In mathematics , operatortheory is the branch of functional analysis that focuses on bounded linear operator s, but which includes closed operator s and nonlinear operator s. Operatortheory also includes the study of linear algebra algebra s of operators. Single operatortheory Single operatortheory deals with the properties and classification of single operators. For example, the classification of normal operator s in terms of their spectrum of an operator spectra falls into this category. Operator algebras The theory of operator algebra s brings algebra over a field algebra s of operators such as C algebra s to the fore. See also Invariant subspace Functional calculus Spectral theory Resolvent formalism Compact operator Fredholm theory of integral equation s Integral operator Fredholm operator Self adjoint operator Unbounded operator Differential operator Umbral calculus Contraction mapping Positive operator on a Hilbert space Perron Frobenius theorem Generalizations Nonnegative operator on a ordered vector space partially ordered vector space External links http www.mathphysics.com opthy OpHistory.html History of OperatorTheory Mathanalysis stub Category Operatortheory de Operatorenrechnung fa ko kk nl Operatorentheorie pt Teoria dos operadores ru tt uk vi L thuy t to n t ... more details
Integral Equations and OperatorTheory is a Academic journal journal dedicated to operatortheory and its applications to engineering and other mathematics mathematical sciences . As some approaches to the study of integral equations theoretically and numerically constitute a subfield of operatortheory, the journal also deals with the theory of integral equations and hence of differential equation s. The journal consists of two sections a main section consisting of refereed papers and a second consisting of short announcements of important results, open problems, information, etc. It has been published monthly by Springer Verlag since 1978. The journal is also available online by subscription. External links http link.springer.de link service journals 00020 Journal homepage Category Mathematics journals Category Publications established in 1978 ... more details
In mathematics , operator K theory is a variant of K theory on the Category mathematics category of Banach algebras In most applications, these Banach algebras are C algebras . Its basic feature that distinguishes it from algebraic K theory is that it has a Bott periodicity . So there are only two K groups, namely math K 0 math , equal to algebraic math K 0 math , and math K 1 math . As a consequence of the periodicity theorem, it satisfies excision . This means that it associates to an Extension mathematics extension of C algebra s to a long exact sequence , which, by Bott periodicity, reduces to an exact cyclic 6 term sequence. Operator K theory is a generalization of topological K theory , defined by means of vector bundle s on locally compact Hausdorff space s. Here, an n dimensional vector bundle over a topological space X is associated to a projection in math M n C X math , where math ... methods into the study of C algebras was the Fredholm index Given a bounded linear operator on a Hilbert ..., which, as it turns out, reflects the defect on the operator i.e. the extent to which it is not invertible ... Singer index theorem index theory of Atiyah and Singer, where the topological index of the manifold can be expressed via the index of elliptic operators on it. Later on, BDF theory Brown, Douglas ... normal operator s up to certain natural equivalence. These ideas, together with G.A. Elliott Elliott s classification of AF C algebra s via K theory led to a great deal of interest in adapting methods such as K theory from algebraic topology into the study of operator algebras. This, in turn, led to K homology , Gennadi Kasparov Kasparov s bivariant KK Theory , and, more recently, Alain Connes Connes and Nigel Higson Higson s E theory. References Citation last1 Rordam first1 M. last2 Larsen first2 Finn last3 Laustsen first3 N. title An introduction to K theory for C sup sup algebras publisher ... 978 0 521 78944 8 year 2000 volume 49 Category K theory Category Article Feedback 5 algebra stub ... more details
