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In operator theory, a multiplication operator is a linear operator T 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,## Example

^{2} defined on [−1, 3]). Indeed, for any complex number λ, the operator T-λ is given by
## See also

- $T(varphi)(x)\; =\; f(x)\; varphi\; (x)\; quad$

for all φ in the function space 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 self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L^{2} space.

Consider the Hilbert space X=L^{2}[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. Define the operator:

- $T(varphi)(x)\; =\; x^2\; varphi\; (x)\; quad$

- $(T-lambda)(varphi)(x)\; =\; (x^2-lambda)\; varphi(x).\; quad$

- $(T-lambda)^\{-1\}(varphi)(x)\; =\; frac\{1\}\{x^2-lambda\}\; varphi(x)\; quad$

which is another multiplication operator.

This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.

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Last updated on Monday October 30, 2006 at 16:51:47 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday October 30, 2006 at 16:51:47 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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