  Quadratic forms of Hermitian matrix resolvents involve the solutions of
shifted linear systems. Efficient iterative solutions use the shift-invariance
property of Krylov subspaces The Hermitian Lanczos method reduces a given
vector and matrix to a Jacobi matrix (real symmetric tridiagonal matrix with
positive super and sub-diagonal entries) and approximates the quadratic form
using the Jacobi matrix. This study develops a shifted Lanczos method that
deals directly with the Hermitian matrix resolvent. We derive a matrix
representation of a linear operator that approximates the resolvent by solving
a Vorobyev moment problem associated with the shifted Lanczos method. We show
that an entry of the Jacobi matrix resolvent can approximate the quadratic
form, matching the moments. We give a sufficient condition such that the method
does not break down, an error bound, and error estimates. Numerical experiments
on matrices drawn from real-world applications compare the proposed method with
previous methods and show that the proposed method outperforms well-established
methods in solving some problems.
