Conside the differential equation $\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator on $\mathcal{H}$. With the initial condition $x(0)=x_0$, one can have $x(t)=e^{At}x_0$ (Is this legal in the infinite dimension case?). With the spectral method (under the assumption that $x_0$ is an eigenvector of $A$), one has the estimate $\|x(t)\|\leq e^{\omega t}\|x_0\|.$
Here is my question:
With some additional assumption, can one estimate $\|x(t)\|$ without using the spectral method, say, simply taking the inner product?
I have thought that the following sub-questions may be helpful:
What is $\frac{d}{dt}\langle x(t),x(t)\rangle$?
In the finite dimension case, this can be done with the product rule. What about the infinite dimension case?