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Douglas' Lemma
In operator theory, an area of mathematics, Douglas' lemma relates factorization, range inclusion, and majorization of Hilbert space operators. It is generally attributed to Ronald G. Douglas, although Douglas acknowledges that aspects of the result may already have been known. The statement of the result is as follows: Theorem: If A and B are bounded operators on a Hilbert space H, the following are equivalent: # \operatorname A \subseteq \operatorname B # AA^* \leq \lambda^2 BB^* for some \lambda \geq 0 # There exists a bounded operator C on H such that A=BC. Moreover, if these equivalent conditions hold, then there is a unique operator C such that * \Vert C \Vert^2 = \inf \ * \ker A = \ker C * \operatorname C \subseteq \overline . A generalization of Douglas' lemma for unbounded operators on a Banach space was proved by Forough (2014). See also * Positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linea ... [...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] |
Matrix Decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A \mathbf = \mathbf, the matrix ''A'' can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix ''L'' and an upper triangular matrix ''U''. The systems L(U \mathbf) = \mathbf and U \mathbf = L^ \mathbf require fewer additions and multiplications to solve, compared with the original system A \mathbf = \mathbf, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses ''A'' as ''QR'' with ''Q'' an orthogonal matrix and ''R'' an upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majorization
In mathematics, majorization is a preorder on vectors of real numbers. Let ^_,\ i=1,\,\ldots,\,n denote the i-th largest element of the vector \mathbf\in\mathbb^n. Given \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below (or equivalently, we say that \mathbf is weakly majorized (or dominated) by \mathbf from below) denoted as \mathbf \succ_w \mathbf if \sum_^k x_^ \geq \sum_^k y_^ for all k=1,\,\dots,\,d. If in addition \sum_^d x_i^ = \sum_^d y_i^, we say that \mathbf majorizes (or dominates) \mathbf , written as \mathbf \succ \mathbf , or equivalently, we say that \mathbf is majorized (or dominated) by \mathbf. The order of the entries of the vectors \mathbf or \mathbf does not affect the majorization, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). As a consequence, majorization is not a partial order, since \mathbf \succ \mathbf and \mathbf \succ \mathbf do not imply \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald G
Ronald is a masculine given name derived from the Old Norse ''Rögnvaldr'', Hanks; Hardcastle; Hodges (2006) p. 234; Hanks; Hodges (2003) § Ronald. or possibly from Old English '' Regenweald''. In some cases ''Ronald'' is an Anglicised form of the Gaelic ''Raghnall'', a name likewise derived from ''Rögnvaldr''. The latter name is composed of the Old Norse elements ''regin'' ("advice", "decision") and ''valdr'' ("ruler"). ''Ronald'' was originally used in England and Scotland, where Scandinavian influences were once substantial, although now the name is common throughout the English-speaking world. A short form of ''Ronald'' is ''Ron''. Pet forms of ''Ronald'' include ''Roni'' and ''Ronnie''. ''Ronalda'' and ''Rhonda'' are feminine forms of ''Ronald''. '' Rhona'', a modern name apparently only dating back to the late nineteenth century, may have originated as a feminine form of ''Ronald''. Hanks; Hardcastle; Hodges (2006) pp. 230, 408; Hanks; Hodges (2003) § Rhona. The names ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Operator
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \langle Ax, x\rangle \in \mathbb and \langle Ax, x\rangle \geq 0, where \mathop(A) is the domain of A. Positive-semidefinite operators are denoted as A\ge 0. The operator is said to be positive-definite, and written A>0, if \langle Ax,x\rangle>0, for all x\in\mathop(A) \setminus \. In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism. Cauchy–Schwarz inequality If A \geq 0, then :\left, \langle Ax,y\rangle \^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle. Indeed, let \varepsilon > 0. Applying Cauchy–Schwarz inequality to the inner product : (x,y)_\varepsilon \stackrel\ \langle (A+\varepsilon\cdot \mathbf)x,y\rangle as \varepsilon \downarrow 0 proves the claim. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
