Misplaced Pages

#459540

53-608: [REDACTED] Look up Trace , trace , traces , or tracing in Wiktionary, the free dictionary. Trace may refer to: Arts and entertainment [ edit ] Music [ edit ] Trace (Son Volt album) , 1995 Trace (Died Pretty album) , 1993 Trace (band) , a Dutch progressive rock band The Trace (album) , by Nell Other uses in arts and entertainment [ edit ] Trace (magazine) , British hip-hop magazine Trace (manhwa) ,

106-498: A continuous linear functional φ T {\displaystyle \varphi _{T}} on B 1 {\displaystyle B_{1}} by φ T ( A ) = Tr ⁡ ( A T ) . {\displaystyle \varphi _{T}(A)=\operatorname {Tr} (AT).} This correspondence between bounded linear operators and elements φ T {\displaystyle \varphi _{T}} of

159-460: A positive bounded linear operator on H {\displaystyle H} . The trace of A {\displaystyle A} is denoted by Tr ⁡ ( A ) {\displaystyle \operatorname {Tr} (A)} and defined as independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator T : H → H {\displaystyle T:H\rightarrow H}

212-488: A 1988 Tunisian film Trace, a fictional character in the game Metroid Prime Hunters Trace, the protagonist of Axiom Verge Trace, another name for Portgas D. Ace , a fictional character in the manga One Piece Trace, the main brand for a number of music channels such as Trace Urban Language [ edit ] Trace (deconstruction) , a concept in Derridian deconstruction Trace (linguistics) ,

265-415: A Dutch progressive rock band The Trace (album) , by Nell Other uses in arts and entertainment [ edit ] Trace (magazine) , British hip-hop magazine Trace (manhwa) , a Korean internet cartoon Trace (novel) , a novel by Patricia Cornwell The Trace (film) , a 1994 Turkish film The Trace (video game) , 2015 video game Sama (film) , alternate title The Trace ,

318-423: A Korean internet cartoon Trace (novel) , a novel by Patricia Cornwell The Trace (film) , a 1994 Turkish film The Trace (video game) , 2015 video game Sama (film) , alternate title The Trace , a 1988 Tunisian film Trace, a fictional character in the game Metroid Prime Hunters Trace, the protagonist of Axiom Verge Trace, another name for Portgas D. Ace , a fictional character in

371-455: A certain set of operators in a Hilbert space Trace operator , a restriction-to-boundary operator in a Sobolev space Physical sciences [ edit ] TRACE (Transition Region and Coronal Explorer), a NASA satellite Trace element , an element which composes less than 0.1% of a sample Trace evidence , material found at a crime scene Trace fossil , fossil record of biological activity Trace radioisotope , an element that

424-755: A draft animal's gear Track and trace , in goods distribution and logistics Trace, a type of eyeblink conditioning An Italian-derived synonym for star fort , in fortification Trade reporting and compliance engine , in the U.S. financial markets, a reporting system for US bond transactions administered by the FINRA The Trace (website) , a website covering gun issues in the United States See also [ edit ] Traceability Tracer (disambiguation) Tracing (disambiguation) Tracking (disambiguation) DTrace Human-Trace (Ichnos-Anthropos) Topics referred to by

477-678: A pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of ℓ ∞ ( N ) , {\displaystyle \ell ^{\infty }(\mathbb {N} ),} the compact operators that of c 0 {\displaystyle c_{0}} (the sequences convergent to 0), Hilbert–Schmidt operators correspond to ℓ 2 ( N ) , {\displaystyle \ell ^{2}(\mathbb {N} ),} and finite-rank operators to c 00 {\displaystyle c_{00}} (the sequences that have only finitely many non-zero terms). To some extent,

530-415: A printed circuit board Stack trace , report of the active steps of a computer program's execution Trace cache , a specialized CPU cache to speed up executable instruction fetch Mathematics [ edit ] Trace (linear algebra) , the sum of the elements on the main diagonal of a square matrix or a linear transformation Field trace , a particular trace in field theory Trace class ,

