Multiplication Operators on Hardy and Weighted Bergman Spaces over Planar Regions Yi Yan Abstract. This paper studies some aspects of commutant theory and func- tional calculus for multiplication operators by bounded analytic functions on Hardy and weighted Bergman spaces over bounded planar regions. Mul- tiplication operators by univalent functions are shown to commute only with multiplication operators. This result is generalized to a tuple of operators, and a sufficient condition is given for irreducibility of that induced by finite Blaschke products. Operators defined by fairly general ancestral functions are shown to commute with no nonzero compact operators, and these include the ones by monomial functions over annuli. For such operators over annuli, we characterize a certain dense subalgebra of the commutant. Norm and se- quential weak closures of the analytic functional calculus algebra generated by a multiplication operator are characterized and essential spectral mapping properties obtained. Generalizing the similarity for finite Blaschke products to a larger class of weighted Bergman spaces, the commutant classification of these operators is obtained and seen strictly finer than the similarity clas- sification, among other related results. Mathematics Subject Classification (2010). 47B35, 30H05. Keywords. Multiplication operator, Composition operator, Hardy space, Weighted Bergman space, Commutant, Functional calculus, Blaschke product. 1. Introduction A planar region is understood to be a nonempty connected open subset of the complex plane C. Let Ω always denote a bounded planar region without assuming finite connectivity nor boundary regularity, and D denote the open unit disc. Fix a power index p, p 2 [1; 1) unless specified otherwise. This work is part of the author's Ph. D. dissertation under the guidance and encouragement of Professors Tyrone Duncan and Albert Sheu. The author also wishes to thank the referees for their careful review of the manuscript and many helpful suggestions. 2 Yan Let w be a nonnegative function in L1(Ω; da), da being the Lebesgue area measure on Ω, such that on every K ⊂ Ω compact, Z w−sda < 1 (1.1) K for some s = s(K) 2 (0; 1). For example, condition (1.1) holds for moduli of nonvanishing continuous functions or nonzero analytic functions on Ω. The col- lection W(Ω) of all such functions w form a convex cone. The weighted Bergman space Ap(Ω; wda) consists of analytic functions h on Ω satisfying Z 1=p khk := jhjpwda < 1 (1.2) Ω p where w 2 W(Ω). Some special cases are the weighted Bergman spaces Ar(D) 2 r with standard Bergman weights wr(z) = (1+r)(1−jzj ) , r > −1, and the spaces A2(D; jq2jda) for polynomials q arising in the (one-variable) essential normality problem of cyclic Bergman submodules [19, Sec. 4.1]. Following [31], the Hardy space Hp(Ω;!) for a fixed reference point ! 2 Ω consists of analytic functions h on Ω for which the subharmonic function jhjp admits a harmonic majorant on Ω. Equipped with the norm khk := u(!)1=p (1.3) where u is the least harmonic majorant for jhjp, Hp(Ω;!) is a Banach space, and a Hilbert space if p = 2. Let H1(Ω) be the Banach algebra of bounded analytic functions on Ω equipped with the sup-norm k:k1. Consider the Banach space (Lemma 2.1) X = Ap(Ω; wda) or Hp(Ω;!): It is clear that H1(Ω) ⊂ X, and that every f 2 H1(Ω) defines a multiplication operator Tf 2 L(X), Tf h = fh, h 2 X. (By the closed graph theorem and a duality argument, H1(Ω) is indeed the multiplier algebra of X.) These operators over general regions were studied in [3] on Ap(Ω; da), [8] on Hp(Ω;!), [4] on A2(Ω; da), and so on. After establishing some basic lemmas in Section 2, a characterization of 0 2 the commutant fTf g is generalized in Section 3 from H (D) [9] to X, from which the commutant inclusion result of [9] follows. This characterization and ? an explicit description of the annihilator (Tf X) serve as the starting point in a duality approach to the commutant problems treated in this paper. For any 0 univalent symbol ξ, fTξg is shown to consist only of multiplication operators, with a partial converse. This result is then generalized to multiple operators, which is of interest in view of Theorems 6.5 and 7.10. Let b be a finite Blaschke ∼ L product with degree n. Despite the unitary equivalence Tb = n Tz on the Hardy 2 L space H (D) [15], the similarity Tb ∼ n Tz on the weighted Bergman space 2 Ar(D) [28], and the Riemann surface representations [9, 