Question: Let ( C [ 0 , 1 ] , ) be a complex Banach space. For the unbounded closed linear differential operator A, defined by

Let (C[0,1],) be a complex Banach space. For the unbounded closed linear differential operator A, defined by Ax:=dt2d2x with domain

D(B):={xC2[a,b]x(0)=x(1)=0} . Show that (A)=p(A)={(n2)}nN , where each eigenvalue (n2),nN, has geometric multiplicity 1 with their corresponding one-dimensional eigenspace span(sin(nt)) .

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