The Conjugate Operator Method

Abstract

Let H be a self-adjoint operator in a Hilbert space $\fancyscript {H}$ R(z) = (H − z)⁻¹ its resolvent and λ a real number in the spectrum of H. Since ∥R(λ+iμ)∥ = |μ|⁻¹, R(λ + iμ) cannot have limits in B($\fancyscript {H}$) as μ → ± 0. However, for certain vectors f ∈ $\fancyscript {H}$, the function F(z) = $\langle {f,R(z)f} \rangle$, which is defined and holomorphic for z outside the spectrum of H, could have a limit as z converges to λ from the upper or lower half-plane (these two limits will be different in general).