In class we recently learned that for a linear operator $\displaystyle T: V \rightarrow V$ and function $\displaystyle g(t) = a_0 + a_1t + \dots + a_nt^n$ one can define the operator $\displaystyle g(T) = a_0I + a_1T + \dots + a_nT^n$ (where $\displaystyle I$ is the identity transformation). We also recently learned about the exponential of a matrix. My question is that for a linear operator $\displaystyle T: V \rightarrow V$ can the operator $\displaystyle e^T$ be defined? (For example, like how $\displaystyle e^A$ is defined for a matrix $\displaystyle A$) (I tried searching for information on it but all I found was information on the exponential of a matrix). Thanks.