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).[1]

Multiplication operators generalize the notion of operator given by a diagonal matrix.[2] 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.[3]

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

Properties

  • A multiplication operator T f {\displaystyle T_{f}} on L 2 ( X ) {\displaystyle L^{2}(X)} , where X is σ {\displaystyle \sigma } -finite, is bounded if and only if f is in L ∞ ( X ) {\displaystyle L^{\infty }(X)} . In this case, its operator norm is equal to ‖ f ‖ ∞ {\displaystyle \|f\|_{\infty }} .[1]
  • The adjoint of a multiplication operator T f {\displaystyle T_{f}} is T f ¯ {\displaystyle T_{\overline {f}}} , where f ¯ {\displaystyle {\overline {f}}} is the complex conjugate of f. As a consequence, T f {\displaystyle T_{f}} is self-adjoint if and only if f is real-valued.[4]
  • The spectrum of a bounded multiplication operator T f {\displaystyle T_{f}} is the essential range of f; outside of this spectrum, the inverse of ( T f − λ ) {\displaystyle (T_{f}-\lambda )} is the multiplication operator T 1 f − λ . {\displaystyle T_{\frac {1}{f-\lambda }}.} [1]
  • Two bounded multiplication operators T f {\displaystyle T_{f}} and T g {\displaystyle T_{g}} on L 2 {\displaystyle L^{2}} are equal if f and g are equal almost everywhere.[4]

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

References

  1. ^ a b c Arveson, William (2001). A Short Course on Spectral Theory. Graduate Texts in Mathematics. Vol. 209. Springer Verlag. ISBN 0-387-95300-0.
  2. ^ Halmos, Paul (1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19. Springer Verlag. ISBN 0-387-90685-1.
  3. ^ Weidmann, Joachim (1980). Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics. Vol. 68. Springer Verlag. ISBN 978-1-4612-6029-5.
  4. ^ a b Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. (2023). Operator Theory by Example. Oxford Graduate Texts in Mathematics. Vol. 30. Oxford University Press. ISBN 9780192863867.