Prob. 10, Sec. 3.9, in Kreyszig's Functional Analysis Book: The null space and adjoint of the right-shift operator

Question

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$, let $T \colon H \to H$ be defined as follows:

Since span of $(e_n)$ is dense in $H$, for every $x \in H$, we have $$x = \sum_{n=1}^\infty \langle x, e_n \rangle e_n.$$ Now let $$Tx \colon= \sum_{n=1}^\infty \langle x, e_n \rangle e_{n+1}.$$ Thus, we have $$Te_n = e_{n+1} \ \mbox{ for each } \ n = 1, 2, 3, \ldots. $$ Moreover, $T$ is linear.

What is the null space of $T$?

What is the Hilbert adjoint operator $T^*$ of $T$?

We show that $T$ is bounded as follows:

Since the series $\sum \langle x, e_n \rangle e_n$ converges, we must have $$\sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 < +\infty. $$ So the series $\sum \langle x, e_n \rangle e_{n+1}$ also converges.

For each $n = 1, 2, 3, \ldots$, let $$x_n \colon= \sum_{j=1}^n \langle x, e_j \rangle e_j.$$ Then using the orthonormality, we have $$\Vert x_n \Vert^2 = \sum_{j=1}^n \vert \langle x, e_j \rangle \vert^2, $$ and so $$\Vert x \Vert^2 = \lim_{n\to \infty} \Vert x_n \Vert^2 = \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2. $$ Similarly, $$\Vert Tx \Vert^2 = \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2. $$ Thus $$\Vert Tx \Vert = \Vert x \Vert \ \mbox{ for all } \ x \in H.$$ So $T$ is bounded with $\Vert T \Vert = 1$. Hence $T^*$ exists.

Is this procedure correct?

Suppose that $T x = 0$. Then $$\Vert T x \Vert^2 = \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 = 0, $$ whence $$\langle x, e_n \rangle = 0 \ \mbox{ for each } \ n=1, 2, 3, \ldots, $$ and so $x= 0$. So $T$ is injective.

Am I going right?

Now how to find $T^*$?

My effort:

Let $x \colon= \sum \langle x, e_n \rangle e_n, y \colon= \sum \langle y, e_n \rangle e_n \in H$. Then $\langle x, y \rangle = \sum \langle x, e_n \rangle \overline{ \langle y, e_n \rangle }$.

Moreover, by the definitoon of $T^*$, we have $$\langle Tx , y \rangle = \langle x, T^* y \rangle.$$ Or, $$\left\langle \sum_{n=1}^\infty \langle x,e_n \rangle e_{n+1}, \sum_{n=1}^\infty \langle y, e_n \rangle e_n \right\rangle = \left\langle \sum_{n=1}^\infty \langle x,e_n \rangle e_{n}, \sum_{n=1}^\infty \langle y, e_n \rangle T^* e_n \right\rangle.$$ Or, $$ \sum_{n=1}^\infty \langle x,e_n \rangle \overline{ \langle y, e_{n+1} \rangle} = \sum_{m=1}^\infty \sum_{n=1}^\infty \langle x, e_m \rangle \overline{ \langle y, e_n \rangle } \langle e_m, T^* e_n \rangle.$$ What next?

What if $H$ is not separable?

After reading @Andre's comment:

For each $m, n \in \mathbb{N}$, we have $$\langle T e_m, e_n \rangle = \langle e_m, T^* e_n \rangle.$$ Or, $$\langle e_{m+1}, e_n \rangle = \langle e_m, T^* e_n \rangle.$$ Thus, for each $m, n \in \mathbb{N}$, we have $$ \langle T^* e_n, e_m \rangle = \begin{cases} 0 & \ \mbox{ if } \ m+1 \not= n; \\ 1 & \ \mbox{ if } \ m+1 = n. \end{cases} $$ What next?

Answer 1

Note that your computations show that $$ \|Tx\|^2=\left\|\sum_n\langle x,e_n\rangle\,e_{n+1}\right\|^2=\sum_n|\langle x,e_n\rangle|^2=\|x\|^2, $$ so $\|Tx\|=\|x\|$ for all $x$ and $T$ is an isometry. In particular, $T$ is bounded and injective.

For the adjoint note that, since $\langle T^*e_n,e_m\rangle=1$ only when $m+1=n$ (and zero elsewhere), $$ \langle T^*e_n,y\rangle=\sum_{m=1}^\infty\overline{\langle y,e_m\rangle}\,\langle T^*e_n,e_m\rangle=\begin{cases}0,&\ n=1,\\ \ \\\overline{\langle y,e_{n-1}\rangle},&\ n\ne1\end{cases} $$ so if we write $e_0=0$, $$ \langle T^*e_n,y\rangle=\langle e_{n-1},y\rangle. $$ As we can do this for any $y$, we get that $T^*e_n=e_{n-1}$. So $$ T^*x=\sum_{n=1}^\infty\langle x,e_n\rangle\,e_{n-1}. $$