The discussion centers on the properties of linear operators and the specific characteristics of an operator U defined by the equation U{f(x)*g(x)} = U{f(x)}*U{g(x)}. Participants clarify that such operators are often referred to as additive operators or homomorphisms, with the terms "linear" and "multiplicative" frequently used to describe them. The conversation emphasizes that the term "homomorphism" serves as a generic descriptor for structure-preserving maps, applicable when the context is clear. The distinction between linearity and multiplicativity is highlighted as essential in understanding these operators.
PREREQUISITESMathematicians, students of functional analysis, and anyone studying operator theory will benefit from this discussion, particularly those interested in the properties of linear and multiplicative operators.
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.