In operator theory, a multiplication operator is a linear operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,

<math display="block">T_f\varphi(x) = f(x) \varphi (x) \quad </math>

for all in the domain of , and all in the domain of (which is the same as the domain of ).

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 L<sup>2</sup> space.

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

Properties

  • A multiplication operator <math>T_f</math> on <math>L^2(X)</math>, where is <math>\sigma</math>-finite, is bounded if and only if is in <math>L^\infty(X)</math>. (The backward direction of the implication does not require the <math>\sigma</math>-finiteness assumption.) In this case, its operator norm is equal to <math>\|f\|_\infty</math>.
  • The adjoint of a multiplication operator <math>T_f</math> is <math>T_\overline{f}</math>, where <math>\overline{f}</math> is the complex conjugate of . As a consequence, <math>T_f</math> is self-adjoint if and only if is real-valued.
  • The spectrum of a bounded multiplication operator <math>T_f</math> is the essential range of ; outside of this spectrum, the inverse of <math>(T_f - \lambda)</math> is the multiplication operator <math>T_{\frac{1}{f - \lambda.</math>
  • Two bounded multiplication operators <math>T_f</math> and <math>T_g</math> on <math>L^2</math> are equal if and are equal almost everywhere.

Example

Consider the Hilbert space of complex-valued square integrable functions on the interval . With , define the operator

<math display="block">T_f\varphi(x) = x^2 \varphi (x) </math>

for any function in . This will be a self-adjoint bounded linear operator, with domain all of and with norm . Its spectrum will be the interval (the range of the function defined on ). Indeed, for any complex number , the operator is given by

<math display="block">(T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x). </math>

It is invertible if and only if is not in , and then its inverse is

<math display="block">(T_f - \lambda)^{-1}(\varphi)(x) = \frac{1}{x^2-\lambda} \varphi(x),</math>

which is another multiplication operator.

This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any L<sup>p</sup> space.

See also

  • Shift operator
  • Transfer operator
  • Decomposition of spectrum (functional analysis)

References

Bibliography