Multiplication operator

In operator theory, a multiplication operator is a linear operator {{math|T_f}} defined on some vector space of functions and whose value at a function {{mvar|φ}} is given by multiplication by a fixed function {{mvar|f}}. That is,

$T_f\varphi(x) = f(x) \varphi (x) \quad$

for all {{mvar|φ}} in the domain of {{math|T_f}}, and all {{mvar|x}} in the domain of {{mvar|φ}} (which is the same as the domain of {{mvar|f}}).{{cite book|last=Arveson|first=William|authorlink = William Arveson|title=A Short Course on Spectral Theory|year=2001|series=Graduate Texts in Mathematics|volume=209|publisher=Springer Verlag|isbn=0-387-95300-0}}

Multiplication operators generalize the notion of operator given by a diagonal matrix.{{cite book|last=Halmos|first=Paul|authorlink=Paul Halmos|title=A Hilbert Space Problem Book|series=Graduate Texts in Mathematics|volume=19|publisher=Springer Verlag|year=1982|isbn=0-387-90685-1}} 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.{{cite book|last=Weidmann|first=Joachim|title=Linear Operators in Hilbert Spaces|series=Graduate Texts in Mathematics|volume=68|publisher=Springer Verlag|year=1980|isbn=978-1-4612-6029-5}}

These operators are often contrasted with composition operators, which are similarly induced by any fixed function {{mvar|f}}. 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 $T_f$ on $L^2(X)$ , where {{mvar|X}} is $\sigma$ -finite, is bounded if and only if {{mvar|f}} is in $L^\infty(X)$ . In this case, its operator norm is equal to $\|f\|_\infty$ .
The adjoint of a multiplication operator $T_f$ is $T_\overline{f}$ , where $\overline{f}$ is the complex conjugate of {{mvar|f}}. As a consequence, $T_f$ is self-adjoint if and only if {{mvar|f}} is real-valued.{{cite book|last1=Garcia|first1=Stephan Ramon|author1link = Stephan Ramon Garcia|last2=Mashreghi|first2=Javad|author2link = Javad Mashreghi|last3=Ross|first3=William T.|title=Operator Theory by Example|year=2023|series=Oxford Graduate Texts in Mathematics|volume=30|publisher=Oxford University Press|isbn=9780192863867}}
The spectrum of a bounded multiplication operator $T_f$ is the essential range of {{mvar|f}}; outside of this spectrum, the inverse of $(T_f - \lambda)$ is the multiplication operator $T_{\frac{1}{f - \lambda}}.$
Two bounded multiplication operators $T_f$ and $T_g$ on $L^2$ are equal if {{mvar|f}} and {{mvar|g}} are equal almost everywhere.

Example

Consider the Hilbert space {{math|1=X = L²[−1, 3]}} of complex-valued square integrable functions on the interval {{closed-closed|−1, 3}}. With {{math|1=f(x) = x²}}, define the operator

$T_f\varphi(x) = x^2 \varphi (x)$

for any function {{mvar|φ}} in {{mvar|X}}. This will be a self-adjoint bounded linear operator, with domain all of {{math|1=X = L²[−1, 3]}} and with norm {{math|9}}. Its spectrum will be the interval {{closed-closed|0, 9}} (the range of the function {{math|x↦ x²}} defined on {{closed-closed|−1, 3}}). Indeed, for any complex number {{mvar|λ}}, the operator {{math|T_f − λ}} is given by

$(T_f - \lambda)(\varphi)(x) = (x^2-\lambda) \varphi(x).$

It is invertible if and only if {{mvar|λ}} is not in {{closed-closed|0, 9}}, and then its inverse is

$(T_f - \lambda)^{-1}(\varphi)(x) = \frac{1}{x^2-\lambda} \varphi(x),$

which is another multiplication operator.

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

References

Bibliography

{{cite book|last=Conway|first=J. B.|authorlink = John B. Conway|title=A Course in Functional Analysis|year=1990|series=Graduate Texts in Mathematics|volume=96|publisher=Springer Verlag|isbn=0-387-97245-5}}

Category:Operator theory

Category:Linear operators

Multiplication operator

Properties

Example

See also

References

Bibliography