In operatortheory , a bounded operator T X &rarr Y between normed vector space s X and Y is said to be a contraction if its operator norm T &le 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. Contractions on a Hilbert space Consider the special case where T is a contraction acting on a Hilbert space math mathcal H math . We define some basic objects associated with T . The defect operators of T are the operators D sub T sub 1 &minus T T sup ½ sup and D sub T sub 1 &minus TT sup ½ sup . The square root is the square root of a matrix positive semidefinite one given by the spectral theorem . The defect spaces math mathcal D T math and math mathcal D T math are the ranges Ran D sub T sub and Ran D sub T sub respectively. The defect indices of T are the pair math mbox dim mathcal D T, mbox dim mathcal ... P U sup n sup P for all n 0. The operator U is called a dilation operatortheory dilation of T and is uniquely ... 0 486 66289 6 Category Operatortheory ... T Gamma oplus U math where U is a unitary operator and &Gamma is completely non unitary in the sense ... in 1953, states that for any contraction T on a Hilbert space H , there is a unitary operator U ... containing H . A direct construction of a unitary dilation of a contraction operator T may be obtained as follows. Let V be the operator on math H oplus H math defined by the matrix math V begin ..., therefore an unitary dilation of T , sometimes called the Julia operator of T . Contractions on Hilbert spaces can be viewed as the operator analogs of cos&thinsp &theta and are called operator angles in some contexts. The explicit description of contractions leads to operator parametrizations of positive and unitary matrices. See also Contraction mapping . Note that a contraction operator ... more details
In operatortheory , a dilation of an operator T on a Hilbert space H is an operator on a larger Hilbert space K , whose restriction to H is T . More formally, let T be a bounded operator on some Hilbert space H , and H be a subspace of a larger Hilbert space H . A bounded operator V on H is a dilation of T if math P H V H T math where math P H math is projection on H . V is said to be a unitary dilation respectively, normal, isometric, etc if V is unitary respectively, normal, isometric, etc . T is said to be a compression of V . If an operator T has a spectral set math X math , we say that V is a normal boundary dilation or a normal math partial X math dilation if V is a normal dilation of T and math sigma V in partial X math . Some texts impose an additional condition. Namely, that a dilation satisfy the following calculus property math P H f V H f T math where f T is some specified functional calculus for example, the polynomial or H sup &infin sup calculus . The utility of a dilation ... Maps and Operator Algebras 2002, ISBN 0 521 81669 6 Category Operatortheory Category Unitary operators ... show that every contraction operator on a Hilbert space has a unitary dilation. A possible construction of this dilation is as follows. For a contraction T , the operator math D T I T T frac 1 2 math is positive, where the continuous functional calculus is used to define the square root. The operator D sub T sub is called the defect operator of T . Let V be the operator on math H oplus H math defined ..., that V is unitary, therefore an unitary dilation of T . This operator V is sometimes called the Julia operator of T . Notice that when T is a real scalar, say math T cos theta math , we have ... matrix describing rotation by . For this reason, the Julia operator V T is sometimes called ... property for a dilation. Indeed, direct calculation shows the Julia operator fails to be a degree ... algebra , any operator T with math X math as a spectral set will have a normal math partial X math dilation ... more details
wiktionary Operatoroperator operators Operator may refer to tocright Music Operator band , an American hard rock band Operator Motown song Operator Motown song , a 1965 song recorded by Mary Wells and Brenda Holloway Operator That s Not the Way It Feels , a 1972 song by Jim Croce from You Don t Mess Around with Jim Operator Midnight Star song Operator Midnight Star song 1984 Operator A Girl Like Me , a 2008 song by Shiloh Operator , a 1970 song by the Grateful Dead from American Beauty album American Beauty Operator , a 1975 song by the Manhattan Transfer from The Manhattan Transfer album The Manhattan Transfer Operator , a 1993 song by Blue System from Backstreet Dreams album Backstreet Dreams Operator , a 1995 song by Real McCoy from Another Night Real McCoy album Another Night Computers Computer operatorOperator programming , a type of computer program function Operator extension , an extension for the Firefox web browser, for reading microformats Operator YAPO or OperaTor, a