
Right shift operator
Dear Colleagues,
Could you please help me in solving the following problem:
Let $\displaystyle (e_{n})$ be a total orthonormal sequence in a separable Hilbert space $\displaystyle H$ and define the right shift operator to be the linear operator $\displaystyle T$ form $\displaystyle H$ into $\displaystyle H$ such that $\displaystyle Te_{n}=e_{n+1}$ for $\displaystyle n=1,2,3,...$, what is the Hilbertadjoint operator $\displaystyle T^{*}$ of $\displaystyle T$.
Best Regards.

What did you try?
You only have to know the definition of the adjoint of an operator and work with the infinite sum.