Traducción - Griego-Inglés - Ευθέως λοιπόν, καλή δύναμη για μη παραίτηση κι...

Την αγάπη μου!!! at the end means like.....

ending a letter " Love....John " ?

Yes. "Την αγάπη μου" means "My Love". It's like that wish at the end of a letter.

Like this:

"Dear Nick,

Blah blah blah.

Love,
Alice"

So, if I understood, as nothing to do with "love", "my love"...."agapi mou"....but it's like a kind cumpliment...generally.even to friends
Doesn't mean she is adressing to her love right?

Yes is just a friendly wish. She doesn't address anyone with that. She's just saying "you have my love" as we say "you have my sympathy" or "you have my respect". Something like a way to close a letter to a friend.

In Greek "love" doesn't necessarily mean something erotic so you can say "you have my love" to any friend. It's often used, even on the radio by the broadcaster when finishing his program for example, like this: "Stay tuned for more music and talk to you again on Sunday. You have my love." Something like that.

Euxaristo poly :-)

Δεν κάνει τίποτα. :-)

Is like "Φίλη μου" that not always mean "my girlfriend" ?

I think that even in English you can say "my girlfriend" without meaning that you are a romantic relationship with her. Isn't that right? Although to be honest I've only seen it in plural "Little Mary was playing Monopoly with her girlfriends."

But in any case in Greek "φίλη μου" could mean either a girl you are in a romantic relationship with, or just a female friend. Although we usually differentiate between the two by using a different article. If you say "μια φίλη μου" (a friend of mine) you usually mean just a female friend. If on the other hand you say "η φίλη μου" (the friend of mine) it hints that you might mean that she's your mate, especially if you are a man.

Still, I don't think this is in any way related to the usage of the term "αγάπη" as we talked about above.