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Element of *-algebra where x* equals x
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a ∗ {\displaystyle a=a^{*}} ). Let A {\displaystyle
Self-adjoint_element
that a = b b ∗ {\displaystyle a=bb^{*}} . There exists a (unique) self-adjoint element c ∈ A s a {\displaystyle c\in {\mathcal {A}}_{sa}} such that a =
Positive_element
Topological complex vector space
is usually denoted ≥ {\displaystyle \geq } . In this ordering, a self-adjoint element x ∈ A {\displaystyle x\in A} satisfies x ≥ 0 {\displaystyle x\geq
C*-algebra
is a self-adjoint element, then at least for every odd n ∈ N {\displaystyle n\in \mathbb {N} } there is a uniquely determined self-adjoint element b ∈
Continuous functional calculus
Continuous_functional_calculus
Result about when a matrix can be diagonalized
perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral
Spectral_theorem
mathematics, an element of a *-algebra is called normal if it commutates with its adjoint. Let A {\displaystyle {\mathcal {A}}} be a *-Algebra. An element a ∈ A
Normal_element
Matrix equal to its conjugate-transpose
Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transpose—that is, its element in the i-th row and j-th
Hermitian_matrix
Type of vector space in math
the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra. An element A of B(H) is called 'self-adjoint' or 'Hermitian'
Hilbert_space
Distinguished element of a Lie algebra's center
multiplication to simplify. As an example of how the Casimir element acts on a representation, consider the adjoint representation of s l 2 ( C ) {\displaystyle {\mathfrak
Casimir_element
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
{A} } with entries a i j {\displaystyle a_{ij}} is called Hermitian or self-adjoint if A = A H {\displaystyle \mathbf {A} =\mathbf {A} ^{\mathrm {H} }} ;
Conjugate_transpose
Matrix operation which flips a matrix over its diagonal
resulting in an isomorphism between the transpose and adjoint of u. The matrix of the adjoint of a map is the transposed matrix only if the bases are
Transpose
Linear operator defined on a dense linear subspace
Every self-adjoint operator is maximal symmetric. The converse is false. An operator is called essentially self-adjoint if its closure is self-adjoint. An
Unbounded_operator
Branch of functional analysis
arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. If T is a self-adjoint operator on a finite-dimensional
Borel_functional_calculus
theorem. If A {\displaystyle A} is a self-adjoint algebra of operators in B ( H ) {\displaystyle B(H)} , then each element a {\displaystyle a} in the unit
Kaplansky_density_theorem
earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a
Regular element of a Lie algebra
Regular_element_of_a_Lie_algebra
into a C* -algebra B {\displaystyle {\mathcal {B}}} , and A is a self-adjoint element of A {\displaystyle {\mathcal {A}}} satisfying m ≤ {\displaystyle
Bhatia–Davis_inequality
Formulation of quantum mechanics on a Hilbert Space
observables of a quantum system are defined to be the (possibly unbounded) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } . A state
Dirac–von_Neumann_axioms
Construction in functional analysis, useful to solve differential equations
the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H)
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Typically linear operator defined in terms of differentiation of functions
the adjoint operator. When T ∗ {\displaystyle T^{*}} is defined according to this formula, it is called the formal adjoint of T. A (formally) self-adjoint
Differential_operator
Existence of certain infima or suprema of a given poset
for any x. Dually, the existence of an upper adjoint for j is equivalent to X having a greatest element. Another simple mapping is the function q: X →
Completeness_(order_theory)
Theorem about the dual of a Hilbert space
} Self-adjoint operators A continuous linear operator A : H → H {\displaystyle A:H\to H} is called self-adjoint if it is equal to its own adjoint; that
Riesz_representation_theorem
Matrix whose conjugate transpose is its negative (additive inverse)
thought of as skew-adjoint (since they are like 1 × 1 {\displaystyle 1\times 1} matrices), whereas real numbers correspond to self-adjoint operators. For
Skew-Hermitian_matrix
Operator in probability theory
Since Cov is symmetric in its arguments, the covariance operator is self-adjoint. Even more generally, for a probability measure P on a Banach space B
Covariance_operator
Theorem in functional analysis
the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below. Let A be a n × n Hermitian matrix.
