Research Projects

This project aims to study the existence of dilation (self-adjoint, positive, isometric, and unitary) of sequences {A_n} of bounded operators on Hilbert space H for n greater than or equal to zero. The goal is to obtain Schaffer type construction for those sequences that obey isometric dilation. Sz.-Nagy's dilation theorem proves that a self-adjoint sequence has self-adjoint dilation B with the spectrum of B in X if and only if c₀+c₁x+c₂x²+c₃x³.....+ c_n x^n is positive whenever c₀+c₁A+c₂A²+c₃A³.....+ c_n A^n is a positive operator for every n. This result is an application of Stinespring's dilation theorem for completely positive maps. The conditions for the existence of positive, isometric, and unitary dilation will be obtained by studying suitable completely positive maps on the C*-algebra of continuous functions. Techniques of completely positive maps and positive definite kernel are employed to establish the existence of such moment dilation. The motivation for moment dilation is mainly drawn from classical moment problems like Hamburger, Stieltjes, and Hausdorff moment problems. In the case of one-dimensional Hilbert spaces, the problem coincides with the classical one. However, the example of Bisgard shows that there is a sequence of self-adjoint 2 by 2 matrices that do not have a bounded dilation. To investigate the possibility of locally bounded dilation, techniques of locally completely positive are used. Results on Stinespring type dilation and Radon-Nikodym theorem are used to prove the possibility of locally bounded dilation.