ABSTRACT
Q20(1) Show that the spectral functions of a projector P ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1032.tif"/> , of the zero operator 0 and of the identity operator I I ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1033.tif"/> are given, respectively by 1 F ^ 0 ( τ ) = { 0 ^ τ < 0 I I ^ τ ≥ 0 . F ^ I I ^ ( τ ) = { 0 ^ τ < 1 I I ^ τ ≥ 1 . F ^ P ^ ( τ ) = { 0 τ < 0 I I ^ − P ^ 0 ≤ τ < 1 I I ^ τ ≥ 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0416.tif"/> SQ20(1) We are dealing with operators with a discrete spectrum. The spectral functions of these operators are therefore of the form of Eq. (20.19), i.e., it is piecewise constant, being 0 ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1034.tif"/> at –∞ and I I ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1035.tif"/> at and with discontinuities at the eigenvalues.