Min-max_theorem
Set of eigenvalues of a matrix
\sigma _{\mathrm {ess} ,k}(A),\ 1\leq k\leq 5} , coincide in the case of self-adjoint operators. The essential spectrum σ e s s , 1 ( A ) {\displaystyle \sigma
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Class of ordinary differential equations
The differential equation (1) is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear homogenous ordinary differential equations
Sturm–Liouville_theory
(on a complex Hilbert space) continuous linear operator
= U − 1 {\displaystyle U^{\ast }=U^{-1}} Hermitian operators (i.e., self-adjoint operators): N ∗ = N {\displaystyle N^{\ast }=N} skew-Hermitian operators:
Normal_operator
Partially ordered set in which all subsets have both a supremum and infimum
y\iff x\leq g(y)} where f is called the lower adjoint and g is called the upper adjoint. By the adjoint functor theorem, a monotone map between any pair
Complete_lattice
involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain
Spectral_triple
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
n×n self-adjoint real matrices, as above. The Jordan algebra of n×n self-adjoint complex matrices, as above. The Jordan algebra of n×n self-adjoint quaternionic
Jordan_algebra
Probability problem
}^{\infty }x^{n}\,d\mu (x)} suggests that μ is the spectral measure of a self-adjoint operator. (More precisely stated, μ is the spectral measure for an operator
Hamburger_moment_problem
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
Jordan_operator_algebra
Mathematical use of "for all"
its domain. The left adjoint of this functor is the existential quantifier ∃ f {\displaystyle \exists _{f}} and the right adjoint is the universal quantifier
Universal_quantification
Type of continuous linear operator
and T : H → H {\displaystyle T:H\to H} is compact and self-adjoint, then every nonzero element of the spectrum of T {\displaystyle T} is an eigenvalue
Compact_operator
Mathematical category
\operatorname {Presh} (D)} that admits a finite-limit-preserving left adjoint. C {\displaystyle C} is the category of sheaves on a Grothendieck site
Topos
Random matrix with gaussian entries
the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution
Gaussian_ensemble
In mathematics, element with a multiplicative inverse
algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if
Unit_(ring_theory)
Mathematical use of "there exists"
elementary topoi, the existential quantifier can be understood as the left adjoint of a functor between power sets, the inverse image functor of a function
Existential_quantification
Operator generalizing the Laplacian in differential geometry
consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions f {\displaystyle f} and
Laplace–Beltrami_operator
Integral expressing the amount of overlap of one function as it is shifted over another
(1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers
Convolution
Natural number
group SO(8). The special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent
8
Net in a normed algebra
For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive
Approximate_identity
Open convex self-dual cones
single element a in E, including recursion formulas for L(am), also hold in EC. Since for b in E, L(b) is still self-adjoint on EC, the adjoint relation
Symmetric_cone
Group representation
transpose. Since the adjoint of a matrix is the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, ρ ∗ ( g ) {\displaystyle
Dual_representation
Function that is its own inverse
conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution. The definition of involution extends readily to
Involution_(mathematics)
Mathematical set of all subsets of a set
elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function
Power_set
Glossary of terms used in branch of mathematics
adjoint of G and G is called the upper adjoint of F. Greatest element. For a subset X of a poset P, an element a of X is called the greatest element of
Glossary_of_order_theory
is an idempotent p that is self-adjoint (p* = p). A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal)
Baer_ring
Writing Lie algebra sets as matrices
Lie algebras. In quantum theory, one considers "observables" that are self-adjoint operators on a Hilbert space. The commutation relations among these operators
Lie_algebra_representation
Branch of functional analysis
context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann
Operator_algebra
Matrices similar to diagonal matrices
Kuroda, showed the following: For any p > 1 {\displaystyle p>1} , any self-adjoint operator T {\displaystyle T} on a Hilbert space H {\displaystyle H}
Diagonalizable_matrix
Self-self morphism
are the involutions; i.e., the functions coinciding with their inverses. Adjoint endomorphism Epimorphism (surjective homomorphism) Frobenius endomorphism
Endomorphism
General concept and operation in mathematics
