On the spectral decomposition of differentiation operators (Q2729600)

In Hilbert space \({\mathcal A}= L^2(0,\infty)\) the differentiation operator \(Af= if'\), \(\text{Dom }A= \{f\in{\mathcal A}\mid f\in \text{AC}(0,\infty), f'\in{\mathcal A}, f(0)= 0\}\) is considered. To this operator a chain of operators \(A^\gamma\) in \({\mathcal A}^\gamma= L^2(0, \gamma)\), \(\gamma\in\Gamma= [\alpha, \infty)\), \(\alpha> 0\) is introduced. For the operators of this chain the spectral decompositions are constructed, where as scaled subspace the defect subspace for the first operator in the chain is chosen. Using projective limit the spectral decomposition of the operator \(A\) is obtained.