Covert link field as link field text format?
If you need to change the HTML you can Rewrite the output using tokens, see more info here: Raw Field Value in Views and in this question and answer: Add link for each column

Then you can emphasize a word in the output to make it italic, or wrap it with some HTML element with a class and italicize it with CSS in your theme. 

If you want to make the whole link text italic you can add a class to that field and style it with CSS. 

If you just want to reword the link text you can use the Rewrite results as mentioned.