SPECTRA OF WEIGHTED COMPOSITION OPERATORS ON WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS R. Aron and M. Lindstr¨ om Abstract. We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functions H ∞ v p when the symbol φ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe. 1. Introduction. We denote by H (D) the space of holomorphic functions on the open unit disk D and consider the weighted Bergman spaces of infinite order H ∞ v = {f ∈ H (D): ||f || v := sup z∈D v(z )|f (z )| < ∞} and H 0 v = {f ∈ H ∞ v : lim |z|→1 v(z )|f (z )| =0}, endowed with the norm ‖·‖ v , where the weight function v : D → R + is radial, continuous, nonincreasing with respect to |z | and satisfy lim |z|→1 v(z )=0. The associated weight ˜ v is defined by ˜ v(z ) = (sup{|f (z )| : f ∈ H ∞ v , ||f || v ≤ 1}) −1 . Then ˜ v is also a weight. If we take ˜ v instead of v, both the spaces H ∞ v and H 0 v and the norm || · || v do not change. Further, given z ∈ D, the element δ z ∈ (H ∞ v ) ∗ defined by δ z (f )= f (z ) satisfies ||δ z || v =1/˜ v(z ), and for each z ∈ D there is f z ∈ H ∞ v , ||f z || v ≤ 1, such that |f z (z )| =1/˜ v(z ). These remarks and more about the associated weight ˜ v can be found in [BBT]. In this paper we are interested in the Bergman spaces of infinite order for the weights v p (z ) = (1 −|z | 2 ) p , p> 0, on D. Let us point out that ˜ v p (z )= v p (z ) and notice also that the norm || · || v p is finer than the compact-open topology. Recall that the polynomials are dense in H 0 v p and 1991 Mathematics Subject Classification. Primary 47B38 Secondary 46E15. Key words and phrases. Weighted Bergman spaces of infinite order, weighted composition operator, spectrum, essential norm, essential spectral radius. The research of the second author was partially supported by the Academy of Finland Project No. 51906; the research of this paper was carried out while this author was visiting Kent State University, whose hospitality is acknowledged with thanks. Typeset by A M S-T E X