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Home > Art, Film & Photography > Functional Calculus: Holomorphic Functional Calculus, Borel Functional Calculus, Continuous Functional Calculus
Functional Calculus: Holomorphic Functional Calculus, Borel Functional Calculus, Continuous Functional Calculus

Functional Calculus: Holomorphic Functional Calculus, Borel Functional Calculus, Continuous Functional Calculus


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About the Book

Chapters: Holomorphic Functional Calculus, Borel Functional Calculus, Continuous Functional Calculus. Source: Wikipedia. Pages: 30. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function of a complex argument z and an operator T, the aim is to construct an operator which in a sense extends the function from complex argument to operator argument. This article will discuss the case where T is a bounded linear operator on some Banach space. In particular, T can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction. In this section T will be assumed to be a n x n matrix with complex entries. If a given function is of certain special type, there are natural ways of defining (T). For instance, if is a complex polynomial, one can simply substitute T for z and define where T = I, the identity matrix. This is the polynomial functional calculus. It is a homomorphism from the ring of polynomials to the ring of n x n matrices. Extending slightly from the polynomials, if is holomorphic everywhere, i.e. an entire function, with MacLaurin series mimicking the polynomial case suggests we define Since the MacLaurin series converges everywhere, the above series will converge, in a chosen operator norm. An example of this is the exponential of a matrix. Replacing z by T in the MacLaurin series of f(z) = e gives The requirement that the MacLaurin series of converges everywhere can be relaxed somewhat. From above it is evident that all that is really needed is the radius of convergence of the MacLaurin series be greater than the operator...More: http: //booksllc.net/?id=313170


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Product Details
  • ISBN-13: 9781157520016
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Holomorphic Functional Calculus, Borel Functional Calculus, Continuous Functional Calculus
  • Width: 229 mm
  • ISBN-10: 1157520014
  • Publisher Date: 15 Sep 2010
  • Binding: Paperback
  • Spine Width: 2 mm
  • Weight: 59 gr


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