This operator is invertible I'm thinking if the self-adjoint operators are invertible. I'm really stuck, I don't know even how to begin,I need a hint or or a counter-example if it's not true.
Thanks

The constant zero operator is self-adjoint, but clearly not invertible.
EDIT:
More generally, for an operator on finite dimensional real vector space represented in some bases by a matrix A it will be self-adjoint if the matrix is symmetric (in the complex case, if the matrix is Hermitian) and it will be invertible if the matrix is invertible. It is very easy to construct symmetric non-zero matrices that are not invertible, for example diagonal matrices that have at least one zero on the diagonal.
