Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T:H\rightarrow H$ such that $Te_n=e_{n+1}$, for $n=1, 2, \ldots.$ Find the range, null space, norm and Hilbert-adjoint operator of $T$.
Right Shift Operator
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functional-analysis
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3Well, with this hypothesis I can write any $x\in H$ as, \begin{align*} \displaystyle x=\sum_{k=1}^\infty \langle x, e_k\rangle e_k, \end{align*} $\langle x, e_k\rangle$ are the Fourier coefficientes of $x\in H$ with respect to the orthonormal sequence $(e_n)$. For finding the $R(T)$ we aply $T$ in the equality above, \begin{align*} \displaystyle Tx=\sum_{k=1}^\infty \langle x, e_k\rangle Te_k=\sum_{k=1}^\infty \langle x, e_k\rangle e_{k+1}. \end{align*} This lead us to conjecture, \begin{align*} \displaystyle R(T)=\overline{\textrm{span}(e_2, e_3, \ldots)}. \end{align*} But how to prove this? – 2012-11-15