Perturbation Theory for Linear Operators

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