# Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$

where $T_k(f):=\frac{1}{2} k^2((T+i k)^{-1} + (T-ik)^{-1})(f) \rightarrow T(f),$ for $f \in D(T).$ Notice, that $T_k$ are bounded operators.

Now, assume that our operator has a purely discrete spectrum $\lambda_0,\lambda_1,\lambda_2 ,....$ and $f:=\sum_{n=0}^{\infty} a_n e_n$, where $e_n$ are eigenfunctions to our operator. How do I show that $$e^{iT}(f)= \sum_{n=0}^{\infty}e^{i \lambda_n} a_n e_n?$$

For $\lambda \in \mathbb{R}$ and $k > 0$, $$\frac{k^{2}}{2}\left[\frac{1}{\lambda+ik}+\frac{1}{\lambda-ik}\right] =\lambda \frac{k^{2}}{\lambda^{2}+k^{2}}$$ So the above function converges to $\lambda$ as $k\rightarrow\infty$. You have $$e^{iT_{k}}x = \sum_{n}\exp\left\{i\lambda_{n}\frac{k^{2}}{\lambda_{n}^{2}+k^{2}}\right\}a_{n}e_{n}.$$ Therefore, $$\left\|e^{iT_{k}}x-\sum_{n}e^{i\lambda_{n}}a_{n}e_{n}\right\|^{2} = \sum_{n}\left|e^{i\lambda_{n}}-\exp\left(i\lambda_{n}\frac{k^{2}}{\lambda_{n}^{2}+k^{2}}\right)\right|^{2}|a_{n}|^{2}\|e_{n}\|^{2}$$ The exponential terms on the right are uniformly bounded and converge to $0$ as $k\rightarrow\infty$ for each fixed $n$. And $$\|x\|^{2}=\sum_{n}|a_{n}|^{2}\|e_{n}\|^{2} < \infty.$$ So you can apply the Lebesgue dominated convergence theorem to see that the right side converges to $0$ as $k\rightarrow\infty$. And that's what you want.
• I took "where en are eigenfunctions to our operator" to mean the eigenfunctions for $T$. Then the second line follows because of how $T_{k}$ is defined in terms of $T$. The first part of the second line was a statement about functions, not about operators. – Disintegrating By Parts Aug 11 '14 at 2:01
• Maybe I am blind, but I still don't see it. $e^{iT_k}(x):=(\sum_{n=0}^{\infty} \frac{T_k^n}{n!})(x)$ and $x = \sum_{n=0}^{\infty}a_n e_n$ and probably your $x$ is my $f$. But still, I just don't see what has happened there, sorry. – user168865 Aug 11 '14 at 9:37
• @user168865 : $T_{k}e_{m} = [\lambda_{m}k^{2}/(\lambda_{m}^{2}+k^{2})]e_{m}$ and $T_{k}^{n}e_{m}=[\lambda_{m}k^{2}/(\lambda_{m}^{2}+k^{2})]^{n}e_{m}$. Yes, I see I switched out $f$ for $x$ in notation, and added confusion. – Disintegrating By Parts Aug 11 '14 at 11:24