Thread: Right Shift Operator

Let (e_n) be a total orthonormal sequence in a seperable Hilbert Space H and define the right shift operator to be the linear operator
T:H \longrightarrow H such that Te_n=e_{n+1} for n=1,2,\cdots.

Explain the name.
Find the range, null space, norm and Hilbert Adjoint operator of T.

Originally Posted by frater_cp

Let (e_n) be a total orthonormal sequence in a seperable Hilbert Space H and define the right shift operator to be the linear operator
T:H \longrightarrow H such that Te_n=e_{n+1} for n=1,2,\cdots.

Explain the name.
Find the range, null space, norm and Hilbert Adjoint operator of T.

can anyone help me solve this problem?

The name should be obvious in that the effect of the operator on each basis element is to shift the index "forward" by one. Example: $\displaystyle Te_1=e_2$.

To find the range, you do the same as any other function. Take an element of the codomain (in this case H) and see if there is an element in the domain which is mapped to it. Example: Take $\displaystyle 3e_4+e_7$. Then an inspired guess tells us $\displaystyle T(3e_3+e_6)=3e_4+e_7$, so we can conclude this is in the range. On the other hand, $\displaystyle e_1$ has no preimage.

For finding the nullspace, here is a hint: if an element $\displaystyle x=\displaystyle\sum_{i=1}^na_ie_i$ maps to zero, what does that imply about the $\displaystyle a_i$. Use the linear independence of basis vectors!

For the norm, can you show this is an isometry?

For the adjoint, it suffices to find an operator A such that $\displaystyle \langle Tx,y\rangle=\langle x,Ay\rangle$ for all $\displaystyle x,y\in H$. Can you show that the operator defined by $\displaystyle Ae_i=e_{i-1}$ if $\displaystyle i>1$ and $\displaystyle Ae_1=0$ works?