portable implementation of the Opera web browser Science and mathematics Operator mathematics , a function between vector spaces Operator biology , a segment of DNA regulating the activity of genes Operator linguistics , a special category including wh interrogatives Operator physics , mathematical operators in quantum physics Fiction Operator Ghost in the Shell Operator Ghost in the Shell , a fictional group in the Ghost in the Shell series Operator The Matrix Operator The Matrix , a crew position in The Matrix franchise Other uses Operator profession Telephone operator , person or company offering telephone services Operator military , a soldier in a special operations force Network operator , a phone carrier Operator sternwheeler Operator sternwheeler , an early 20th century ship on the Skeena River Operator Grammar , a theory of human language See also Operation disambiguation Operator precedence grammar , a grammar for formal languages Operator No. 5 , a pulp fiction hero from the 1930s ... more details
Infobox Single See Wikipedia WikiProject Songs Name Operator, Operator Cover Artist Eddy Raven Album Love and Other Hard Times B side Just for the Sake of the Thrill ref name raven Released 1985 Format 7 single Recorded Genre Country music Country Length 3 04 Label RCA Records RCA Writer Larry Willoughby , Janet Willoughby Producer Paul Worley , Eddy Raven Last single She s Gonna Win Your Heart br 1984 This single Operator, Operator br 1985 Next single I Wanna Hear It from You br 1985 Misc Operator, Operator originally titled Heart on the Line Operator, Operator is a country music song co written and recorded by Larry Willoughby , a cousin of country music singer Rodney Crowell , and Janet Willoughby. He released the song in 1983 from the album Building Bridges , and took it to number 65 on the Hot Country Songs charts. ref name whitburn cite book last Whitburn first Joel title Hot Country Songs 1944 to 2008 publisher Record Research, Inc date 2008 page 469 isbn 0 89820 177 2 ref The Oak Ridge Boys also recorded it under the original title, as the b side to their 1983 single Love Song The Oak Ridge Boys song Love Song . ref Whitburn, p. 303 ref A 1985 recording by Eddy Raven , under the title Operator, Operator , appeared on his album Love and Other Hard Times . This version went to number 9 on the same chart. ref name raven Whitburn, p. 340 ref Chart performance Larry Willoughby class wikitable sortable align left Chart 1983 align center Peak br position align left U.S. Billboard Hot Country Singles align center 65 Eddy Raven class wikitable sortable align left Chart 1985 align center Peak br position align left U.S. Billboard Hot Country Singles align center 9 align left Canadian RPM Country Tracks align center 8 References reflist Category 1983 singles Category 1985 singles Category Eddy Raven songs Category Larry Willoughby songs Category Songs produced by Paul Worley 1980s country song stub ... more details
In mathematics , especially operatortheory , a convexoid operator is a bounded operator bounded linear operator T on a complex Hilbert space H such that the closure of the numerical range coincides with the convex hull of its spectrum. An example of such an operator is a normal operator or some of its generalization . It is not known whether a paranormal operator is a convexoid or not, I think. Taku. A closely related operator is a spectraloid operator an operator whose spectral radius coincides with its numerical radius . In fact, an operator T is convexoid if and only if math T lambda math is spectraloid for every complex number math lambda math . This result is due to Furuta, I believe Taku See also Aluthge transform References T. Furuta. http www.projecteuclid.org DPubS?service UI&version 1.0&verb Display&handle euclid.pja 1195526397 Certain convexoid operators Category Operatortheory mathanalysis stub ... more details
Unreferenced stub auto yes date December 2009 Disputed date March 2008 In theoretical physics , a disorder operator is an Operator mathematics operator that creates a discontinuity of the ordinary order operator s or a monodromy for their values. For example, a t Hooft operator is a disorder operator. So is the Jordan Wigner transformation . DEFAULTSORT Disorder Operator Category Quantum field theory Category Statistical mechanics Phys stub ... more details
