with a notion of closeness
\}\cup \Gamma } is a topology on X . {\displaystyle X.} Many sets of linear operators in functional analysis are endowed with topologies that are defined
Topological_space
Branch of topology
generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because
General_topology
Type of convergence in Hilbert spaces
space definition of weak convergence. Dual topology Operator topologies – topologies on the set of operators on a Hilbert space "redirect". dept.math.lsa
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Type of vector space in math
topological vector space – Space with topology generated by convex sets Operator topologies – Topologies on operators on a Hilbert space Rigged Hilbert space –
Hilbert_space
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that
Kuiper's_theorem
Typically linear operator defined in terms of differentiation of functions
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first
Differential_operator
Cellular service provider
A mobile network operator (MNO), also known as a mobile network provider, mobile network carrier, mobile, wireless service provider, wireless carrier,
Mobile_network_operator
Notion of convergence in mathematics
is found in general topology Cylinder set – Natural basic set in product spaces List of topologies – List of concrete topologies and topological spaces
Pointwise_convergence
Topological complex vector space
linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed
C*-algebra
All points and limit points in a subset of a topological space
can put other topologies rather than the standard one. If X = R {\displaystyle X=\mathbb {R} } is endowed with the lower limit topology, then cl X (
Closure_(topology)
Bounded operators with sub-unit norm
In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. Every
Contraction_(operator_theory)
Hungarian and American mathematician and physicist (1903–1957)
*-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant
John_von_Neumann
preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞(X, μ), the following topologies agree on norm
Abelian_von_Neumann_algebra
Set with operations obeying given axioms
structure. Vertex operator algebra Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology. Algebraic structures
Algebraic_structure
Mathematical structure that describes the dynamics in a Markovian open quantum system
anti-commutator. Operator topologies – Topologies on operators on a Hilbert space Von Neumann algebra – *-algebra of bounded operators on a Hilbert space
Quantum_Markov_semigroup
space reliant on the tree topology. Both the Scott and Tree topologies exhibit continuity with respect to the binary operators of application ( f applied
Computable_topology
Former telephony occupation
telephony, companies used manual telephone switchboards, and switchboard operators connected calls by inserting a pair of phone plugs into the appropriate
Switchboard_operator
Concept in set theory
in topology List of topologies – List of concrete topologies and topological spaces Arkhangelskii, A.V.; Pontryagin, L.S. (1990). General Topology. Vol
Baire_space_(set_theory)
Type of adjustable-speed drive
drive topologies, simulation and control techniques, and control hardware and software. VFDs include low- and medium-voltage AC–AC and DC–AC topologies. Pulse-width
Variable-frequency_drive
Topics referred to by the same term
Society of Toxicology Sound on tape, in television broadcasting Strong operator topology, in mathematics Sot (village), Vojvodina, Serbia Sodankylä Airfield
Sot
Axioms for defining a topology
any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to
Kuratowski_closure_axioms
strong operator topology. 1) If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1
Kaplansky_density_theorem
Topological space that is homeomorphic to a metric space
operator topology is metrizable (see Proposition II.1 in ). Non-normal spaces cannot be metrizable; important examples include the Zariski topology on
Metrizable_space
Function over linear operators
trace. Whereas the trace is a scalar-valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in
Partial_trace
Space with topology generated by convex sets
convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on
Locally convex topological vector space
Locally_convex_topological_vector_space
Mathematical function with no sudden changes
on the elements of the topology T X {\displaystyle T_{X}} ), but the continuity of f {\displaystyle f} depends on the topologies used on X {\displaystyle
Continuous_function
Call assistance services
Operator assistance refers to service provided by a telephone operator to the calling party of a telephone call. This included telephone calls made from
Operator_assistance
closure of S. Closure operator See Kuratowski closure axioms. Coarser topology If X is a set, and if T1 and T2 are topologies on X, then T1 is coarser
Glossary_of_general_topology
weak topologies coincide, the weak-operator topology on Φ(A)− coincides with the ultraweak topology. Thus the weak-operator and ultraweak topologies on
Universal representation (C*-algebra)
Universal_representation_(C*-algebra)
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
realised as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology. Of these the spin factors can be constructed
Jordan_algebra
usually assumed that the topologies are in non-decreasing order. However some authors prefer the associated closure operators { k i } i ∈ I {\displaystyle
Polytopological_space
Concept in topology
The unitary group of a separable Hilbert space (with the strong operator topology) is a Polish group. The group of homeomorphisms of a compact metric
Polish_space
Type of continuous linear operator
mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional operator such as a matrix. In
