Hilbert–Schmidt Operators
In mathematics, Hilbert–Schmidt may refer to * a Hilbert–Schmidt operator; ** a Hilbert–Schmidt integral operator; * the Hilbert–Schmidt theorem In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is ...

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 ...
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Hilbert–Schmidt Operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_H, where \ is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm \, \cdot\, _\text is identical to the Frobenius norm. , , ·, , is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if \_ and \_ are such bases, then \sum_i \, Ae_i\, ^2 = \sum_ \left, \langle Ae_i, f_j\rangle \^2 = \sum_ \left, \langle e_i, A^*f_j\rangle \^2 = \sum_j\, A^* f_j\, ^2. If e_i = f_i, then \sum_i \, Ae_i\, ^2 = \sum_i\, A^* e_i\, ^2. As for any bounded operato ...
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Hilbert–Schmidt Integral Operator
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in ''n''-dimensional Euclidean space R''n'', a Hilbert–Schmidt kernel is a function ''k'' : Ω × Ω → C with :\int_ \int_ , k(x, y) , ^ \,dx \, dy < \infty (that is, the ''L''²(Ω×Ω; C) norm of ''k'' is finite), and the associated Hilbert–Schmidt integral operator is the operator ''K'' : ''L''²(Ω; C) → ''L''²(Ω; C) given by :$(K\; u)\; (x)\; =\; \backslash int\_\; k(x,\; y)\; u(y)\; \backslash ,\; dy.$ Then ''K'' is a with Hilbert–Schmidt norm :$\backslash Vert\; K\; \backslash Vert\_\backslash mathrm\; =\; \backslash Vert\; k\; \backslash Vert\_.$ Hilbert–Schmidt integral ...
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