In mathematics , especially operatortheory , a hyponormal operator is a generalization of a normal operator . In general, a bounded linear operator T on a complex Hilbert space H is said to be p hyponormal math 0 p le 1 math if math T T p ge TT p math That is to say, math T T p TT p math is a positive operator. If math p 1 math , then T is called a hyponormal operator. If math p 1 2 math , then T is called a semi hyponormal operator. Moreoever, T is said to be log hyponormal if it is invertible and math log T T ge log TT . math An invertible p hyponormal operator is log hyponormal. On the other hand, not every log hyponormal is p hyponormal. The class of semi hyponormal operators was introduced by Xia, and the class of p hyponormal operators was studied by Aluthge, who used what is today called the Aluthge transformation . Every subnormal operator in particular, a normal operator is hyponormal, and every hyponormal operator is a paranormal operator paranormal convexoid operator . Not every paranormal operator is, however, hyponormal. Let T be a hyponormal operator. If math T T TT math is compact, then T is normal. Maybe the statement isn t quite accurately stated. See also Putnam s inequality References http www.jstor.org pss 2162263 Category Operatortheory mathanalysis stub ... more details
Unreferenced date December 2009 In mathematics , in the area of functional analysis and operatortheory , the Volterra operator , named after Vito Volterra , represents the operation of indefinite integration , viewed as a bounded linear operator on the space L sup 2 sup 0,1 of complex valued square integrable function s on the interval 0,1 . It is the operator corresponding to the Volterra integral equation s. Definition The Volterra operator, V , may be defined for a function f s L sup 2 sup 0,1 and a value t 0,1 , as math V f t int 0 t f s , ds . math Properties V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint math V f t int t 1 f s , ds . math V is a Hilbert Schmidt operator , hence in particular is compact operator compact . V has no eigenvalue s and therefore, by the spectral theory of compact operators , its spectrum functional analysis spectrum V 0 . V is a quasinilpotent operator that is, the spectral radius , V , is zero , but it is not nilpotent . The operator norm of V is exactly V sup 2 sup sub sub . DEFAULTSORT Volterra Operator Category Operatortheory ... more details
Unreferenced date December 2009 In operatortheory , a multiplication operator is a linear operator T defined on some function space vector space of functions and whose value at a function is given by multiplication by a fixed function f . That is, math T varphi x f x varphi x quad math for all in the function space and all x in the domain mathematics domain of which is the same as the domain of f . This type of operators is often contrasted with composition operator s. Multiplication operators generalize the notion of operator given by a diagonal matrix . More precisely, one of the results of operatortheory is a spectral theorem , which states that every self adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an Lp space L sup 2 sup space . Example Consider the Hilbert space X L sup 2 sup &minus 1, 3 of complex number complex valued square integrable functions on the interval mathematics interval &minus 1, 3 . Define the operator math T varphi x x 2 varphi x quad math for any function in X . This will be a self adjoint operator self adjoint bounded linear operator with operator norm norm 9. Its spectrum of an operator spectrum will be the interval 0, 9 the range mathematics range of the function x x sup 2 sup defined on &minus 1, 3 . Indeed, for any complex number , the operator T is given by math T lambda varphi x x 2 lambda varphi x . quad math It is invertible function invertible if and only if is not in 0, 9 , and then its inverse is math T lambda 1 varphi x frac 1 x 2 lambda varphi x quad math which is another multiplication operator. This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space . See also translation operator shift operator Decomposition of spectrum functional analysis DEFAULTSORT Multiplication Operator Category Functional analysis ... more details
Expert subject date November 2010 Unreferenced date November 2010 Given a unital algebra unital C algebra math mathcal A math a closed subspace Disambiguation needed date June 2011 S containing 1 is called an operator system . One can associate to each subspace math mathcal M subseteq mathcal A math of a unital C algebra an operator system via math S mathcal M mathcal M mathbb C 1 math . Category Operator theory math stub ... more details
