|
Subnormal Operator
In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols. Definition Let ''H'' be a Hilbert space. A bounded operator ''A'' on ''H'' is said to be subnormal if ''A'' has a normal extension. In other words, ''A'' is subnormal if there exists a Hilbert space ''K'' such that ''H'' can be embedded in ''K'' and there exists a normal operator ''N'' of the form :N = \begin A & B\\ 0 & C\end for some bounded operators :B : H^ \rightarrow H, \quad \mbox \quad C : H^ \rightarrow H^. Normality, quasinormality, and subnormality Normal operators Every normal operator is subnormal by definition, but the converse is not true in general. A simple class of examples can be obtained by weakening the properties of unitary operators. A unitary operator is an isometry with dense range. Consider n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reducing Subspace
In linear algebra, a reducing subspace W of a linear map T:V\to V from a Hilbert space V to itself is an invariant subspace of T whose orthogonal complement W^\perp is also an invariant subspace of T. That is, T(W) \subseteq W and T(W^\perp) \subseteq W^\perp. One says that the subspace W reduces the map T. One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible. If V is of finite dimension r and W is a reducing subspace of the map T:V\to V represented under basis B by matrix M \in\R^ then M can be expressed as the sum M = P_W M P_W + P_ M P_ where P_W \in\R^ is the matrix of the orthogonal projection from V to W and P_ = I - P_ is the matrix of the projection onto W^\perp. (Here I \in \R^ is the identity matrix.) Furthermore, V has an orthonormal basis B' with a subset that is an orthonormal basis of W. If Q \in \R^ is the transition matrix from B to B' then with respect to B' the matrix Q^MQ representing T is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi-definite Hermitian matrix in the complex case), both square and of the same size. Intuitively, if a real n\times n matrix A is interpreted as a linear transformation of n-dimensional space \mathbb^n, the polar decomposition separates it into a rotation or reflection U of \mathbb^n, and a scaling of the space along a set of n orthogonal axes. The polar decomposition of a square matrix A always exists. If A is invertible, the decomposition is unique, and the factor P will be positive-definite. In that case, A can be written uniquely in the form A = U e^X , where U is unitary and X is the unique self-adjoint logarithm of the matrix P. This decomposition is useful in computing the fundamental group of (matrix) Lie groups. The polar deco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasinormal Operator
In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator. Every quasinormal operator is a subnormal operator. Every quasinormal operator on a finite-dimensional Hilbert space is normal. Definition and some properties Definition Let ''A'' be a bounded operator on a Hilbert space ''H'', then ''A'' is said to be quasinormal if ''A'' commutes with ''A*A'', i.e. :A(A^*A) = (A^*A) A.\, Properties A normal operator is necessarily quasinormal. Let ''A'' = ''UP'' be the polar decomposition of ''A''. If ''A'' is quasinormal, then ''UP = PU''. To see this, notice that the positive factor ''P'' in the polar decomposition is of the form (''A*A''), the unique positive square root of ''A*A''. Quasinormality means ''A'' commutes with ''A*A''. As a consequence of the continuous functional calculus for self-adjoint operators, ''A'' commutes with ''P'' = (''A*A'') also, i.e. :U P P = P U P.\, So ''UP = PU'' on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Dilation
In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''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 :P_H \; V , _H = T where P_H is an orthogonal 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 X, we say that ''V'' is a normal boundary dilation or a normal \partial X dilation if ''V'' is a normal dilation of ''T'' and \sigma(V)\subseteq \partial X. Some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property: :P_H \; f(V) , _H = f(T) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unilateral Shift
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. Definition Functions of a real variable The shift operator (where ) takes a function on R to its translation , : T^t f(x) = f_t(x) = f(x+t)~. A practical operational calculu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Of A Function
In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given two sets and , a binary relation between and is a (total) function (from to ) if for every in there is exactly one in such that relates to . The sets and are called domain and codomain of , respectively. The image of is then the subset of consisting of only those elements of such that there is at least one in with . Terminology As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the codomain. More modern books, if they use the word "range" at all, generally use it to mean what is now called the image. To avoid any confusion, a number of modern books don't use the word "range" at all. Elaboration and example Given a functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. A unitary element is a generalization of a unitary operator. In a unital algebra, an element of the algebra is called a unitary element if , where is the identity element. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator on a ... [...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] |
Normal Extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. Definition Let ''L/K'' be an algebraic extension (i.e. ''L'' is an algebraic extension of ''K''), such that L\subseteq \overline (i.e. ''L'' is contained in an algebraic closure of ''K''). Then the following conditions, any of which can be regarded as a definition of ''normal extension'', are equivalent: * Every embedding of ''L'' in \overline induces an automorphism of ''L''. * ''L'' is the splitting field of a family of polynomials in K\left \right/math>. * Every irreducible polynomial of K\left \right/math> which has a root in ''L'' splits into linear factors in ''L''. Other properties Let ''L'' be an extension of a field ''K''. Then: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. Details Let ''S''1 be the circle, with the standard Lebesgue measure, and ''L''2(''S''1) be the Hilbert space of square-integrable functions. A bounded measurable function ''g'' on ''S''1 defines a multiplication operator ''Mg'' on ''L''2(''S''1). Let ''P'' be the projection from ''L''2(''S''1) onto the Hardy space ''H''2. The ''Toeplitz operator with symbol g'' is defined by :T_g = P M_g \vert_, where " , " means restriction. A bounded operator on ''H''2 is Toeplitz if and only if its matrix representation, in the basis , has constant diagonals. Theorems * Theorem: If g is continuous, then T_g - \lambda is Fredholm if and only if \lambda is not in the set g(S^1). If it is Fredholm, its index is minus the winding number of the curve traced out by g with respect to the origin. For a proof, see . He attributes the theorem to Mark Krein, Harold Widom, and A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