583-528: A sample Trace evidence , material found at a crime scene Trace fossil , fossil record of biological activity Trace radioisotope , an element that is found in small quantities because it undergoes radioactive decay Seismic trace , in seismology, record of ground movement from a seismograph Trace (precipitation) , in earth science, an amount of precipitation that falls that is too small to be measured with standard units Places [ edit ] Trace, West Virginia The Trace (Land Between

SECTION 10

#1732766166460

636-595: A sequence ( α i ) i {\displaystyle \left(\alpha _{i}\right)_{i}} of non-negative numbers with α i → 0 {\displaystyle \alpha _{i}\to 0} such that T x = ∑ i α i ⟨ x , v i ⟩ u i  for all  x ∈ H . {\displaystyle Tx=\sum _{i}\alpha _{i}\langle x,v_{i}\rangle u_{i}\quad {\text{ for all }}x\in H.} Making

689-550: A sign Sign-trace, a detectable amount conceived of by Béatrice Galinon-Mélénec Mathematics, science, and technology [ edit ] Computing and electronics [ edit ] TRACE, a request method in the HTTP protocol Traces, the equivalence classes of strings of a trace monoid , studied in trace theories of concurrent computation Digital traces , the traces of activities and behaviours that people leave when they interact in digital environments Packet trace,

742-698: A stream in Missouri Trace Creek (Cub Creek) , a stream in Missouri Trace Creek (Twelvemile Creek) , a stream in Missouri Trace Lake , a lake in Minnesota Other uses [ edit ] Trace (name) , a given name, nickname, and surname A synonym for trail , as in Natchez Trace Trace (tack) , part of a draft animal's gear Track and trace , in goods distribution and logistics Trace,

795-403: A syntactic placeholder resulting from a transformation TRACE (psycholinguistics) , a psycholinguistic model of speech perception Trace (semiology) , the history carried by a sign Sign-trace, a detectable amount conceived of by Béatrice Galinon-Mélénec Mathematics, science, and technology [ edit ] Computing and electronics [ edit ] TRACE, a request method in

848-407: A timestamped sequence of packets captured on a computer network with a sniffer or similar tools Signal trace , a printed or etched wire on a printed circuit board Stack trace , report of the active steps of a computer program's execution Trace cache , a specialized CPU cache to speed up executable instruction fetch Mathematics [ edit ] Trace (linear algebra) , the sum of

901-432: A trace class operator. That is, suppose K {\displaystyle K} is a continuous symmetric positive-definite kernel on L 2 ( [ a , b ] ) {\displaystyle L^{2}([a,b])} , defined as then the associated Hilbert–Schmidt integral operator T K {\displaystyle T_{K}} is trace class, i.e., Every finite-rank operator

954-567: A trace-class operator in a separable Hilbert space H , {\displaystyle H,} and let { λ n ( A ) } n = 1 N ≤ ∞ {\displaystyle \{\lambda _{n}(A)\}_{n=1}^{N\leq \infty }} be the eigenvalues of A . {\displaystyle A.} Let us assume that λ n ( A ) {\displaystyle \lambda _{n}(A)} are enumerated with algebraic multiplicities taken into account (that is, if

1007-532: A type of eyeblink conditioning An Italian-derived synonym for star fort , in fortification Trade reporting and compliance engine , in the U.S. financial markets, a reporting system for US bond transactions administered by the FINRA The Trace (website) , a website covering gun issues in the United States See also [ edit ] Traceability Tracer (disambiguation) Tracing (disambiguation) Tracking (disambiguation) DTrace Human-Trace (Ichnos-Anthropos) Topics referred to by

1060-438: Is ‖ T ‖ 2 = [ Tr ⁡ ( T ∗ T ) ] 1 / 2 = ( ∑ i α i 2 ) 1 / 2 . {\displaystyle \|T\|_{2}=\left[\operatorname {Tr} \left(T^{*}T\right)\right]^{1/2}=\left(\sum _{i}\alpha _{i}^{2}\right)^{1/2}.} Also,

