Trace formulas for some operators related to quadrature domains in Riemann surfaces
Let \(T\) be a linear bounded operator acting on a Hilbert space \(H\). We assume that the self-commutator \([T^*, T]\) has finite rank and that the subspace \(K\) generated by the range of the commutators \([T^{*n}T, T^*]\), \(n\geq 1\), is also finite-dimensional. Such an operator is called by the author to be of finite type. A simple argument shows that such an operator \(T\) is classsified up to unitary equivalence by the couple of finite rank (restricted) operators \(D= [T^*,T]|_K\) and \(S= T^*|_K\). The algebraic equation \(\text{det}((\overline z- S)(z- S^*)- D)= 0\) defines a system \(L\) of planar curves. In the best understood cases (\(T\) subnormal or \([T^*, T]\) of rank one), this is the essential spectrum of \(T\). The main result of the present note is a trace formula which states: \[ 2\pi i\text{\,trace}[f(T, T^*), h(T, T^*)]= \int_L m(z) f(z)\,dh(z), \] where \(f\), \(h\) are analytic functions in \(T\), \(T^*\) and \(m\) is an integer multiplicity (independent of \(f\) and \(h\)). The above formula is a refinement of Carey and Pincus' trace formula for semi-normal operators. The geometry of quadrature domains and their related Riemann surfaces comes naturally into the discussion.
scientific article; zbMATH DE number 2021094
scientific article