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Reducing Subspaces for a Class of Multiplication Operators
Author(s) -
Zhu Kehe
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001198
Subject(s) - linear subspace , bounded function , combinatorics , analytic function , unit disk , hilbert space , mathematics , blaschke product , multiplication (music) , product (mathematics) , complex plane , space (punctuation) , unit (ring theory) , bergman space , bounded operator , physics , mathematical analysis , pure mathematics , linguistics , philosophy , geometry , mathematics education
Let D be the open unit disk in the complex plane C. The Bergman spaceL 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 nand g ( z ) = ∑ n ‐ 0 ∞b n z nare two functions inL 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 nb ¯ nn + 1We study multiplication operators onL a 2 ( D )induced by analytic functions. Thus for φ ∈ H ∞(D), the space of bounded analytic functions in D, we defineM φ : L a 2 ( D ) → L a 2 ( D )byM φ f = φ f ,f ∈ L a 2 ( D )It is easy to check that M ϕ is a bounded linear operator onL a 2 ( D )with ‖ M φ ‖=‖φ‖ ∞ =sup{|φ( z )|: z ∈D}.