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

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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:

\(\ds \forall \mathbf a \in \ell^2 : \forall \phi \in \map {CL} {\ell^2, \C}: \, \) \(\ds \size {\map \phi {R^n \mathbf a} }^2\) \(=\) \(\ds \size {\sum_{k \mathop = 1}^\infty \map {\mathbf a} k \paren{\map {\mathbf x_\phi} {n + k} }^* }^2\)
\(\ds \) \(\le\) \(\ds \norm {\mathbf a}^2_2 \sum_{k \mathop = 1}^\infty \size {\map {\mathbf x_\phi} {n + k} }^2\) Cauchy-Schwarz Inequality
\(\ds \leadsto \ \ \) \(\ds \lim_{n \mathop \to \infty } \size {\map \phi {R^n \mathbf a} }^2\) \(\le\) \(\ds \lim_{n \mathop \to \infty} \norm {\mathbf a}^2_2 \sum_{k \mathop = 1}^\infty \size {\map {\mathbf x_\phi} {n + k} }^2\)
\(\ds \) \(=\) \(\ds \norm {\mathbf a}^2_2 \lim_{n \mathop \to \infty} \sum_{k \mathop = 1}^\infty \size {\map {\mathbf x_\phi} {n + k} }^2\) Multiple Rule for Real Sequences
\(\ds \) \(=\) \(\ds \norm {\mathbf a}^2_2 \sum_{k \mathop = 1}^\infty \lim_{n \mathop \to \infty} \size {\map {\mathbf x_\phi} {n + k} }^2\) Exchange of Limits
\(\ds \) \(=\) \(\ds 0\) Definition of P-Sequence Space, Terms in Convergent Series Converge to Zero


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

$\blacksquare$


Sources

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