Let An:l2l2be the multiplier shift operator, An(x1,x2,,)=(0,0,,x1,x2,)

where every entry from 0 to n are replaced by 0.

Find the resolvent set of An, (An):={CAnIisone-to-oneandR(AnI)=l2} and the spectrum of An, (An):=C(A):={CAnIisnotone-to-one}

Also find, p(An):={CAnIisnotone-to-oneand}, c(An):={CAnIisone-to-oneandR(AnI)=l2,R(AnI)=l2}, c(An):={CAnIisone-to-oneandR(AnI)=l2}, the point, continuous, and residual spectrum of A_n, respectively.