# shift operators in $\ell^{p}$

Let $\mathbbmss{F}$ be $\mathbbmss{R}$ or $\mathbbmss{C}$, and let $1\leq p\leq\infty$, let $\ell^{p}(\mathbbmss{F}),\|\cdot\|_{p}$ be as in the parent entry.

The right and left shift operators $S_{r},S_{l}\colon\ell^{p}(\mathbbmss{F})\to\ell^{p}(\mathbbmss{F})$ as defined as follows. For $a=(a_{1},a_{2},\ldots)\in\ell^{p}(\mathbbmss{F})$,

 $S_{r}(a)=(0,a_{1},a_{2},\ldots)$

and

 $S_{l}(a)=(a_{2},a_{3},\ldots).$

## Properties

1. 1.

$S_{l}\circ S_{r}$ is the identity, but $S_{r}\circ S_{l}$ is not.

2. 2.

$S_{r}$ is an isometry; $\|S_{r}(a)\|=\|a\|$, and $\|S_{l}(a)\|_{p}\leq\|a\|$. Both shift operators are therefore bounded (and continuous).

Title shift operators in $\ell^{p}$ ShiftOperatorsInellp 2013-03-22 15:17:40 2013-03-22 15:17:40 matte (1858) matte (1858) 4 matte (1858) Definition msc 46B99 msc 54E50