In functional analysis , a discipline within mathematics, an operator space is a Banach space given together with an isometric embedding into the space B H of all bounded operators on a Hilbert space H . ref cite book url http books.google.com books?id 0pKL o7WUOAC&pg PA1&dq Operator space title Introduction to Operator Space Theory last Pisier first Gilles publisher Cambridge University Press page 1 isbn 9780521811651 year 2003 accessdate 2008 12 18 ref The category mathematics category of operator spaces includes operator algebra s. ref cite book url http books.google.com books?id lwprbgvFA4IC&pg PP11&dq 22Operator space 22 title Operator Algebras and Their Modules An Operator Space Approach authors Blecher, David P. and Christian Le Merdy publisher Oxford University Press page First page of Preface isbn 9780198526599 year 2004 accessdate 2008 12 18 nopp true ref References Reflist 2 Category Banach spaces Category Operatortheory mathanalysis stub ... more details
theory. Universitext Tracts in Mathematics. Springer Verlag, New York, 1993. xvi 223 pp. ISBN 0 387 94067 7. Category Operatortheory Category Functional analysis Category Dynamical systems ...for information about the operator of composition function composition composition of relations In mathematics , the composition operator math C phi math with symbol math phi math is a linear operator defined by the rule math C phi f f circ phi math where math f circ phi math denotes function composition . In physics , and especially the area of dynamical systems , the composition operator is usually referred to as the Koopman operator ref B.O. Koopman, Hamiltonian systems and transformations in Hilbert space , 1931 Proceedings of the National Academy of Sciences of the USA , 17 , pp.315 318. ref ref Pierre Gaspard, Chaos, scattering and statistical mechanics , 1998 Cambridge University Press ref . It is the left adjoint of the Frobenius Perron or transfer operator . In the language of category theory , the composition operator is a pull back on the space of measurable function s it is adjoint to the transfer operator in the same way that the pull back is adjoint to the push forward the composition operator is the inverse image functor . The domain of a composition operator is usually taken to be some Banach space , often consisting of holomorphic function s for example, some Hardy ... to how the Spectrum functional analysis spectral properties of the operator depend on the function space. Other questions include whether math C phi math is compact operator compact or trace class ..., composition operators commonly occur in the study of shift operator s, for example, in the Beurling ... spin lattice s. Composition operators appear in the theory of Aleksandrov Clark measure s. The eigenvalue equation of the composition operator is Schr der s equation . The study of composition ... operator References references C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic ... more details
In operatortheory , a Toeplitz operator is the dilation operatortheory compression of a multiplication operator on the circle to the Hardy space . Details Let S sup 1 sup be the circle, with the standard Lebesgue measure, and L sup 2 sup S sup 1 sup be the Hilbert space of square integrable functions. A bounded measurable function g on S sup 1 sup defines a multiplication operator M sub g sub on L sup 2 sup S sup 1 sup . Let P be the projection from L sup 2 sup S sup 1 sup onto the Hardy space H sup 2 sup . The Toeplitz operator with symbol g is defined by math T g P M g vert H 2 , math where means restriction. A bounded operator on H sup 2 sup is Toeplitz if and only if its matrix representation, in the basis z sup n sup , n 0 , has constant diagonals. References citation last1 B ttcher first1 A. author1 link Albrecht B ttcher last2 Silbermann first2 B. year 2006 title Analysis of Toeplitz Operators edition 2nd publisher Springer Verlag series Springer Monographs in Mathematics isbn 9783540324348 . citation first1 Marvin last1 Rosenblum first2 James last2 Rovnyak title Hardy Classes and OperatorTheory year 1985 publisher Oxford University Press . Reprinted by Dover Publications, 1997, ISBN 9780486695365. DEFAULTSORT Toeplitz Operator Category Operatortheory mathanalysis stub ... more details
