Multiplication Theorems and Square Theorems; Bourlet Operators
A “constructive theory of linear operators” should deal with approximation and expansion of linear operators in terms of “simple” linear operators, in an analogy to the constructive theory of functions. We know that polynomials are simple in a fundamental sense, trigonometric, Bessel, etc. functions simple at least in a historic sense. But when is a linear operator on an abstract space “simple”? We shall seek the answer in demanding that beside linearity, Ω should have some simple multiplicative property, i.e. we must be able to say something about Ω(uv); this indicates that we cannot formulate our criterion in terms of a space; we must have an algebra.
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