more generally model categories. Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D HomD(F(c), d) ≅ HomC(c, G(d)), in
Duality_(mathematics)
Correspondence between properties of a category and its opposite
In this context, the duality is often called Eckmann–Hilton duality. Adjoint functor Dual object Duality (mathematics) Opposite category Pulation square
Dual_(category_theory)
When one nuclear reaction causes more
000,000 neutron lifetimes can pass. The average (also referred to as the adjoint unweighted) prompt neutron lifetime takes into account all prompt neutrons
Nuclear_chain_reaction
Interaction of a quantum system with a classical observer
John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". These observables
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
First case of a Lie group that is both compact and non-abelian
{su}}(2)={\mathfrak {sl}}(2;\mathbb {C} )~.} (Skew self-adjoint matrices with trace zero plus self-adjoint matrices with trace zero gives all matrices with
Representation theory of SU(2)
Representation_theory_of_SU(2)
Mathematical operator
lower adjoint of a Galois connection between P and A, with the upper adjoint being the embedding of A into P. Furthermore, every lower adjoint of an embedding
Closure_operator
Eigenvector, eigenvalue, eigenfunction Hermitian operator self-adjoint operator, Hermitian adjoint Hilbert matrix Shift operator Symmetric matrix Parseval's
List of functional analysis topics
List_of_functional_analysis_topics
Type of graph in mathematics and physics
{c} =S(k){\hat {\mathbf {c} }}} . For self-adjoint matching conditions S {\displaystyle S} is unitary. An element of σ ( u v ) ( v w ) {\displaystyle \sigma
Quantum_graph
Axiom of set theory
which satisfies the appropriate solution set condition has a left adjoint (the Freyd adjoint functor theorem). There are several weaker statements unprovable
Axiom_of_choice
Study of vector bundles, principal bundles, and fibre bundles
example is the lowercase a adjoint bundle ad ( P ) {\displaystyle \operatorname {ad} (P)} constructed using the adjoint representation ρ : G → Aut
Gauge_theory_(mathematics)
self-adjoint A self-adjoint operator is a bounded operator whose adjoint is itself. More generally, a closed densely defined operator is called self-adjoint
Glossary of functional analysis
Glossary_of_functional_analysis
Universal construction of a complex Lie group from a real Lie group
iX, where u is a unitary operator in the compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex
Complexification_(Lie_group)
Mathematical study of linear operators
perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral
Operator_theory
Mathematical objects that generalise the notion of Hilbert spaces
of a C*-algebra A {\displaystyle A} is said to be positive if it is self-adjoint with non-negative spectrum.) An analogue to the Cauchy–Schwarz inequality
Hilbert_C*-module
Nilpotent subalgebra of a Lie algebra
{\mathfrak {g}}} that consists of semisimple elements (an element is semisimple if the adjoint endomorphism induced by it is diagonalizable). A Cartan subalgebra
Cartan_subalgebra
Apparent lack of definite state before measurement of quantum systems
theory was based in turn on the theory of projection-valued measures for self-adjoint operators that had been recently developed (by von Neumann and independently
Quantum_indeterminacy
Differential operator in mathematics
v\,dx=\int _{\Omega }v\,\Delta u\,dx,} so the Laplacian is formally self-adjoint. Taking u = v {\displaystyle u=v} gives the energy identity ∫ Ω u Δ u
Laplace_operator
Mathematical compact operator
be composed with a positive operator with trivial kernel to produce a self-adjoint operator. Such operators arose naturally in the work on integral operators
Symmetrizable compact operator
Symmetrizable_compact_operator
operators on a Kähler manifold relating the Dolbeault operators and their adjoints, contraction and wedge operators of the Kähler form, and the Laplacians
Kähler_identities
Calculation rule in quantum mechanics
{\displaystyle |\psi \rangle } (see Bra–ket notation), corresponds to a self-adjoint operator A {\displaystyle A} whose spectrum is discrete if: the measured
Born_rule
Exterior algebraic map taking tensors from p forms to n-p forms
k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This
Hodge_star_operator
Matrix equal to its transpose
element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint
Symmetric_matrix
Type of matrix representation
{\displaystyle U} is unitary, and X {\displaystyle X} is the unique self-adjoint logarithm of the matrix P {\displaystyle P} . This decomposition is useful
Polar_decomposition
Relativistic quantum mechanical wave equation
particular group element. The gauge field is a matrix valued gauge field A μ a {\displaystyle A_{\mu }^{a}} which transforms in the adjoint representation
Dirac_equation
