This book is devoted to recent developments concerning linear operators, covering topics such as the Cauchy problem, Riesz basis, frames, spectral theory and applications to the Gribov operator in Bargmann space. Also, integral and integro-differential equations as well as applications to problems in mathematical physics and mechanics are discussed. ContentsIntroductionLinear operatorsBasic notations and resultsBasesSemi-groupsDiscrete operator and denseness of the generalized eigenvectorsFrames in Hilbert spacesSummability of seriesν-convergence operatorsΓ-hypercyclic set of linear operatorsAnalytic operators in Béla Szökefalvi-Nagy’s senseBases of the perturbed operator T(ε) Frame of the perturbed operator T(ε) Perturbation method for sound radiation by a vibrating plate in a light fluid Applications to mathematical models Reggeon field theory