Unitary equivalence: a new approach to the Laplace transform
and the Hardy operator

Abstract: The thesis consists of three parts. In part one we compose the Laplace
transform L with a special involution T on L2(R+) and show that the
resulting operators TL and LT are unitary equivalent to multiplication by
the gamma function on L2(R). Further, with the unitary equivalence between
TL and multiplication by the gamma function we are able to derive the exact
constant of the norm of the Laplace transform on L2(R+). We end part one
with an estimate for the Laplace transform on LP(R+), 1<p&lt;2.

in part two it is shown that the hardy operator h on l2(r+) is unitary
equivalent to multiplication by the function 1/(1/2+iw) on l2(r). we then
consider the hardy minus identity operator h-i and prove, with the unitary
equivalence between h and multiplication by 1/(1/2+iw), that the unit sphere
of the space of all bounded linear operators on l2(r+) contains an interval
with the ends i and h-i. in addition, we show that i-h=-(h-i) on l2(r+) is
unitary equivalent to an operator i-a, where a is a convolution operator, on
l2(r). moreover, i-a is a shift isometry in an orthonormal basis,
constituted of laguerre functions, in l2(r).

finally, in part three we derive a sharp estimate for h-i on the cone of
decreasing functions in lp(r+) valid for all integers p bigger or equal than
2.