Multiplication operator

In <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>, a multiplication operator is an operator Tf defined on some <a href="/facts/Function_space/lLUE2R1t">vector space of functions</a> 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 <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of Tf, and all x in the domain of φ (which is the same as the domain of f).
Multiplication operators generalize the notion of operator given by a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a>. More precisely, one of the results of <a href="/facts/Operator_theory/Ays1ofel">operator theory</a> is a <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> that states that every <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a> on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> is <a href="/facts/Self-adjoint_operator/VrH1nEAK">unitarily equivalent</a> to a multiplication operator on an <a href="/facts/Lp_space/gbad0Rv1">L2 space</a>.
These operators are often contrasted with <a href="/facts/Composition_operator/j2Tzx5w4">composition operators</a>, which are similarly induced by any fixed function f. They are also closely related to <a href="/facts/Toeplitz_operator/loEQuV97">Toeplitz operators</a>, which are <a href="/facts/Dilation_(operator_theory)/VaxIVN9T">compressions</a> of multiplication operators on the circle to the <a href="/facts/Hardy_space/aN45dgx7">Hardy space</a>.

Multiplication operator open-in-new

Multiplication operator