The Joint Spectrum Of A Tuple Of Projection Operators

We study the joint spectrum of projection operators in a separable complex Hilbert space.Firstly,we calculate the joint spectrum of a pair of projection operators.We char-acterize the joint spectrum for a tuple[I,P,Q]in which I is the identy and P?Q is a pair of regular projections.As a consequence,we investigate the spectrum of P+Q,where P?Q is a pair of regular projection operators.We obtain that,if P?Q is a pair of regular projection operators,then ?(P+Q)is contained in[0,2],symmetric with center 1,and does not contain 0 or 1 as an isolated point.Conversely,if C is contained in[0,2],symmetric with center 1,and does not contain 0 or 1 as an isolated point,then there exist a separable Hilbert space K and a pair of regular projection operators P?Q,such that?(P+Q)=C.Secondly,this paper gives a sufficient and necessary condition for the invertibility of the sum and difference of two projection operators.Lastly,we also calculate the joint spectrum of a 3-tuple of projection operators[P,Q,R]for P?Q?R satisfying some specific conditions.