We know the Pauli spin matrices (along with identity matrix) are a basis for the C2 vector space. All of them are self adjoint. However, since they are a basis for the entire C2 operator space, linear combinations can give rise to operators that are not self adjoint.
What are the conditions on the coefficients of this basis, such that I get the most general self adjoint operator space on C2?
It is a simple solution that I missed. On taking a,b,c and d as coefficients, and equating the general operator to its adjoint, one gets the condition that all coefficients must be real.
Hence, any linear combination of the Pauli matrices and the identity with real coefficients is an operator.