shift operators in ${\mathrm{\ell}}^{p}$
Let $\mathbb{F}$ be $\mathbb{R}$ or $\u2102$, and let $1\le p\le \mathrm{\infty}$, let ${\mathrm{\ell}}^{p}(\mathbb{F}),\parallel \cdot {\parallel}_{p}$ be as in the parent entry.
The right and left shift operators ${S}_{r},{S}_{l}:{\mathrm{\ell}}^{p}(\mathbb{F})\to {\mathrm{\ell}}^{p}(\mathbb{F})$ as defined as follows. For $a=({a}_{1},{a}_{2},\mathrm{\dots})\in {\mathrm{\ell}}^{p}(\mathbb{F})$,
$${S}_{r}(a)=(0,{a}_{1},{a}_{2},\mathrm{\dots})$$ 
and
$${S}_{l}(a)=({a}_{2},{a}_{3},\mathrm{\dots}).$$ 
Properties

1.
${S}_{l}\circ {S}_{r}$ is the identity, but ${S}_{r}\circ {S}_{l}$ is not.

2.
${S}_{r}$ is an isometry; $\parallel {S}_{r}(a)\parallel =\parallel a\parallel $, and ${\parallel {S}_{l}(a)\parallel}_{p}\le \parallel a\parallel $. Both shift operators are therefore bounded (and continuous^{}).
Title  shift operators in ${\mathrm{\ell}}^{p}$ 

Canonical name  ShiftOperatorsInellp 
Date of creation  20130322 15:17:40 
Last modified on  20130322 15:17:40 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  4 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 46B99 
Classification  msc 54E50 