Compact_operator
Mathematical concept
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its
Gauss–Kuzmin–Wirsing_operator
Matrix decomposition
\mathbf {M} } . Compact operators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series expression
Singular_value_decomposition
Mathematical theorem
other words, the double commutant is the closure of A in the weak operator topology. See also the Kaplansky density theorem in the von Neumann algebra
Jacobson_density_theorem
to obtain information on the operator norm on an operator, acting from one Hilbert space into another, when the operator can be decomposed into almost
Cotlar–Stein_lemma
\geq \alpha \}}}} . A topology on a given set can be defined in several ways (in terms of open sets, by specifying a closure operator, etc.) Filters give
Filters_in_topology
Analyzes the topology of a manifold by studying differentiable functions on that manifold
661–692. doi:10.4310/jdg/1214437492. Roe, John (1998). Elliptic Operators, Topology and Asymptotic Method. Pitman Research Notes in Mathematics Series
Morse_theory
Method of describing higher-order polyhedra
specified, in this article (and in the literature on Conway operators in general) topology is the primary concern. Polyhedra with genus 0 (i.e. topologically
Conway_polyhedron_notation
Surjective bounded operator on a Hilbert space preserving the inner product
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples
Unitary_operator
_{t}(p)\,dt\rightarrow p} in the strong operator topology or equivalently the weak operator topology (these topologies coincide on unitaries, hence involutions
Ergodic_flow
Linear operator related to topological vector spaces
nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately
Nuclear_operator
Roughly, the number of k-dimensional holes on a topological surface
New York: Springer, ISBN 0-387-90894-3. Roe, John (1998), Elliptic Operators, Topology, and Asymptotic Methods, Research Notes in Mathematics Series, vol
Betti_number
Correspondence in functional analysis
A} , the closure of Φ ( A ) {\displaystyle \Phi (A)} in the weak operator topology is called the enveloping von Neumann algebra of A {\displaystyle A}
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Theorem in functional analysis
these topologies if and only if it also converges to f {\displaystyle f} in the other topology (the conclusion follows because two topologies are equal
Banach–Alaoglu_theorem
Topic in mathematics
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A : H → H {\displaystyle A\colon H\to
Hilbert–Schmidt_operator
Topics referred to by the same term
Terytorialnej, a Polish reserve force Wot, Nepal Wang-an Airport, Taiwan Weak operator topology, in functional analysis Web of Things, objects connected to the Web
WOT
Type of spatial relationship
topological operators are inherently complex and their implementation requires care to be taken with usability and conformance to standards. Digital topology DE-9IM
Geospatial_topology
Type of algebra
C*-subalgebra of bounded operators on HU. The enveloping von Neumann algebra of A is defined to be the closure of πU(A) in the weak operator topology (equivalently
Enveloping von Neumann algebra
Enveloping_von_Neumann_algebra
Idempotent linear transformation from a vector space to itself
object. A projection on a vector space V {\displaystyle V} is a linear operator P : V → V {\displaystyle P\colon V\to V} such that P 2 = P {\displaystyle
Projection_(linear_algebra)
a homomorphism into the bounded invertible operators which is continuous for the strong operator topology, and such that sup g ∈ G ‖ T g ‖ B ( H ) {\displaystyle
Uniformly bounded representation
Uniformly_bounded_representation
Value approached by a mathematical object
topological spaces. If X {\displaystyle X} is a topological space with topology τ {\displaystyle \tau } , and { a n } n ≥ 0 {\displaystyle \{a_{n}\}_{n\geq
Limit_(mathematics)
Topics referred to by the same term
(as opposed to weak convergence). The convergence of operators in the strong operator topology. This disambiguation page lists articles associated with
Strong_convergence
Vector space with a notion of nearness
X {\displaystyle X} has infinitely many distinct vector topologies: Some of these topologies are now described: Every linear functional f {\displaystyle
Topological_vector_space
Linear operator whose graph is closed
branch of mathematics, a closed linear operator or often a closed operator is a partially defined linear operator whose graph is closed (see closed graph
Closed_linear_operator
topology corresponds to the coproduct of C*-algebras, the tensor product of algebras. In a more specialized setting, compactifications of topologies correspond
Noncommutative_topology
strong operator topology to H. Moreover, if Vf(z) = f(H(z)), then VHV−1 – H is an operator with smooth kernel, so a Hilbert–Schmidt operator. The Hardy
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
computer-aided design, the Euler operators modify the graph of connections to add or remove details of a mesh while preserving its topology. They are named by Baumgart
Euler operator (digital geometry)
Euler_operator_(digital_geometry)
Branch of functional analysis
operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator
Borel_functional_calculus
Mathematical concept
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions;
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Overview of and topical guide to algebraic structures
are *-algebras on a Hilbert space which are equipped with the weak operator topology. Some algebraic structures find uses in disciplines outside of abstract
Outline of algebraic structures
Outline_of_algebraic_structures
Coarsest topology making certain functions continuous
