The multiplication operator is a mathematical symbol or function used to indicate the operation of multiplying two or more numbers or variables. In most contexts, it is represented by the symbol "×" or "*". The multiplication operator can be used in arithmetic, algebra, and other areas of mathematics to combine values.
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For a finite almost-everywhere measurable function on a sigma-finite measure space, the multiplication operator on L2 space has maximal operator domainIt is a closed operator: if and in L2 space, suitable subsequences converge almost everywhere, giving . Its operator domain is a dense subset, since truncating to approximates it in L2 space.
If is essentially bounded, is a bounded linear operator and its operator norm is the essential supremum of . The upper bound follows by integrating . For the matching lower bound, use a normalized indicator of a finite-positive-measure subset where is within of its essential supremum; sigma-finiteness supplies such a subset. Thus the measure-space hypothesis also justifies the claimed operator norm, rather than merely closedness.