Abstract. We completely characterize the spectrum of a weighted composition operator \W on \HtD when \ph has Denjoy--Wolff point $a$ with $0<|\ph '(a)|< 1$, the iterates, $\ph_n$, converge uniformly to $a$, and $\psi $ is in \Hi(the space of bounded analytic functions on \D ) and continuous at $a$. We also give bounds and some computations when $|a|=1$ and $\ph '(a)=1$ and, in addition, show that these symbols include all linear fractional \ph that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi $ so that \W is seminormal.

DOI: 10.14232/actasm-014-542-y

AMS Subject Classification (1991): 47B33, 47B35, 47A10, 47B20, 47B38

Keyword(s): weighted composition operator, spectrum of an operator, hyponormal operator

Received May 21, 2014, and in revised form September 18, 2014. (Registered under 42/2014.)