|
Selfadjoint Operator
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements is referred to as A subset \mathcal \subseteq \mathcal that is closed under the involution *, i.e. \mathcal = \mathcal^*, is called A special case of particular importance is the case where \mathcal is a complete normed *-algebra, that satisfies the C*-identity (\left\, a^*a \right\, = \left\, a \right\, ^2 \ \forall a \in \mathcal), which is called a C*-algebra. Especially in the older literature on *-algebras and C*-algebras, such elements are often called Because of that the notations \mathcal_h, \mathcal_H or H(\mathcal) for the set of self-adjoint elements are also sometimes used, even in the more recent literature. Examples * Each positive element of a C*-algebra is * For each element a of a *-algebra, the elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real-valued Function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real functions'') and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Algebraic structure Let (X,) be the set of all functions from a set to real numbers \mathbb R. Because \mathbb R is a field, (X,) may be turned into a vector space and a commutative algebra over the reals with the following operations: *f+g: x \mapsto f(x) + g(x) – vector addition *\mathbf: x \mapsto 0 – additive identity *c f: x \mapsto c f(x),\quad c \in \mathbb R – scalar multiplication *f g: x \mapsto f(x)g(x) – pointwise multiplication These operations extend to partial functions from to \mathbb R, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for all x, y ∊ ''V''. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nth Root
In mathematics, an th root of a number is a number which, when raised to the power of , yields : r^n = \underbrace_ = x. The positive integer is called the ''index'' or ''degree'', and the number of which the root is taken is the ''radicand.'' A root of degree 2 is called a ''square root'' and a root of degree 3, a '' cube root''. Roots of higher degree are referred by using ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. The computation of an th root is a root extraction. For example, is a square root of , since , and is also a square root of , since . The th root of is written as \sqrt /math> using the radical symbol \sqrt. The square root is usually written as , with the degree omitted. Taking the th root of a number, for fixed , is the inverse of raising a number to the th power, and can be written as a fractional exponent: \sqrt = x^. For a positive real number , \sqrt denotes the positive square root of and \sqrt /math> denotes the pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive And Negative Parts
In mathematics, the positive part of a real number, real or extended real number line, extended real-valued function (mathematics), function is defined by the formula f^+(x) = \max(f(x),0) = \begin f(x) & \text f(x) > 0 \\ 0 & \text \end Intuitively, the graph of a function, graph of f^+ is obtained by taking the graph of f, 'chopping off' the part under the -axis, and letting f^+ take the value zero there. Similarly, the negative part of is defined as f^-(x) = \max(-f(x),0) = -\min(f(x),0) = \begin -f(x) & \text f(x) 0]f \\ f^- &= -[f<0]f. \end One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function. Measure-theoretic properties Given a sigma-algebra, measurable space , an extended real-valued function is measurable function, measurable if and only if its positive and negative parts are. Therefore, if such a function is measurab ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, the sum is defined (a, b) + (c, d) to be (a + c, b + d); in other words, addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. Direct sums can also be formed with any finite number of summands; for example, A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). That relies on the fact that the direct sum is associative up to isomorphism. That is, (A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to ''Proportionality (mathematics), proportionality''. Examples in physics include the linear relationship of voltage and Electric current, current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are ''Nonlinear system, nonlinear''. Generalized for functions in more than one dimension (mathematics), dimension, linearity means the property of a function of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied. Definition A binary operation * on a set ''S'' is ''commutative'' if x * y = y * x for all x,y \in S. An operation that is not commutative is said to be ''noncommutative''. One says ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product (mathematics)
In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 21 is the product of 3 and 7 (the result of multiplication), and x\cdot (2+x) is the product of x and (2+x) (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a '' multiple''. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the ''commutative law'' of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functional Calculus
In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. In advanced theory, the applications of this functional calculus are so natural that they are often not even mentioned. It is no overstatement to say that the continuous functional calculus makes ''the'' difference between C*-algebras and general Banach algebras, in which only a holomorphic functional calculus exists. Motivation If one wants to extend the natural functional calculus for polynomials on the spectrum \sigma(a) of an element a of a Banach algebra \mathcal to a functional calculus for continuous functions C(\sigma(a)) on the spectrum, it seems obvious to approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements converges to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
