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Abstract Index Group
In operator theory, a branch of mathematics, every Banach algebra can be associated with a group called its abstract index group. Definition Let ''A'' be a Banach algebra and ''G'' the group of invertible elements in ''A''. The set ''G'' is open and a topological group. Consider the identity component :''G''0, or in other words the connected component containing the identity 1 of ''A''; ''G''0 is a normal subgroup of ''G''. The quotient group :Λ''A'' = ''G''/''G''0 is the abstract index group of ''A''. Because ''G''0, being the component of an open set, is both open and closed in ''G'', the index group is a discrete group. Examples Let ''L''(''H'') be the Banach algebra of bounded operators on a Hilbert space. The set of invertible elements in ''L''(''H'') is path connected. Therefore, Λ''L''(''H'') is the trivial group. Let T denote the unit circle in the complex plane. The algebra ''C''(T) of continuous functions from T to the complex numbers is a Banach algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winding Number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kehe Zhu
Kaph (also spelled kaf) is the eleventh letter of the Semitic abjads, including Phoenician kāp , Hebrew kāf , Aramaic kāp , Syriac kāp̄ , and Arabic kāf (in abjadi order). The Phoenician letter gave rise to the Greek kappa (Κ), Latin K, and Cyrillic К. Origin of kaph Kaph is thought to be derived from a pictogram of a hand (in both modern Arabic and modern Hebrew, kaph כף means "palm" or "grip"), though in Arabic the ''a'' in the name of the letter (كاف) is pronounced longer than the ''a'' in the word meaning "palm" (كَف). D46 Hebrew kaf Hebrew spelling: Hebrew pronunciation The letter kaf is one of the six letters that can receive a dagesh kal. The other five are bet, gimel, daleth, pe, and tav (see Hebrew alphabet for more about these letters). There are two orthographic variants of this letter that alter the pronunciation: Kaf with the dagesh When the kaph has a "dot" in its center, known as a dagesh, it represents a voiceless velar plosive () ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Index
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The ''index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The '' index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atkinson's Theorem
In operator theory, Atkinson's theorem (named for Frederick Valentine Atkinson) gives a characterization of Fredholm operators. The theorem Let ''H'' be a Hilbert space and ''L''(''H'') the set of bounded operators on ''H''. The following is the classical definition of a Fredholm operator: an operator ''T'' ∈ ''L''(''H'') is said to be a Fredholm operator if the kernel Ker(''T'') is finite-dimensional, Ker(''T*'') is finite-dimensional (where ''T*'' denotes the adjoint of ''T''), and the range Ran(''T'') is closed. Atkinson's theorem states: :A ''T'' ∈ ''L''(''H'') is a Fredholm operator if and only if ''T'' is invertible modulo compact perturbation, i.e. ''TS'' = ''I'' + ''C''1 and ''ST'' = ''I'' + ''C''2 for some bounded operator ''S'' and compact operators ''C''1 and ''C''2. In other words, an operator ''T'' ∈ ''L''(''H'') is Fredholm, in the classical sense, if and only if its projection in the Calkin algebra is invertible. Sketch of proof The outline of a proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator On Hilbert Space
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach. For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces. More generally, the compactness assumption can be dropped. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous 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 under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calkin Algebra
In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of ''B''(''H''), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space ''H'', by the ideal ''K''(''H'') of compact operators. Here the addition in ''B''(''H'') is addition of operators and the multiplication in ''B''(''H'') is composition of operators; it is easy to verify that these operations make ''B''(''H'') into a ring. When scalar multiplication is also included, ''B''(''H'') becomes in fact an algebra over the same field over which ''H'' is a Hilbert space. Properties * Since ''K''(''H'') is a maximal norm-closed ideal in ''B''(''H''), the Calkin algebra is simple. In fact, ''K''(''H'') is the only closed ideal in ''B''(''H''). * As a quotient of a C*-algebra by a two-sided ideal, the Calkin algebra is a C*-algebra itself and there is a short exact sequence ::0 \to K(H) \to B(H) \to B(H)/K(H) \to 0 :which induces a six-term cyclic exact sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
