Hilbert space and orthonormal basis. Let $H$ be a Hilbert space and let ${e_n} ,\   n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(e_n)=e_1,  n=1,2,3,\ldots$$ 
$$(b)\quad T(e_n)=e_{n+1}, n=1,2,3,\ldots$$
$$(c)\quad T(e_n)=e_{n-1} , n=2,3,4,\ldots , \,\,T(e_1)=0$$
I think $(a)$ is not true because $e_1$ can not span the range space. I really don't know how to approach to this problem. Could you please give me some hints? Thank you very much.

You're questions been answered, but if you're curious about examples of bounded linear operators that satisfy the last two resp.  Look no further than $l^2(\mathbb{N})$ and consider the left and right shift operators on the standard basis.  That is 
$$T_L( (a_1, a_2, ... a_n , ... )) = (a_2, a_3, ... a_n,\   ... ) \\
T_R( (a_1, a_2, ... a_n , ... )) = (0, a_1,\  ... \ a_n,\  ... ) $$
