A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if .
In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.
Authors:Péter Berkes, Mihály Nyerges, and János Váczi
Hungarian soccer and sponsorship market is a relatively new and unexplored subject of research in the field of sports sponsorship in view of the fact that most studies have focused on the major European soccer leagues so far. This paper focuses on the Hungarian soccer sponsorship market, which gives a variety of comparisons to other studies (Chadwick — Thwaites 2005; Couvelaere — Richelieu 2005; Bühler 2006) on soccer sponsorship focusing on the major soccer markets. A comprehensive overview of current literature on sport sponsorship in general, and soccer sponsorship in particular provided the theoretical base for this study, revealing that this specific research theme has little empirical evidence. Two issues repeatedly brought up by researchers as being important are the sponsoring companies’ objectives, and measuring the return on investment. The main objective of this paper, focusing on the Hungarian soccer sponsorship market, is to evaluate the range of soccer sponsorship objectives and the range of evaluation tools that sponsors use to measure the effectiveness of their sponsorships. The representatives of sponsor companies (N = 103) were asked to rate the importance of sponsorship objectives and the importance of the evaluation techniques which are used to measure the effectiveness of their soccer sponsorships.