Exponential of an Operator

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.

Re: Exponential of an Operator

What is $\displaystyle e$ here and how is $\displaystyle e^A$ defined for a matrix A?

Re: Exponential of an Operator

$\displaystyle e$ here is the constant, $\displaystyle e = 2.718...$ (the one used for the exponential function $\displaystyle f(x) = e^x$.

And for a matrix $\displaystyle A$, $\displaystyle e^A$ is defined as follows:

$\displaystyle e^A = \sum_{n=0}^\infty \frac{A^n}{n!}$

Re: Exponential of an Operator

Quote:

Originally Posted by

**Assassin0071** And for a matrix $\displaystyle A$, $\displaystyle e^A$ is defined as follows:

$\displaystyle e^A = \sum_{n=0}^\infty \frac{A^n}{n!}$

Thanks.

Quote:

Originally Posted by

**Assassin0071** 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).

Are you asking about operators on infinite-dimensional vector spaces? Because on a finite-dimensional space, every linear operator is given by a matrix, so the exponentiation of an operator is determined by the exponentiation of its matrix.