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

Let $𝔽$ be $ℝ$ or $ℂ$, and let $1\le p\le \mathrm{\infty }$,let ${\mathrm{\ell }}^p(𝔽),\parallel \cdot {\parallel }_p$ be as in the parent entry.

The right and left shift operators$S_r,S_l:{\mathrm{\ell }}^p(𝔽)\to {\mathrm{\ell }}^p(𝔽)$as defined as follows. For $a=(a_1,a_2,\mathrm{\dots })\in {\mathrm{\ell }}^p(𝔽)$,

$$S_r(a)=(0,a_1,a_2,\mathrm{\dots })$$

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 operatorsare therefore bounded (and continuous).