What are eigenvalues/eigenfunctions of a "pointwise product" operator

Question

Let us consider the Hilbert space $l^2([0,1])$ with inner product $<u,v>=\int_0^1 u(x)v(x)\mathrm dx$. We define a pointwise product operator $A$ as $(A\circ u)(x)=a(x)\cdot u(x)$, where "$\cdot$" denotes the scalar product, and $a$ is a bounded function. It can be verified that $A$ is a linear and self-adjoint operator. Then, there should be countable real eigenvalues $\lambda_n$ and eigenfunctions $\phi_n$ so that $(A\circ u)(x)=\sum_n \lambda_n <\phi_n,u>\phi_n(x)$. But it is not clear for me what they are.