Hardy space and essential spectrum of an operator

Abstract

We give the definitionofHardy space, the unilateral shift operator and the spectrum with the essential spectrum of an operator. The real sequence are also defined. Furthermore we prove when a trace class operator on an Hilbert space is also trace class. We also consider the special function like Airy function and Bessel function and we consider their Tracy-Widom system and Airy function asymptotic formular.