The discussion revolves around the properties of linear operators and the potential naming of an operator U that satisfies the property U{ f(x)*g(x) } = U{f(x)}*U{g(x)}. Participants explore the definitions and classifications of such operators, including their linearity and multiplicativity.
Participants do not reach a consensus on the definitions and classifications of operators, with multiple competing views regarding the terminology and properties of the operators discussed.
There is ambiguity regarding the definitions of linearity and multiplicativity, as well as the conditions under which the term "homomorphism" is applied. The discussion does not resolve these ambiguities.
The most common usage of the word "linear" requires another property beyond this one!Swapnil said:If an operator T is linear then for functions g(x) anf f(x),
T{ f(x) + g(x) } = T{f(x)} + T{g(x)}
Linear and multiplicative are common adjectives for such an operator. It might also be called a homomorphism.I was wondering, is there a name for operator U which has the property,
U{ f(x)*g(x) } = U{f(x)}*U{g(x)}
Wait! I am getting mixed messages. You implied that additive operators might also be called homomorphisms. But then at the end you said that linear and multiplicative operators can also be called homomorphisms?Hurkyl said:The most common usage of the word "linear" requires another property beyond this one!
Operators with this property are also often called additive operators or homomorphisms, or other things.Linear and multiplicative are common adjectives for such an operator. It might also be called a homomorphism.