Linear operator exponentials?

Question

A physics book has the following line in it: $$ f(k)=e^{k\frac{d}{dx}}f(0). $$ This is, of course, the "correct" Taylor expansion if we write out the series expansion of $e$ and assume that $k$ is a constant. I have heard of the exponentiation of matrices, and the question here is: can we also have the exponents of any general linear operators in Banach spaces? Just cite any relevant references. Also, is the above notation legal in math?