Orphan date February 2009 A quantum field theory of General Relativity provides Operator mathematics operators that measure the geometry of space time . The volume operator math V R math of a region math R math is defined as the operator that yields the expectation value of a volume measurement of the region math R math , given a state math psi math of quantum General Relativity. I.e. math lang psi, V R psi rang math is the expectation value for the volume of math R math . Loop Quantum Gravity for example provides volume operators, area operators and length operators for regions, surfaces and path respectively. http www.arxiv.org abs gr qc 9711031 Article by A. Ashtekar and J. Lewandowski construction of a volume operator in Loop Quantum Gravity Category Quantum field theory ... more details
In mathematics , especially operatortheory , a paranormal operator is a generalization of a normal operator . More precisely, a bounded linear operator T on a complex Hilbert space H is said to be paranormal if math T 2x ge Tx 2 , math for every unit vector x in H . The class of paranormal operators was introduced by V. Istratescu in 1960s, though the term paranormal is probably due to Furuta. ref V. Istratescu. http projecteuclid.org DPubS Repository 1.0 Disseminate?view body&id pdf 1&handle euclid.pjm 1102992095 On some hyponormal operators ref ref name Furuta Every hyponormal operator in particular, a subnormal operator , a quasinormal operator and a normal operator is paranormal. If T is a paranormal, then T sup n sup is paranormal. ref name Furuta Furuta, Takayuki. http projecteuclid.org DPubS Repository 1.0 Disseminate?view body&id pdf 1&handle euclid.pja 1195521514 On the Class of Paranormal Operators ref On the other hand, Paul Halmos Halmos gave an example of a hyponormal operator T such that T sup 2 sup isn t hyponormal. Consequently, not every paranormal operator is hyponormal. ref P.R.Halmos, A Hilbert Space Problem Book 2nd edition, Springer Verlag, New York, 1982. ref A Compact operator compact paranormal operator is normal. ref Furuta, Takayuki. http www.journalarchive.jst.go.jp jnlpdf.php?cdjournal pjab1945&cdvol 47&noissue SupplementI&startpage 888&lang en&from jnlabstract Certain Convexoid Operators ref References reflist Category Operatortheory mathanalysis stub ... more details
mechanics References cite book last Blackadar first Bruce title Operator Algebras Theory of C Algebras ... year 2005 isbn 3540284869 Category Operatortheory Category Functional analysis Category Operator algebras fr Alg bre d op rateurs nl Operator algebra ja ru ...In functional analysis , an operator algebra is an algebra over a field algebra of continuous function topology continuous linear operator s on a topological vector space with the multiplication given by the composition of mappings. Although it is usually classified as a branch of functional analysis, it has direct applications to representation theory , differential geometry , quantum statistical mechanics and quantum field theory . Such algebras can be used to study wiktionary arbitrary arbitrary sets of operators with little algebraic relation simultaneously . From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non commutative ring mathematics ring s. An operator algebra is typically required to be closed in a specified operator topology inside the algebra of the whole continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties ... contexts for example, algebras of pseudo differential operator s acting on spaces of distributions , the term operator algebra is usually used in reference to algebras of bounded operator s on a Banach ... with the operator norm mathematics norm topology. In the case of operators on a Hilbert space ... examples are self adjoint operator algebras, meaning that they are closed under taking adjoints. These include ... operator algebras can be regarded as the algebra of Complex numbers complex valued continuous functions ... measurable space . Thus, general operator algebras are often regarded as a noncommutative generalizations ... non classical and or pathological objects by noncommutative operator algebras. Examples of operator ... more details
