OPERATOR HILBERTOVE MATRICE I LIBERIN OPERATOR NA PROSTORIMA HOLOMORFNIH FUNKCIJA

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OPERATOR HILBERTOVE MATRICE I LIBERIN OPERATOR NA PROSTORIMA HOLOMORFNIH FUNKCIJA

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Title: OPERATOR HILBERTOVE MATRICE I LIBERIN OPERATOR NA PROSTORIMA HOLOMORFNIH FUNKCIJA
Author: Karapetrović, Boban
Abstract: In this thesis, we study the in nite Hilbert matrix viewed as an operator, called the Hilbert matrix operator and denoted by H and Libera operator, denoted by L, on the classical spaces of holomorphic functions on the unit disk in the complex plane. It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space Ap into Ap if and only if 2 < p < 1. Also, it is known that the norm of the Hilbert matrix operator H on the Bergman space Ap is equal sin 2 p , when 4 p < 1, and it was conjectured that kHkAp!Ap = sin 2 p ; when 2 < p < 4. In this thesis we prove this conjecture. We nd the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap; , kHkAp; !Ap; sin ( +2) p ; for 1 < + 2 < p: We show that if 4 2( + 2) p, then kHkAp; !Ap; = sin ( +2) p ; while in the case 2 +2 < p < 2( +2), upper bound for the norm kHkAp; !Ap; , better then known, is obtained. We prove that the Hilbert matrix operator H is bounded on the Besov spaces Hp;q; if and only if 0 < p; ; = 􀀀 􀀀 1 p + 1 < 1. In particular, operator H is bounded on the Bergman space Ap; if and only if 1 < + 2 < p and it is bounded on the Dirichlet space Dp = Ap; 1 if and only if maxf􀀀1; p 􀀀 2g < < 2p 􀀀 2. We also show that if > 2 and 0 < " 􀀀 2, then the logarithmically weighted Bergman space A2 log is mapped by the Hilbert matrix operator H into the space A2 log 􀀀2􀀀" . If 2 R, then the Hilbert matrix operator H maps logarithmically weighted Bloch space Blog into Blog +1. We also prove that operator H maps logarithmically weighted Hardy-Bloch space B1 log , when 0, into B1 log 􀀀1 and that this result is sharp. Also, we have that the space VMOA is not mapped by the Hilbert matrix operator H into the Bloch space B. On the other hand, we nd that the Libera operator L is bounded on the Besov space Hp;q; if and only if 0 < p; ; = 􀀀 􀀀 1 p + 1. Then, we prove that if > 1, then the logarithmically weighted Bergman space A2 log is mapped by the Libera operator L into the space A2 log 􀀀1 , while if 2 R, then the Libera operator L maps logarithmically weighted Bloch space Blog into itself. If > 0, we have that operator L maps logarithmically weighted Hardy-Bloch space B1 log into B1 log 􀀀1 and we show that this result is sharp. The well known conjecture due to Korenblum about maximum principle in Bergman space Ap states: Let 0 < p < 1. Then there exists a constant 0 < c < 1 with the following property. If f and g are holomorphic functions in the unit disk D, such that jf(z)j jg(z)j for all c < jzj < 1, then kfkAp kgkAp . Hayman proved Korenblum's conjecture for p = 2 and Hinkkanen generalized this result, by proving conjecture for all 1 p < 1. The case 0 < p < 1 of conjecture still remains open. In this thesis we resolve this case of the Korenblum's conjecture, by proving that Korenblum's maximum principle in Bergman space Ap does not hold when 0 < p < 1.
URI: http://hdl.handle.net/123456789/4497
Date: 2017

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