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I have started operator theory in Functional Analysis and have got stuck here:

If $(e_n)_n $is a total orthonormal sequence in a separable Hilbert Space $H$ and the right shift operator $T:H\to H$ is defined by $T(e_n)=e_{n+1}$ for $n=1,2\dots$ .

Show that $T$ is not normal.

Step 1:I need to find $T^*$ i.e. the Hilbert -Adjoint Operator first.I am unable to find the operator $T^*$.

Step 2: Need to show that $TT^*\neq T^*T$.

However I am stuck on the first one.Please help.

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You have $\langle T^*e_n,e_k\rangle=\langle e_n,Te_k\rangle=\langle e_n,e_{k+1}\rangle=\delta_{n,k+1}=\delta_{n-1,k}$, which implies $T^*e_n=e_{n-1}$ if $n\geq 2$, and $T^*e_1=0$. Then compare $T^*Te_1$ with $TT^*e_1$.

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Define for all $\xi \in H = l_2(\mathbb{N})$ the operator $(S\xi)(n) := \xi(n+1)$.

We then calculate for $\xi,\eta \in H$ on the one hand $$ \langle \xi, S \eta \rangle = \sum_{n = 1}^\infty \xi(n)\, S \eta(n) = \sum_{n = 1}^\infty \xi(n)\, \eta(n + 1) = 0 \cdot \eta(1) + \sum_{n = 2}^\infty \xi(n - 1)\, \eta(n)\\ = T\xi(1)\, \eta(1) + \sum_{n = 2}^\infty T\xi(n)\, \eta(n) = \sum_{n = 1}^\infty T\xi(n) \, \eta(n) = \langle T \xi, \eta \rangle. $$ On the other hand, we have $$ \langle \xi, T^* \eta \rangle = \langle T \xi, \eta \rangle. $$

Uniqueness of the adjoint then gives $S = T^*$.

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