Difference of Composition Operators on Hardy Space

Suppose φ is an analytic self-map of open unit disk D{double-struck} and w is an analytic function on D{double-struck}. Then a weighted composition operator induced by φ with weight w is given by (W_w,φ f)(z) = w(z) f (φ(z)), for z ∈ D{double-struck} and f analytic on D{double-struck}. We find a sufficient condition under which two composition operators lie in the same path component of C(H²), and we find a sufficient condition for the difference of such operators to be compact on H²(D{double-struck}). Then we provide another example that answers a question raised by Shapiro and Sundberg [18] negatively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H²(D{double-struck}).