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,

T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad }
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).

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

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

T f φ ( x ) = x 2 φ ( x ) {\displaystyle T_{f}\varphi (x)=x^{2}\varphi (x)}
for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of X = L2[−1, 3] and with norm 9. Its spectrum will be the interval [0, 9] (the range of the function x↦ x2 defined on [−1, 3]). Indeed, for any complex number λ, the operator Tf − λ is given by
( T f − λ ) ( φ ) ( x ) = ( x 2 − λ ) φ ( x ) . {\displaystyle (T_{f}-\lambda )(\varphi )(x)=(x^{2}-\lambda )\varphi (x).}

It is invertible if and only if λ is not in [0, 9], and then its inverse is

( T f − λ ) − 1 ( φ ) ( x ) = 1 x 2 − λ φ ( x ) , {\displaystyle (T_{f}-\lambda )^{-1}(\varphi )(x)={\frac {1}{x^{2}-\lambda }}\varphi (x),}
which is another multiplication operator.

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

See also

Notes

References