Preconditioners in spectral approximation

Abstract

Let $\mathcal{H}$ be a complex separable Hilbert space, and let $A$ be a bounded self-adjoint operator on $\mathcal{H}$. Consider the orthonormal basis $\mathcal{B}=\{e_1,e_2,\dots \}$ and the projection $P_n$ of $\mathcal{H}$ onto the finite-dimensional subspace spanned by the first $n$ elements of $\mathcal{B}$. The finite-dimensional truncations $A_n=P_{n}A{P}_n$ shall be regarded as a sequence of finite matrices by restricting their domains to $P_n\left(\mathcal{H}\right)$ for each $n$. Many researchers used the sequence of eigenvalues of $A_n$ to obtain information about the spectrum of $A$. But in many situations, these $A_n$’s need not be simple enough to make the computations easier. The natural question Can we use some simpler sequence of matrices $B_n$ instead of $A_n$? is addressed in this article. The notion of preconditioners and their convergence in the sense of eigenvalue clustering are used to study the problem. The connection between preconditioners and compact perturbations of operators is identified here. The usage of preconditioners in the spectral gap prediction problems is also discussed. The examples of Toeplitz and block Toeplitz operators are considered as an application of these results. Finally, some future possibilities are discussed.