In my readings through Banach Algebra I noticed something. For real functions for example we know that the functional equation
$$ f(x+y)=f(x)f(y) $$
Has as solution $f(x) = e^x$. Now since bounded operators in $L^p(\mu)$ form a Banach Algebra as well I observed that a similar relationship holds for the translation operator. If $(T_yf)(x) = f(x -y)$ we have the relationship
$$ T_{x+y} = T_x T_y $$
Which resembles pretty much the first functional equation I saw. From a random reading yesterday about quantum mechanics I saw that the translation operator can be expressed as something like an exponential as well. I wonder if therefore there's some techniques for solving functional equations in Banach Algebra that would allow to represent operators using functional equations.
What you are looking for is the called "One-parameter Semigroup" in mathematics. For example, Peter Lax's Functional analysis text, Chapter 34 showed that if $T_0=Id$ and $\forall x,y\ge0, T_xT_y = T_{x+y}$ and $T$ is continuous at $0$ (that is $\lim_{x\rightarrow 0} T_x = Id$), then $T$ must be of the form $T_t = e^{tG}$ where $G$ is known as the infinitesimal generator of the one-parameter semi-group. The proof is essentially to write $G$ as log of $T_t$, and it's not hard to show the power series for the usual log converges around $1$, because $T_t$ will be close to $Id$, hence its norm is not far from $1$.