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Math Help - Right Shift Operator

  1. #1
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    Cool 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.
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  2. #2
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    Quote Originally Posted by frater_cp View Post
    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?
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  3. #3
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    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: 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 3e_4+e_7. Then an inspired guess tells us T(3e_3+e_6)=3e_4+e_7, so we can conclude this is in the range. On the other hand, e_1 has no preimage.

    For finding the nullspace, here is a hint: if an element x=\displaystyle\sum_{i=1}^na_ie_i maps to zero, what does that imply about the 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 \langle Tx,y\rangle=\langle x,Ay\rangle for all x,y\in H. Can you show that the operator defined by Ae_i=e_{i-1} if i>1 and Ae_1=0 works?
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