Hyponormal and quasinormal weighted composition operators on \(\ell ^ 2\) (Q808410)

For a Hilbert space H of complex-valued functions defined on a set X, a weighted composition operator T on H is usually defined by \(Tf=u(f\circ g)\) for all f in H, where u: \(X\to {\mathbb{C}}\) is a weighted function, and g: \(X\to X\) is a composition function. The author confined himself to the Hilbert space \(\ell^ 2\) of complex-valued functions defined on the integers. For a subset Y of integers, g is defined to be a composition function of Y to the integers, and u is a weighted function of integers to non-zero complex numbers, thus generalizing the usual definition of a weighted composition function. The author has characterized the normality, hyponormality and quasinormality for T and \(T^*\) with given u and g.