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Abstract:
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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 maxf1; 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. |