1113-411: Is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator A on H (and into H ), Tr ⁡ ( A ( x ⊗ y ) ) = ⟨ A x , y ⟩ . {\displaystyle \operatorname {Tr} (A(x\otimes y))=\langle Ax,y\rangle .} Let A {\displaystyle A} be

SECTION 20

#1732766166460

1166-451: Is a positive operator, for any orthonormal basis u i , {\displaystyle u_{i},} one has ∑ i ⟨ T f u i , u i ⟩ = f ( I ) ≤ ‖ f ‖ , {\displaystyle \sum _{i}\langle T_{f}u_{i},u_{i}\rangle =f(I)\leq \|f\|,} where I {\displaystyle I}

1219-703: Is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace of B 1 ( H ) {\displaystyle B_{1}(H)} (when endowed with the trace norm). Given any x , y ∈ H , {\displaystyle x,y\in H,} define the operator x ⊗ y : H → H {\displaystyle x\otimes y:H\to H} by ( x ⊗ y ) ( z ) := ⟨ z , y ⟩ x . {\displaystyle (x\otimes y)(z):=\langle z,y\rangle x.} Then x ⊗ y {\displaystyle x\otimes y}

1272-472: Is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms. The dual space of c 0 {\displaystyle c_{0}} is ℓ 1 ( N ) . {\displaystyle \ell ^{1}(\mathbb {N} ).} Similarly, we have that the dual of compact operators, denoted by K ( H ) ∗ , {\displaystyle K(H)^{*},}

1325-502: Is called trace class if and only if where | T | := T ∗ T {\displaystyle |T|:={\sqrt {T^{*}T}}} denotes the positive-semidefinite Hermitian square root . The trace-norm of a trace class operator T is defined as ‖ T ‖ 1 := Tr ⁡ ( | T | ) . {\displaystyle \|T\|_{1}:=\operatorname {Tr} (|T|).} One can show that

1378-397: Is convergent, and T {\displaystyle T} is finite-rank iff the sequence ( α i ) i {\displaystyle \left(\alpha _{i}\right)_{i}} has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when H {\displaystyle H}

1431-467: Is different from Wikidata All article disambiguation pages All disambiguation pages Trace [REDACTED] Look up Trace , trace , traces , or tracing in Wiktionary, the free dictionary. Trace may refer to: Arts and entertainment [ edit ] Music [ edit ] Trace (Son Volt album) , 1995 Trace (Died Pretty album) , 1993 Trace (band) ,

1484-442: Is different from Wikidata All article disambiguation pages All disambiguation pages Trace class In quantum mechanics, quantum states are described by density matrices , which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators , though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use

1537-566: Is found in small quantities because it undergoes radioactive decay Seismic trace , in seismology, record of ground movement from a seismograph Trace (precipitation) , in earth science, an amount of precipitation that falls that is too small to be measured with standard units Places [ edit ] Trace, West Virginia The Trace (Land Between the Lakes) , a scenic roadway in Kentucky and Tennessee Trace Creek (Castor River) ,

1590-447: Is infinite-dimensional: {  finite rank  } ⊆ {  trace class  } ⊆ {  Hilbert--Schmidt  } ⊆ {  compact  } . {\displaystyle \{{\text{ finite rank }}\}\subseteq \{{\text{ trace class }}\}\subseteq \{{\text{ Hilbert--Schmidt }}\}\subseteq \{{\text{ compact }}\}.} The trace-class operators are given

1643-477: Is isometrically isomorphic to B 1 . {\displaystyle B_{1}.} Recall that the dual of ℓ 1 ( N ) {\displaystyle \ell ^{1}(\mathbb {N} )} is ℓ ∞ ( N ) . {\displaystyle \ell ^{\infty }(\mathbb {N} ).} In the present context, the dual of trace-class operators B 1 {\displaystyle B_{1}}

Trace - Misplaced Pages Continue

1696-455: Is the bounded operators B ( H ) . {\displaystyle B(H).} More precisely, the set B 1 {\displaystyle B_{1}} is a two-sided ideal in B ( H ) . {\displaystyle B(H).} So given any operator T ∈ B ( H ) , {\displaystyle T\in B(H),} we may define

