Application on a Factor Derived from Hilbert and Carleson Measure on Hardy Spaces

For
to a positive Borel measure on the interval
. The Hankel matrix
with entries
, where
, the operator is formally induced

in the space of each analytical function
in the unit disc
. We classify positive Borel measures on [0,1) as such

for all in Hardy spaces
, and we describe those for which
is a bounded* operator from
into
.