Sequence of Natural Powers of Right Shift Operator in 2-Sequence Space Converges in Weak Operator Topology

Theorem
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence normed vector space.

Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.

Let $R \in \map {CL} {\ell^2}$ be the right shift operator over $\ell^2$.

Let $\sequence {R^n}_{n \mathop \in \N}$ be a sequence.

Let $\mathbf 0 \in \map {CL} {\ell^2}$ be the zero mapping.

Then $\sequence {R^n}_{n \mathop \in \N}$ converges to $\mathbf 0$ in the weak operator topology.

Proof
By Representation Theorem:


 * $\ds \forall \phi \in \map {CL} {\ell^2, \C} : \exists \mathbf x_\phi = \sequence {\map {\mathbf x_\phi} k}_{k \mathop \in \N} \in \ell^2 : \forall \mathbf a = \sequence {\map {\mathbf a} k}_{k \mathop \in \N} \in \ell^2 : \map \phi {\mathbf a} = \sum_{k \mathop = 1}^\infty \map {\mathbf a} k \paren{\map {\mathbf x_\phi} k}^*$

where $*$ denotes the complex conjugation.

Furthermore:

Hence, $\sequence {R^n}_{n \mathop \in \N}$ converges to $\mathbf 0 \in \map {CL} {\ell^2}$ in the weak operator topology.