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In operator theory, a multiplication operator is an operator ''T'' 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, : for all φ in the function space and all ''x'' in the domain of φ (which is the same as the domain of ''f''). This type of operators 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, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an ''L''''2'' space. ==Example== Consider the Hilbert space ''X''=''L''2(3 ) of complex-valued square integrable functions on the interval (3 ). Define the operator: : for any function φ in ''X''. This will be a self-adjoint bounded linear operator with norm 9. Its spectrum will be the interval (9 ) (the range of the function ''x''→ ''x''2 defined on (3 )). Indeed, for any complex number λ, the operator ''T''-λ is given by : It is invertible if and only if λ is not in (9 ), and then its inverse is : which is another multiplication operator. This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multiplication operator」の詳細全文を読む スポンサード リンク
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