A theorem of Brown-Halmos type on the Bergman space modulo finite rank operators (Q2404082)

The authors study the following specific algebraic problem for Toeplitz operators on the Bergman space on the unit disk: characterise the conditions under which the operator \(T_fT_g - T_h\) has a finite rank, where \(f\) and \(g\) are bounded harmonic functions, and \(h\) is a bounded \(C^2\)-function with invariant Laplacian in \(L^1\). The main results of the paper state: (1) If the rank of \(T_fT_g - T_h\) is at most one, then either \(f\) is conjugate analytic or \(g\) is analytic; in either case, \(h=fg\) and \(T_fT_g = T_h\). (2) For each \(m >1\), there are functions \(f, g\) and \(h \neq fg\) such that the operator \(T_fT_g - T_h\) has rank \(m\). Several interesting examples are also given.