The transfer operator is different from the Transfer group theory transfer homomorphism . In mathematics , the transfer operator encodes information about an iterated map and is frequently used to study ... of points of X under iteration the study of Chaos theory point dynamics , the transfer operator defines ... Category Operatortheory Category Spectral theory fr Op rateur de transfert ... operator is sometimes called the Ruelle operator , after David Ruelle , or the Ruelle&ndash Perron&ndash Frobenius operator in reference to the applicability of the Frobenius&ndash Perron theorem to the determination of the eigenvalues of the operator. The iterated function to be studied is a map math f X rightarrow X math for an arbitrary set math X math . The transfer operator is defined as an operator ... operator can be shown to be the point set limit of the measure theoretic pushforward of g in essence, the transfer operator is the direct image functor in the category of measurable space s. The left adjoint of the Frobenius&ndash Perron operator is the Koopman operator or composition operator . Applications ... the field of molecular dynamics . It is often the case that the transfer operator is positive, has ..., the transfer operator is sometimes called the Frobenius&ndash Perron operator. The eigenfunction s of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum Hamiltonian quantum theory Hamiltonian , the eigenvalues will typically be very ... operator of the Bernoulli map math b x 2x lfloor 2x rfloor math is exactly solvable and is a classic example of chaos theory deterministic chaos the discrete eigenvalues correspond to the Bernoulli polynomials . This operator also has a continuous spectrum consisting of the Hurwitz zeta function . The transfer operator of the Gauss map math h x 1 x lfloor 1 x rfloor math is called the Gauss&ndash Kuzmin&ndash Wirsing operator Gauss&ndash Kuzmin&ndash Wirsing GKW operator and due to its ... more details
In mathematics , specifically set theory , a dimensional operator on a set E is a function from the subsets of E to the subsets of E . Definition If the power set of E is denoted P E then a dimensional operator on E is a map math d P E rightarrow P E , math that satisfies the following properties for S , T &isin P E S &sube d S d S d d S d is idempotent if S &sube T then d S &sube d T if is the set of finite subsets of S then d S &cup sub A &isin sub d A if x &isin E and y &isin d S &cup x d S , then x &isin d S &cup y . The final property is known as the exchange axiom. ref Julio R. Bastida, Field Extensions and Galois Theory , Addison Wesley Publishing Company, 1984, pp. 212&ndash 213. ref Examples For any set E the identity map on P E is a dimensional operator. The map which takes any subset S of E to E itself is a dimensional operator on E . References references Category Set theory ... more details
Unreferenced date July 2009 In operatortheory , a bounded operator T on a Hilbert space is said to be nilpotent if T sup n sup 0 for some n . It is said to be quasinilpotent or topological nilpotent if its spectrum functional analysis spectrum T 0 . Examples In the finite dimensional case, i.e. when T is a square matrix with complex entries, T 0 if and only if T is similar to a matrix whose only nonzero entries are on the superdiagonal, by the Jordan canonical form . In turn this is equivalent to T sup n sup 0 for some n . Therefore, for matrices, quasinilpotency coincides with nilpotency. This is not true when H is infinite dimensional. Consider the Volterra operator , defined as follows consider the unit square X 0,1 × 0,1 R sup 2 sup , with the Lebesgue measure m . On X , define the kernel function K by math K x,y left begin matrix 1, & mbox if x geq y 0, & mbox otherwise . end matrix right. math The Volterra operator is the corresponding integral operator T on the Hilbert space L sup 2 sup X , m given by math T f x int 0 1 K x,y f y dy. math The operator T is not nilpotent take f to be the function that is 1 everywhere and direct calculation shows that T sup n sup f 0 in the sense of L sup 2 sup for all n . However, T is quasinilpotent. First notice that K is in L sup 2 sup X , m , therefore T is compact operator on Hilbert space compact . By the spectral properties of compact operators, any nonzero in T is an eigenvalue. But it can be shown that T has no nonzero eigenvalues, therefore T is quasinilpotent. DEFAULTSORT Nilpotent Operator Category Operatortheory pl Operator nilpotentny ... more details