1749-795: Is the identity operator: I = ∑ i ⟨ ⋅ , u i ⟩ u i . {\displaystyle I=\sum _{i}\langle \cdot ,u_{i}\rangle u_{i}.} But this means that T f {\displaystyle T_{f}} is trace-class. An appeal to polar decomposition extend this to the general case, where T f {\displaystyle T_{f}} need not be positive. A limiting argument using finite-rank operators shows that ‖ T f ‖ 1 = ‖ f ‖ . {\displaystyle \|T_{f}\|_{1}=\|f\|.} Thus K ( H ) ∗ {\displaystyle K(H)^{*}}

1802-426: Is the rank-one operator given by S x , y ( h ) = ⟨ h , y ⟩ x . {\displaystyle S_{x,y}(h)=\langle h,y\rangle x.} This identification works because the finite-rank operators are norm-dense in K ( H ) . {\displaystyle K(H).} In the event that T f {\displaystyle T_{f}}

1855-802: Is the trace-class operators, denoted by B 1 . {\displaystyle B_{1}.} The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let f ∈ K ( H ) ∗ , {\displaystyle f\in K(H)^{*},} we identify f {\displaystyle f} with the operator T f {\displaystyle T_{f}} defined by ⟨ T f x , y ⟩ = f ( S x , y ) , {\displaystyle \langle T_{f}x,y\rangle =f\left(S_{x,y}\right),} where S x , y {\displaystyle S_{x,y}}

1908-423: The HTTP protocol Traces, the equivalence classes of strings of a trace monoid , studied in trace theories of concurrent computation Digital traces , the traces of activities and behaviours that people leave when they interact in digital environments Packet trace, a timestamped sequence of packets captured on a computer network with a sniffer or similar tools Signal trace , a printed or etched wire on

1961-422: The sequence space ℓ 1 ( N ) . {\displaystyle \ell ^{1}(\mathbb {N} ).} Indeed, it is possible to apply the spectral theorem to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an ℓ 1 {\displaystyle \ell ^{1}} sequence with respect to some choice of

2014-407: The singular values { s m ( A ) } m = 1 M {\displaystyle \{s_{m}(A)\}_{m=1}^{M}} of the compact operator A . {\displaystyle A.} One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces , with trace-class operators as the noncommutative analogue of

2067-418: The Lakes) , a scenic roadway in Kentucky and Tennessee Trace Creek (Castor River) , a stream in Missouri Trace Creek (Cub Creek) , a stream in Missouri Trace Creek (Twelvemile Creek) , a stream in Missouri Trace Lake , a lake in Minnesota Other uses [ edit ] Trace (name) , a given name, nickname, and surname A synonym for trail , as in Natchez Trace Trace (tack) , part of

2120-438: The above heuristic comments more precise, we have that T {\displaystyle T} is trace-class iff the series ∑ i α i {\textstyle \sum _{i}\alpha _{i}} is convergent, T {\displaystyle T} is Hilbert–Schmidt iff ∑ i α i 2 {\textstyle \sum _{i}\alpha _{i}^{2}}

2173-773: The algebraic multiplicity of λ {\displaystyle \lambda } is k , {\displaystyle k,} then λ {\displaystyle \lambda } is repeated k {\displaystyle k} times in the list λ 1 ( A ) , λ 2 ( A ) , … {\displaystyle \lambda _{1}(A),\lambda _{2}(A),\dots } ). Lidskii's theorem (named after Victor Borisovich Lidskii ) states that Tr ⁡ ( A ) = ∑ n = 1 N λ n ( A ) {\displaystyle \operatorname {Tr} (A)=\sum _{n=1}^{N}\lambda _{n}(A)} Note that

Trace - Misplaced Pages Continue

2226-471: The definition of the trace of a matrix . If H {\displaystyle H} is complex, then A {\displaystyle A} is always self-adjoint (i.e. A = A ∗ = | A | {\displaystyle A=A^{*}=|A|} ) though the converse is not necessarily true. Given a bounded linear operator T : H → H {\displaystyle T:H\to H} , each of

