Reducing Subspaces for a Class of Multiplication Operators

Abstract

Let D be the open unit disk in the complex plane C. The Bergman space L a 2 ( D ) is the Hilbert space of analytic functions f in D such that ‖ f ‖ 2 = ∫ D | f ( z ) | 2 d A ( z ) < ∞ where dA is the normalized area measure on D. If f ( z ) = ∑ n - 0 ∞ a n z n and g ( z ) = ∑ n - 0 ∞ b n z n are two functions in L a 2 ( D ) , then the inner product of f and g is given by 〈 f , g 〉 = ∫ D f ( z ) g ( z ) ¯ d A ( z ) = ∑ n = 0 ∞ a n b ¯ n n + 1 We study multiplication operators on L a 2 ( D ) induced by analytic functions. Thus for φ ∈ H ∞(D), the space of bounded analytic functions in D, we define M φ : L a 2 ( D ) → L a 2 ( D ) by M φ f = φ f , f ∈ L a 2 ( D ) It is easy to check that Mϕ is a bounded linear operator on L a 2 ( D ) with ‖Mφ‖=‖φ‖∞=sup{|φ(z)|:z∈D}.