| dbo:abstract |
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential The 'scope' here means the kind of function of an operator which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus one. More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. (en) Der beschränkte Borel-Funktionalkalkül ist ein Hilfsmittel zur Untersuchung von Von-Neumann-Algebren. Dieser Funktionalkalkül ist eine Erweiterung des aus der Theorie der C*-Algebren bekannten stetigen Funktionalkalküls auf beschränkte Borel-Funktionen. Diese Erweiterung des Funktionalkalküls ist in allgemeinen C*-Algebren nicht möglich, man muss sich dafür auf die kleinere Klasse der Von-Neumann-Algebren einschränken. (de) |
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Der beschränkte Borel-Funktionalkalkül ist ein Hilfsmittel zur Untersuchung von Von-Neumann-Algebren. Dieser Funktionalkalkül ist eine Erweiterung des aus der Theorie der C*-Algebren bekannten stetigen Funktionalkalküls auf beschränkte Borel-Funktionen. Diese Erweiterung des Funktionalkalküls ist in allgemeinen C*-Algebren nicht möglich, man muss sich dafür auf die kleinere Klasse der Von-Neumann-Algebren einschränken. (de) In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential (en) |
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Beschränkter Borel-Funktionalkalkül (de) Borel functional calculus (en) |
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