Editing Shift operator (section)

==Properties of the shift operator==

The shift operator acting on real- or complex-valued functions or sequences is a linear operator  which preserves most of the standard [[norm (mathematics)|norms]] which appear in functional analysis. Therefore, it is usually a [[continuous operator]] with norm one.

===Action on Hilbert spaces===

The shift operator acting on two-sided sequences is a [[unitary operator]] on  {{tmath|\ell_2(\Z).}} The shift operator acting on functions of a real variable is a unitary operator on {{tmath|L_2(\R).}}

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:
<math display="block"> \mathcal{F} T^t = M^t \mathcal{F}, </math>
where {{mvar|M<sup> t</sup>}} is the [[multiplication operator]] by {{math|exp(''itx'')}}. Therefore, the spectrum of {{mvar|T{{sup| t}}}} is the unit circle.

The one-sided shift {{mvar|S}} acting on {{tmath|\ell_2(\N)}} is a proper [[isometry]] with [[range of a function|range]] equal to all [[Vector (geometric)|vectors]] which vanish in the first [[coordinate]]. The operator {{mvar|S}} is a [[compression (functional analysis)|compression]] of {{math|''T''{{i sup|−1}}}}, in the sense that
<math display="block">T^{-1}y = Sx \text{ for each } x \in \ell^2(\N),</math>
where {{mvar|y}} is the vector in {{tmath|\ell_2(\Z)}} with {{math|1=''y<sub>i</sub>'' = ''x<sub>i</sub>''}} for {{math|''i'' &ge; 0}} and {{math|1=''y<sub>i</sub>'' = 0}} for {{math|''i'' < 0}}. This observation is at the heart of the construction of many [[unitary dilation]]s of isometries.

The [[Spectrum (functional analysis)|spectrum]] of {{mvar|S}} is the [[unit disk]]. The shift {{mvar|S}} is one example of a [[Fredholm operator]]; it has Fredholm index&nbsp;−1.