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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,
This type of operator is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. 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.
Example edit
Consider the Hilbert space X = L2[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. With f(x) = x2, define the operator
It is invertible if and only if λ is not in [0, 9], and then its inverse is
This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.
See also edit
Notes edit
References edit
- Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5.