eigenfunctions of the adjoint of an operator

If the eigenfunctions of a linear operator are known, is there a way to calculate the eigenfunctions of the corresponding adjoint operator based on the known eigenfunctions? In other words, what's the relation between the eigenfunctions of an operator and its adjoint?

Thanks!

• Also, how about the eigenfunctions of the inverse operator if it's invertible? – chaohuang Apr 16 '12 at 18:23

If you don't specify some conditions on your operator, the answer is no. Consider $H=\ell^2(\mathbb{N})$, and let $T$ be the "reverse shift" operator, given by $$T(a_1,a_2,\ldots)=(a_2,a_3,\ldots).$$ Then it is easy to see that every $\lambda\in\mathbb{C}$ with $|\lambda|<1$ is an eigenvalue with eigenvector $(\lambda,\lambda^2,\lambda^3,\ldots)$. More properly, one is free to choose the first coordinate, but that is irrelevant here.
Now, $T^*$ is the usual shift $$T^*(a_1,a_2,\ldots)=(0,a_1,a_2,\ldots)$$ and it has no eigenvalues (and so no eigenvectors).
In the finite-dimensional case, the eigenfunctions for the adjoint are the dual basis to the basis of eigenfunctions. That is, if $f_1,\ldots,f_n$ are the eigenfunctions for an operator $T$, then the eigenfunctions $g_1,\ldots,g_n$ for the adjoint are defined by the equations $$\langle g_i,f_j\rangle = \delta_{ij},$$ where $\langle-,-\rangle$ is the inner product on the space of functions.
If we write $g_i = a_{i1}f_1 + a_{i2}f_2 + \cdots + a_{in}f_n$, then we can use the above equations to solve for the coefficients $a_{ij}$. The solution is that the matrix of coefficients $a_{ij}$ is the inverse of the matrix whose entries are $\langle f_i,f_j\rangle$.