Perturbation theory for linear operators

adjoint algebraic assume Banach space basis belongs boundary condition bounded operator Cauchy Cauchy sequence closable closed linear manifold coefficients commutes compact complete complex numbers consider continuous function convergence denoted densely defined diagonal differential operator dist domain eigen eigenprojections eigenvalues of T(x eigenvectors equation Example exceptional point exists finite finite-dimensional follows given H₁ H₂ Hence Hilbert space holomorphic family implies inequality integral operator inverse Lemma linear form linear operator M₂ matrix multiplicity nonnegative normed space orthogonal projection P₁ P₂ proof proved PT(¹ relatively bounded Remark replaced resolvent respect satisfied sectorial selfadjoint selfadjoint operator semigroup sequence sesquilinear form singular space H spectral spectrum subset subspace sufficiently symmetric operator T-bounded T₁ T₂ Theorem true u₁ u₂ unitary space vector space zero