truly meshfree, spectral convergent (numerical observations), symmetric (self-adjoint PDEs), integration-free, and easy to learn and implement. The method
Boundary_knot_method
Matroid with no linear representation
matroids also have applications in coding theory. The Vámos matroid has no adjoint. This means that the dual lattice of the geometric lattice of the Vámos
Vámos_matroid
Theory of logic to account for observations from quantum theory
this article assumes the reader is familiar with the spectral theory of self-adjoint operators on a Hilbert space. However, the main ideas can be understood
Quantum_logic
Concept in quantum mechanics
self-adjoint operator A {\displaystyle A} corresponding to the set of real-valued eigenvalues { a n } {\displaystyle \{a_{n}\}} . If the self-adjoint
Complete set of commuting observables
Complete_set_of_commuting_observables
Mathematical result in differential geometry
(Gilkey 1994). If D is a differential operator with adjoint D*, then D*D and DD* are self adjoint operators whose non-zero eigenvalues have the same multiplicities
Atiyah–Singer_index_theorem
Lie algebra all of which elements are semisimple
semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is called toral if the adjoint representation of h {\displaystyle {\mathfrak {h}}} on g {\displaystyle
Toral_subalgebra
Operators useful in quantum mechanics
increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry
Creation and annihilation operators
Creation_and_annihilation_operators
condition. Recall that a projection of a C*-algebra is a self-adjoint idempotent element. A C*-algebra A is an AW*-algebra if for every subset S of
AW*-algebra
Mathematical inequality relating inner products and norms
prove the spectral theorem for self-adjoint operators in the finite-dimensional case. Let A {\displaystyle A} be a self-adjoint operator on a finite-dimensional
Cauchy–Schwarz_inequality
Theorem of convex functions
x+(1-\lambda )y{\bigr )}\leq \lambda f(x)+(1-\lambda )f(y)} for every pair of self‐adjoint operators x and y (with spectra in I) and every scalar λ ∈ [ 0 , 1 ]
Jensen's_inequality
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems
Neumann–Poincaré_operator
Equivalence relation on rings
ring property P {\displaystyle {\mathcal {P}}} is Morita invariant. An element e in a ring R is a full idempotent when e2 = e and ReR = R. P {\displaystyle
Morita_equivalence
Analog of the continuous Laplace operator
discrete Laplacian on an infinite grid is of key interest; since it is a self-adjoint operator, it has a real spectrum. For the convention Δ = I − M {\displaystyle
Discrete_Laplace_operator
Method for approximating eigenvalues
Eugene (2004). "Spectral pollution and second order relative spectra for self-adjoint operators". IMA Journal of Numerical Analysis. 24 (3): 393–416. arXiv:math/0212087
Rayleigh–Ritz_method
Algebraic structure with "nice" duality properties
S-modules to the category of left R-modules has both a left and a right adjoint, called co-restriction and restriction, respectively. The ring extension
Frobenius_algebra
Associative Artinian algebra with a trivial Jacobson radical
n {\displaystyle n} , i.e. A {\displaystyle A} is not nilpotent. Any self-adjoint subalgebra A {\displaystyle A} of n × n {\displaystyle n\times n} matrices
Semisimple_algebra
Conversion of a matrix or a tensor to a vector
X ) = A X − X A {\displaystyle \operatorname {ad} _{A}(X)=AX-XA} (the adjoint endomorphism of the Lie algebra gl(n, C) of all n×n matrices with complex
Vectorization_(mathematics)
Construction in algebra
condition of stability, adr(h)(A) ⊆ A for all h in H, where the right adjoint mapping adr is defined by adr(h)(a) = S(h(1))ah(2) for all a in A, h in
Hopf_algebra
Conjecture on zeros of the zeta function
suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts
Riemann_hypothesis
Number representing a continuous quantity
therefore non-Archimedean ordered fields. Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals
Real_number
Generalized function whose value is zero everywhere except at zero
Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as f = ∑ n = 1 ∞ α n φ n . {\displaystyle
Dirac_delta_function
American mathematician
1142/S0219876204000083 Bourne, David; Elman, Howard; Osborn, John E. (2009), "A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction. {II}
John E. Osborn (mathematician)
John_E._Osborn_(mathematician)
Hilbert space of square-integrable holomorphic functions of n complex variables
creation and annihilation operators would be adjoints of each other. We may now construct self-adjoint "position" and "momentum" operators Aj and Bj
Segal–Bargmann_space
Mapping equal to its square under mapping composition
construction of projection-valued measures. In their sense, a projection is a self-adjoint idempotent linear operator. In differential topology, any fiber bundle
Projection_(mathematics)
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
Girl/Female
Hebrew Biblical
Rock.
