19.3.25 Linear operators

Condition: In the real linear space \( C[0, \pi] \) find the eigenvalues ​​and eigenvectors of the operator \( A x(t)=x^{\prime \prime}(t) \), if the operator is defined on the set \[ D(A)=\left\{x \in C[0, \pi]: x^{\prime \prime} \in C[0, \pi], x(0)=x(\pi)=0\right\} . \]