2279-450: The elements on the main diagonal of a square matrix or a linear transformation Field trace , a particular trace in field theory Trace class , a certain set of operators in a Hilbert space Trace operator , a restriction-to-boundary operator in a Sobolev space Physical sciences [ edit ] TRACE (Transition Region and Coronal Explorer), a NASA satellite Trace element , an element which composes less than 0.1% of

2332-657: The following statements is equivalent to T {\displaystyle T} being in the trace class: Let T {\displaystyle T} be a bounded self-adjoint operator on a Hilbert space. Then T 2 {\displaystyle T^{2}} is trace class if and only if T {\displaystyle T} has a pure point spectrum with eigenvalues { λ i ( T ) } i = 1 ∞ {\displaystyle \left\{\lambda _{i}(T)\right\}_{i=1}^{\infty }} such that Mercer's theorem provides another example of

2385-406: The manga One Piece Trace, the main brand for a number of music channels such as Trace Urban Language [ edit ] Trace (deconstruction) , a concept in Derridian deconstruction Trace (linguistics) , a syntactic placeholder resulting from a transformation TRACE (psycholinguistics) , a psycholinguistic model of speech perception Trace (semiology) , the history carried by

2438-485: The relationships between these classes of operators are similar to the relationships between their commutative counterparts. Recall that every compact operator T {\displaystyle T} on a Hilbert space takes the following canonical form: there exist orthonormal bases ( u i ) i {\displaystyle (u_{i})_{i}} and ( v i ) i {\displaystyle (v_{i})_{i}} and

2491-446: The same term [REDACTED] This disambiguation page lists articles associated with the title Trace . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Trace&oldid=1257568934 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description

2544-446: The same term [REDACTED] This disambiguation page lists articles associated with the title Trace . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Trace&oldid=1257568934 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description

2597-556: The series on the right converges absolutely due to Weyl's inequality ∑ n = 1 N | λ n ( A ) | ≤ ∑ m = 1 M s m ( A ) {\displaystyle \sum _{n=1}^{N}\left|\lambda _{n}(A)\right|\leq \sum _{m=1}^{M}s_{m}(A)} between the eigenvalues { λ n ( A ) } n = 1 N {\displaystyle \{\lambda _{n}(A)\}_{n=1}^{N}} and

2650-532: The term "nuclear operator" in more general topological vector spaces (such as Banach spaces ). Note that the trace operator studied in partial differential equations is an unrelated concept. Let H {\displaystyle H} be a separable Hilbert space , { e k } k = 1 ∞ {\displaystyle \left\{e_{k}\right\}_{k=1}^{\infty }} an orthonormal basis and A : H → H {\displaystyle A:H\to H}

2703-478: The trace norm ‖ T ‖ 1 = Tr ⁡ [ ( T ∗ T ) 1 / 2 ] = ∑ i α i . {\textstyle \|T\|_{1}=\operatorname {Tr} \left[\left(T^{*}T\right)^{1/2}\right]=\sum _{i}\alpha _{i}.} The norm corresponding to the Hilbert–Schmidt inner product

SECTION 50

#1732766166460

2756-497: The trace-norm is a norm on the space of all trace class operators B 1 ( H ) {\displaystyle B_{1}(H)} and that B 1 ( H ) {\displaystyle B_{1}(H)} , with the trace-norm, becomes a Banach space . When H {\displaystyle H} is finite-dimensional, every (positive) operator is trace class and this definition of trace of A {\displaystyle A} coincides with

2809-544: The usual operator norm is ‖ T ‖ = sup i ( α i ) . {\textstyle \|T\|=\sup _{i}\left(\alpha _{i}\right).} By classical inequalities regarding sequences, ‖ T ‖ ≤ ‖ T ‖ 2 ≤ ‖ T ‖ 1 {\displaystyle \|T\|\leq \|T\|_{2}\leq \|T\|_{1}} for appropriate T . {\displaystyle T.} It

#459540