operator theory

(mathematics) The general qualitative study of operators in terms of such concepts as eigenvalues, range, domain, and continuity.

At one level of abstraction an operator is simply a function whose arguments and values are real- (or complex-) valued functions of one or more real variables; in more naive terms an operator is a rule for converting such real-(or complex-) valued functions into others. The following are simple examples: (i) the operator which takes each differentiable real-valued function of one variable into its derivative; (ii) the operator which takes each twice-differentiable function ƒ of one variable into expression (1); (iii) the operator which takes each twice-differentiable function ƒ of three variables into expression (2); and (iv) the operator which takes the continuous function ƒ of one real variable into the function g where relation (3) holds.
1.

2.

3.

Since an operator is a function, the usual functional notation is applicable. L(ƒ) may be used to denote the result of operating on ƒ with the operator L. The set of all functions ƒ for which L(ƒ) is defined is called the domain of L, and the set of all functions g such that L(ƒ) = g for some ƒ in the domain of L is called the range of L. It is obvious that solving a differential or integral equation is equivalent (in many ways) to solving an operator equation L(ƒ) = g, where g and L are given and it is required to find ƒ. Moreover, the operator concept can be very useful both in theory and practice, producing a great variety of illuminating insights. See also Differential equation; Integral equation.

In large part the fruitfulness of the operator concept can be traced to two sources. One of these is the possibility of adding and multiplying operators in such a way that many, though not all, of the laws of ordinary algebra hold. The other is the fact that the ranges and domains of operators behave in many respects like ordinary space and, indeed, may be regarded as contained in infinite dimensional generalizations of the familiar three-dimensional space of solid geometry. This makes it possible to think of an operator as a geometrical transformation and to exploit one's spatial intuition.

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