A . For example, in an SU N gauge theory Yang Mills gauge theory a t Hooft operator creates a magnetic monopole Dirac magnetic monopole with respect to the center group theory center of SU N . If a condensate is present which transforms in a representation group theory representation of SU N which ...In theoretical physics , a t Hooft operator , introduced by Gerard t Hooft in the 1978 paper http igitur ... is a d 3 differential form form and so the t Hooft operator is the integral of A sup mag sup over a d 3 dimensional surface. While the Wilson loop is an order operator , t Hooft operator is an example of a disorder operator because it creates a Gravitational singularity singularity or a Classification ... Mills gauge theory must either contain massless particles, responsible for the interactions between the t Hooft operator and the Wilson loop, or else at least one of the two operators must be confined by an object one dimension higher. He identified the phase in which the t Hooft operator is confined .... Although such mixed phases had not been seen in quantum field theories quantum field theory at the time, they are now know to occur for example in the Argyres Douglas conformal field theory. Therefore ... a given operator is confined, a finite tension surface is created whose boundary is the operator ... phase the d 3 dimensional t Hooft operator is confined, and so the action scales as the area of the d ... of the t Hooft operator scales as the distance squared. In the mixed phase both operators are confined, and so both obey this scaling. On the other hand he claimed that if a given operator is Higgsed ... from the operator. Therefore the action will be proportional to volume of the surface on which the operator ... phase the t Hooft operator is Higgsed, and so the corresponding action fails as the area ... abs hep th 0212264v1 On t Hooft s loop operator H. Reinhardt http igitur archive.library.uu.nl phys ... Quantum field theory ... more details
, K X forms a two sided operator ideal ring theory ideal in B X . math id X math is compact ... is a compact operator if and only if the space is finite dimensional. See also Spectral theory of compact ... date May 2008 Category Compactness mathematics Category Operatortheory de Kompakter Operator fr Op rateur ...In functional analysis , a branch of mathematics , a compact operator is a linear operator L from a Banach ... compact subset of Y . Such an operator is necessarily a bounded operator , and so continuous. Any bounded operator L that has finite rank of a linear operator rank is a compact operator indeed, the class of compact operators is a natural generalisation of the class of finite rank operator s in an infinite dimensional setting. When Y is a Hilbert space , it is true that any compact operator ... as the closure in the operator norm of the finite rank operators. Whether this was true in general ... Per Enflo gave a counter example. The origin of the theory of compact operators is in the theory ... Fredholm integral equation gives rise to a compact operator K on function space s the compactness ... in the numerical solution of such equations. The abstract idea of Fredholm operator is derived from this connection. Equivalent formulations A bounded operator T is compact if and only if any of the following ... a Cauchy sequence Cauchy subsequence . Note that if a linear operator is compact, then it is easy ... , W are Banach spaces, B X , Y is the space of bounded operators from X to Y with the operator ... K X , X , math id X math is the identity operator on X . K X , Y is a closed subspace ... Banach space to the other, and suppose that T sub n sub converges to T with respect to the operator ... operator of index 0. In particular, math operatorname im , id X T math &thinsp is closed ... where M is closed and N is finite dimensional, then nowrap M N is also closed. Any compact operator is strictly singular , but not vice versa. ref N.L. Carothers, A Short Course on Banach Space Theory ... more details
In quantum field theory , an order operator or an order field is a quantum field version of Landau s order parameter whose expectation value characterizes phase transition s. There exists a dual version of it, the disorder operator or disorder field , whose expectation value characterizes a phase transition by indicating the prolific presence of defect or vortex lines in an ordered phase. Books Hagen Kleinert Kleinert, Hagen , gauge field Gauge Fields in condensed matter Condensed Matter , Vol. I, superfluidity SUPERFLOW AND vortex line VORTEX LINES , pp. 1 742, Vol. II, Stress physics STRESSES AND defect DEFECTS , pp. 743 1456, http www.worldscibooks.com physics 0356.htm World Scientific Singapore, 1989 Paperback ISBN 9971 5 0210 0 also available online http www.physik.fu berlin.de kleinert kleiner reb1 contents1.html Vol. I and http www.physik.fu berlin.de kleinert kleiner reb1 contents2.html Vol. II Category Quantum field theory Category Statistical mechanics ... more details
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