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Isometric Operators

  1. Sep 30, 2011 #1
    Hi there,
    This is my first post.
    In operator theory, what we mean by "The operator M_u (the multiplicative operator) acts isometrically from L^1 to L^1". Also, what is the anti-linear isometry.
    Thanks in advance.
  2. jcsd
  3. Sep 30, 2011 #2


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    Science Advisor

    An "isometry" (from "iso" meaning "same" and "metry" meaning distance) is a function that "preserves" distance. If f is an isometry and the distance between a and b is d, then the distance between f(x) and f(y) is also d.

    What is the definition of "linear" you are using?
  4. Sep 30, 2011 #3
    Here is the sentence,
    "We will use the antilinear isometry [itex] J:L^2\rightarrow L^2[/itex], given by [itex]J(f)(z)=\overline{zf(z)}[/itex].
    I think linear means [itex]A(\alpha f+g)=\alpha A f+ Ag[/itex].
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