Operator Theory: Isometric Operators & Anti-Linear Isometry

[LikeMath]

Hi there,
This is my first post.
In operator theory, what we mean by "The operator M_u (the multiplicative operator) acts isometrically from L^1 to L^1". Also, what is the anti-linear isometry.
Thanks in advance.

 

[HallsofIvy]

An "isometry" (from "iso" meaning "same" and "metry" meaning distance) is a function that "preserves" distance. If f is an isometry and the distance between a and b is d, then the distance between f(x) and f(y) is also d.

What is the definition of "linear" you are using?

 

[LikeMath]

Here is the sentence,
"We will use the antilinear isometry $J:L^2\rightarrow L^2$, given by $J(f)(z)=\overline{zf(z)}$.
I think linear means $A(\alpha f+g)=\alpha A f+ Ag$.