the topologies { τ i } {\displaystyle \left\{\tau _{i}\right\}} in the lattice of topologies on X . {\displaystyle X.} That is, the initial topology τ {\displaystyle
Initial_topology
Compact operator for which a finite trace can be defined
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
Trace_class
simplest examples of non-self-adjoint operator algebras. Furthermore, they are closed in the weak operator topology and hyperreflexive with distance constant
Nest_algebra
Continuous dual space endowed with the topology of uniform convergence on bounded sets
identical to the topology induced by the norm on X . {\displaystyle X.} Dual topology Dual system – Dual pair of vector spaces List of topologies – List of concrete
Strong_dual_space
Type of mathematical space
In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite
Compact_space
Representation theory of the symplectic group
{\displaystyle \|U(f)\|,\|V(g)\|\leq 1} and converge in the strong operator topology to the identity operator if the supports of the measures decrease to 0. Since U(f)V(g)
Oscillator_representation
Algebraic structure
P.J. Morandi. The McKinsey–Tarski topology of an interior algebra is the intersection of the former two topologies. Grzegorczyk proved the first-order
Interior_algebra
Topological space characterized by sequences
sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via transfinite
Sequential_space
Discrete (i.e., incremental) version of infinitesimal calculus
original function. In formal terms, the difference quotient is a linear operator which takes a function as its input and produces a second function as its
Discrete_calculus
All points in the topological closure not belonging to the interior
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the
Boundary_(topology)
ball in a normed space is compact in the weak-* topology. adjoint The adjoint of a bounded linear operator T : H 1 → H 2 {\displaystyle T:H_{1}\to H_{2}}
Glossary of functional analysis
Glossary_of_functional_analysis
Type of strongly continuous semigroup
operator topology. The infinitesimal generators of analytic semigroups have the following characterization: A closed, densely defined linear operator
Analytic_semigroup
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
Boy/Male
Muslim
Orator, Preacher, Religious minister
Boy/Male
Muslim/Islamic
Orator Preacher
Girl/Female
Hindu, Indian, Sindhi, Tamil
Magnificent Poetess; Orator
Girl/Female
Arabic
Orator; Preacher
Boy/Male
Hindu, Indian, Malayalam, Marathi
Great Orator
Girl/Female
Biblical
An orator, a word.
Boy/Male
Tamil
Vakpati | வாகà¯à®ªà®¤à®¿
Great orator
Vakpati | வாகà¯à®ªà®¤à®¿
Boy/Male
Biblical
An orator.
Girl/Female
Assamese, Hindu, Indian, Tamil
Magnificent Poetess; Orator
Boy/Male
Arabic
Orator; Speaker
Boy/Male
Arabic, Muslim
Orator; Preacher
Biblical
an orator
Boy/Male
Muslim
Orator, Preacher, Religious minister
Boy/Male
Hindu
Great orator
Girl/Female
Arabic
Orator; Preacher
Boy/Male
Arabic, Indian, Muslim
Orator; Preacher
Boy/Male
Tamil
Orator
Boy/Male
Hindu, Indian, Kannada, Marathi, Tamil, Telugu
Orator
Boy/Male
Arabic
Orator; Speaker
Girl/Female
Biblical
An orator, an interpreter.
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
Girl/Female
Latin American
Young. Jove's child. Feminine of Julius.
Male
English
English surname transferred to unisex forename use, DALE means "dale, valley."
Girl/Female
Hindu, Indian, Tamil
Light
Boy/Male
Arabic, Muslim
King of Kings
Female
Norse
Old Norse name composed of the name of the god Þórr and the word frÃðr "beautiful," hence "Þórr's beauty."
Girl/Female
Australian, Portuguese
From Italy
Girl/Female
Slavic
Free.
Female
English
(Hebrew מַרְתָּה, Aramaic: מַרְתָּ×, Greek: ΜάÏθα): Greek name of Aramaic origin, MARTHA means "lady, mistress." In the bible, this is the name of a sister of Lazarus.
Girl/Female
Tamil
Hindu Goddess Parvati
Boy/Male
Egyptian
God of the moon.
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
OPERATOR TOPOLOGIES
n.
The act or process of operating; agency; the exertion of power, physical, mechanical, or moral.
imp. & p. p.
of Operate
n.
The symbol that expresses the operation to be performed; -- called also facient.
n.
Operation.
n.
In the University of Oxford, an examiner for moderations; at Cambridge, the superintendant of examinations for degrees; at Dublin, either the first (senior) or second (junior) in rank in an examination for the degree of Bachelor of Arts.
n.
Something to be done; some transformation to be made upon quantities, the transformation being indicated either by rules or symbols.
n.
One who performs some act upon the human body by means of the hand, or with instruments.
n.
An officer who is the voice of the university upon all public occasions, who writes, reads, and records all letters of a public nature, presents, with an appropriate address, those persons on whom honorary degrees are to be conferred, and performs other like duties; -- called also public orator.
n.
A mechamical arrangement for regulating motion in a machine, or producing equality of effect.
n.
A dealer in stocks or any commodity for speculative purposes; a speculator.
n.
That which is operated or accomplished; an effect brought about in accordance with a definite plan; as, military or naval operations.
n.
Any methodical action of the hand, or of the hand with instruments, on the human body, to produce a curative or remedial effect, as in amputation, etc.
n.
The officer who presides over an assembly to preserve order, propose questions, regulate the proceedings, and declare the votes.
a.
Alt. of Operatical
n.
One who, or that which, operates or produces an effect.
n.
One fond of his own opinious; one who holds an opinion.
v. t.
To put into, or to continue in, operation or activity; to work; as, to operate a machine.
n.
A laboratory.
n.
Effect produced; influence.
n.
The method of working; mode of action.