Multiplication operator

In operator theory, a multiplication operator is an operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, for all φ in the domain of Tf, and all x in the domain of φ (which is the same as the domain of f).[1]

Multiplication operators generalize the notion of operator given by a diagonal matrix.[2] More precisely, one of the results of operator theory is a spectral theorem that states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space.[3]

These operators are often contrasted with composition operators, which are similarly induced by any fixed function f. They are also closely related to Toeplitz operators, which are compressions of multiplication operators on the circle to the Hardy space.

  1. ^ Arveson, William (2001). A Short Course on Spectral Theory. Graduate Texts in Mathematics. Vol. 209. Springer Verlag. ISBN 0-387-95300-0.
  2. ^ Halmos, Paul (1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19. Springer Verlag. ISBN 0-387-90685-1.
  3. ^ Weidmann, Joachim (1980). Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics. Vol. 68. Springer Verlag. ISBN 978-1-4612-6029-5.