Multiplication operator
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In operator theory, a multiplication operator is an operator T_{f} 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 T_{f}, and all x in the domain of φ (which is the same as the domain of f).
This type of operators 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, which states that every selfadjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L^{2} space.
Contents
Example
Consider the Hilbert space X = L^{2}[−1, 3] of complexvalued square integrable functions on the interval [−1, 3]. With f(x) = x^{2}, define the operator
for any function φ in X. This will be a selfadjoint bounded linear operator, with domain all of X = L^{2}[−1, 3] with norm 9. Its spectrum will be the interval [0, 9] (the range of the function x→ x^{2} defined on [−1, 3]). Indeed, for any complex number λ, the operator T_{f} − λ is given by
It is invertible if and only if λ is not in [0, 9], and then its inverse is
which is another multiplication operator.
This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.
See also
 Translation operator
 Shift operator
 Transfer operator
 Decomposition of spectrum (functional analysis)
Notes
References
 Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96. Springer Verlag. ISBN 0387972455.