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Positive-definite 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 \operatorname(A), \langle Ax, x\rangle \in \mathbb and \langle Ax, x\rangle \geq 0, where \operatorname(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 \. Many authors define a positive operator A to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness. In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism. Cauchy–Schwarz inequality Take ... [...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] |
Hellinger–Toeplitz Theorem
In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot , \cdot \rangle is bounded. By definition, an operator ''A'' is ''symmetric'' if : \langle A x , y \rangle = \langle x , A y\rangle for all ''x'', ''y'' in the domain of ''A''. Note that symmetric ''everywhere-defined'' operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named after Ernst David Hellinger and Otto Toeplitz. This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternatively, it can be argued using the uniform boundedness principle. One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator ''A'' is defined ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