24, 18] for operators 0 0 in the commutant fTbg , there are no explicit global characterizations of fTbg (except for n = 2 on H2(D) [35]). On the other hand, deep results on the reducing 2 subspaces for Tb on the Bergman space A (D) are obtained in [38, 26, 24, 18, 17]. Multiplication Operators over Planar Regions 3 We give a sufficient condition in terms of boundary behavior for irreducibility of 0 2 a k-tuple fTb1 ; :::; Tbk g on X, and leave more results about fTbg on A (D; wda) to Section 7. X-ancestral symbol functions are taken up following C. C. Cowen's original concept [9]. It is proved that Tf commutes with no nonzero compact operators if the cluster set of the X-ancestral f does not exhaust its range. 2 Reducing subspaces of Tzn on Ar(D) are obtained in [38, 33]. Over an an- 2 2 nulus R = fz : 0 < r1 < jzj < r2 < 1g, H (R) and A (R) have the orthonormal k bases fckz : k 2 Zg for which Tzn is a weighted n-step bilateral shift. The 0 structure allows for characterizations of fTzn g and its reducing subspaces [16]. 0 These results motivate the treatment in Section 4 of fTzn g on the Banach space X over R where a matching direct-sum decomposition is not available. We char- 0 acterize a certain subalgebra of fTzn g containing a nontrivial idempotent thus making Tzn reducible, while Tzn does not commute with nonzero compact op- erators. This leads to the result that fTzn1 ; :::; Tznk g is reducible if and only if gcdfn1; :::; nkg > 1. Consequently, if S denotes the restriction to a certain invari- p ant subspace of the analytic Toeplitz operator Tπ! 2 L(H (@D)) of the covering n1 n map π! from D onto R, then the k-tuple of iterates fS ; :::; S k g is reducible if and only if gcdfn1; :::; nkg > 1. The norm and sequential weak closures of the algebra of operators obtained from Tf by the analytic functional calculus are characterized in Section 5, with mapping properties obtained for the essential spectrum and Browder's essential spectrum of operators in the norm closure. On unweighted Bergman and Hardy spaces Axler [3] and Conway [8] actually identify the essential spectrum. How- ever, such results are not available in the literature for weighted spaces nor for Browder's essential spectrum. Section 6 concerns commutant problems on the Hilbert space H2(D). It has been an important problem to find symbol conditions under which the commutant of analytic Toeplitz operators equals that of the operator defined by some inner function. Only sufficient conditions of varied strength are known ([5, 34, 36, 37, 9]), and under these conditions the inner function is a finite Blaschke product. Instead, we obtain a sufficient and necessary condition for the commutant of a 2 family of analytic Toeplitz operators on H (D) to equal that of Tφ for a given inner function φ. The last section treats the corresponding problems on the Hilbert space 2 2 A (D; wda) for w in a subclass Wd(D) ⊂ W(D). After generalizing from Ar(D) 2 L to A (D; wda) the similarity Tb ∼ n Tz [28, 24] for a finite Blaschke product b, 0 we obtain various results on fTbg using the approach in Section 6. The added generality beyond the standard Bergman weights seems justified by these results. In this paper all linear spaces are over the complex number field. For a bounded linear operator T 2 L(E) on a Banach space E with dual space E∗, σ(T ) denotes the spectrum and ρ(T ) its radius, σe(T ) the essential spectrum, and T ∗ 2 L(E∗) the adjoint operator. For a subset O ⊂ L(E) of operators, the commutant is O0 := fT 2 L(E): TS = ST; 8S 2 Og, and the essential 0 commutant is Oe := fT 2 L(E): TS − ST 2 K(E); 8S 2 Og where K(E) is the ideal of compact operators. For a subset W ⊂ E, W ? ⊂ E∗ denotes its 4 Yan annihilator. The maximal ideal space of a commutative unital Banach algebra A is written M(A) and the Gelfand transform of a 2 A isa ^,a ^(M(A)) = σ(a). The unit circle @D is equipped with the normalized linear Lebesgue measure dθ. 2. Preliminary lemmas We first need a key estimate which in particular implies that Ap(Ω; wda) is a Banach space, and that the point evaluation functionals over compact subsets of Ω are uniformly bounded in the dual space Ap(Ω; wda)∗. Lemma 2.1. Fix w 2 W(Ω). Then for every K ⊂ Ω compact, there exists a finite constant CK such that p sup jhj ≤ CK khk; 8h 2 A (Ω; wda): (2.1) K Proof. Choose = (K) > 0 such that K := fz 2 C : d(z; K) < g ⊂ Ω, and let s = s(K) 2 (0; 1) be the corresponding exponent as in (1.1) for the compact K.