A trace formula for Hankel operators

Abstract

We show that if G G is an operator valued analytic function in the open right half plane such that the Hankel operator H G H_G with symbol G G is of trace-class, then G G has continuous extension to the imaginary axis, \[ G ( ∞ ) := lim r → ∞ r ∈ R G ( r ) G(\infty ):=\lim \limits _{r \rightarrow \infty \atop r \in \mathcal {R}} G(r) \] exists in the trace-class norm, and t r ( H G ) = 1 2 t r ( G ( 0 ) − G ( ∞ ) ) \mathrm {tr}(H_G)={1\over 2}\, \mathrm {tr}(G(0)-G(\infty )) .