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+), 1In 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.
Author: Setterqvist, Eric
Source: Lulea University of Technology
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