Girl/Female
British, English
Soft
Boy/Male
Muslim
Sword
Female
Irish
Variant spelling of Irish Éadan, ÉADAOIN means "face" or perhaps "against" or "opposite."
Male
Yiddish
(סֶעף) Variant spelling of Yiddish Zeff, SEFF means "wolf."
Boy/Male
African, Arabic, Hindu, Indian, Muslim, Sindhi, Swahili
Sword; Brave; Sword of Religion
Biblical
a rock
Surname or Lastname
English
English : from Middle English selle, a rough hut of the type normally occupied by animals, hence a topographic name for someone who lived in a hut like this. In many cases the name may have been in effect a metonymic occupational name for a herdsman.Americanized spelling of Hungarian and Hungarian Jewish Széll, a topographic name for someone who lived in a spot exposed to the wind, from Hungarian szél ‘wind’.German : variant of Selle.
Girl/Female
Egyptian
Girl/Female
African, Australian, British, Chinese, Christian, English, French, Greek, Hawaiian, Hebrew
Saviour; Ewe of West Africa; Goddess of the Moon; Cliff; Rock
Boy/Male
British, English, Nigerian, Norwegian
Rock
Female
Egyptian
, a form of Isis.
Boy/Male
Indian
Sword
Male
Welsh
Welsh form of Greek SolomÅn, SELYF means "peaceable."Â
Boy/Male
Welsh
peace'.
Male
English
(סֶלַע) Anglicized form of Hebrew Cela, SELA means "a rock." In the Old Testament bible, this is the name of the capital city of Edom, possibly an early name for Petra. In use as a unisex name.
Surname or Lastname
English (East Anglia)
English (East Anglia) : from the Middle English personal name Saulf, Old English Sǣwulf, composed of the elements sǣ ‘sea’ + wulf ‘wolf’.
Boy/Male
Biblical
A rock.
Boy/Male
Muslim/Islamic
Sword
Boy/Male
British, English, Hebrew
A Tree
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
Girl/Female
Arabic, Indian, Muslim, Sindhi
Break of Dawn; Daybreak; Dawn
Boy/Male
Muslim
Decorate, Beautify
Boy/Male
Anglo Saxon
From the cornered hill.
Boy/Male
Indian, Sanskrit
Friend
Boy/Male
Indian, Sikh
Light of the World
Girl/Female
Indian
Honor
Girl/Female
Indian, Sikh
New Sol
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sikh, Telugu
Speech; Promise
Girl/Female
Indian, Sikh
True
Boy/Male
Indian
Servant of the one
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
SELF ADJOINT-ELEMENT
n.
Self-denial; self-renunciation; self-sacrifice.
n.
The act of estimating one's self; self-esteem.
n.
Self-devotion.
n.
Self-communion.
imp. & p. p.
of Adjoin
n.
Imposture practiced on one's self; self-deceit.
n.
Control of one's self; restraint exercised over one's self; self-command.
n.
The idolizing of one's self; immoderate self-conceit.
n.
Self-deceit.
n.
Restraint over one's self; self-control; self-command.
n.
Self-love.
v. i.
To join one's self.
n.
Enjoyment of one's self; self-satisfaction.
a.
Dependent on one's self; self-depending; self-reliant.
a.
Self-repelling.
n.
The act of governing one's self, or the state of being governed by one's self; self-control; self-command.
a.
Disposed to self-assertion; self-asserting.
n.
Self.
n.
Faith in one's self; self-reliance.
a.
Refusing to gratify one's self; self-sacrificing.