Linear Operator T Property: U{f(x)*g(x)}=U{f(x)}*U{g(x)}

[Swapnil]

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)}

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)}

??

 

[Hurkyl]

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)}

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.

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)}

Linear and multiplicative are common adjectives for such an operator. It might also be called a homomorphism.

 

[Swapnil]

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.

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]

Yes. Homomorphism is a generic term for "structure-preserving map", and it is commonly used when it's clear which structure is meant.