Simple example of functional equation in Banach Algebra and how to solve it 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.
 A: 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 $1$.
A: About the exponential operator, it is well known in quantum mechanics, but I have us it in the past for a numerical method to solve the propagation of laser beams, named "Split-Step Fourier Method".
I left the wikipedia links hoping they will help you to understand how this operator can be used and its equations solved (since its a numerical method, I believe could be an example for a Banach Space):
Shift Operator, Split-Step Method, Baker-Campbell-Hausdorff formula
