On holomorphic functions on a strip in the complex plane.pdf

a r X i v : m a t h / 9 9 0 5 1 2 6 v 1 [ m a t h .C V ] 2 0 M a y 1 9 9 9 On holomorphic functions on a strip in the complex plane Konrad Schmu?dgen AMS Subject class. (1991): Primary 30D05; Secondary 81S05, 47D40, 17B37. Abstract Let f be a holomorphic function on the strip {z ∈ C : ?α Im z α}, α 0, belonging to the class H(α,?α; ε) defined below. It is shown that there exist holomorphic functions w1 on {z ∈ C : 0 Im z 2α} and w2 on {z ∈ C : ?2α Im z 2α} such that w1 and w2 have boundary values of modulus one on the real axis and satisfy the relation w1(z) = f(z?αi)w2(z?2αi) and w2(z+2αi) = f?(z+αi)w1(z) for 0 Im z 2, where f?(z) := f(z?). This leads to a ”polar decomposition” f(z) = uf (z+αi)gf (z) of the function f(z), where uf (z+αi) and gf (z) are holomorphic functions for ?α Im z α such that |uf (x)| = 1 and gf (x) ≥ 0 a.e. on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed. 1 Introduction and Main Results Let ? be a positive number and α and β be real numbers such that α β. Let H(α, β; ?) denote the set of all holomorphic functions h(z) on the strip I(α, β) := {z ∈ C : α Im z β} such that sup αyβ ∞∫ ?∞ |h (x+yi)|2 e?2γx2 dx ∞ for all numbers γ ?. As stated in Lemma 2 below, each function h ∈ H(α, β; ?) admits boundary values h(x+βi) and h(x+αi), x ∈ IR, which satisfy lim y↓β ∫ |h (x+yi)?h (x+βi)|2 e?2γx2 dx = 0, (1) lim y↑α ∫ |h (x+yi)?h (x+αi)|2 e?2γx2 dx = 0 (2) for all γ ?. Throughout this paper, i denotes the complex unit. By some slight abuse of notation, we denote functions in H(α, β; ?) and their boundary values by the same symbol. Our main results are contained in the following Theorem 1. Let ε and α be positive real numbers and let f 6= 0 a function of the class H(α,?α; ε) such that inf{|f(x?αi)|; x ∈ IR} 0 . (3) 1 Then there exist functions w1 ∈ H(2α, 0; ε) and w2 ∈ H(2α,?2α; ε) such that |w1(x)| = |w2(x)| = 1 a.e. on IR and w1(x) = f(x?αi)w2(x?2αi), (4) w2(x) = f?(