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CONCEPT OF AN OPERATOR

The term “operator” is another term for function. An operator assigns an object from one set (the co-domain) to an object from another set (the domain). If we are talking about vector spaces we think of an operator as “operating” on one vector to produce another vector. It is viewed as a kind of black box that operates on vectors to produce other vectors. The black box has an input and an output. We input a vector into the box and it then outputs a vector. An example is the matrix A in the matrix equation y = Ax where A is viewed as a black box that operates on the vector x to produce vector y. Here matrix A maps a vector x from one space (the domain) into the vector y in another space (the co-domain). If we are talking about functional spaces where a function f(x) is viewed as a vector, an operator is viewed as a black box that operates on a function to produce another function. The input to the box is a function and the output is another function.

Example:

which associates with function f a certain function g (where K(x, t) is a definite known function called the kernel).