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This is the Part I paper of our series of full papers on a gauge-invariant{\it linear} perturbation theory on the Schwarzschild background spacetimewhich was briefly reported in our short papers [K.~Nakamura, Class. QuantumGrav. {\bf 38} (2021), 145010; K.~Nakamura, Letters in High Energy Physics {\bf2021} (2021), 215.]. We first review our general framework of thegauge-invariant perturbation theory, which can be easily extended to the {\ithigher-order} perturbation theory. When we apply this general framework toperturbations on the Schwarzschild background spacetime, a gauge-invarianttreatments of $l=0,1$ mode perturbations are required. On the other hand, inthe current consensus on the perturbations of the Schwarzschild spacetime,gauge-invariant treatments for $l=0,1$ modes are difficult if we keep thereconstruction of the original metric perturbations in our mind. Based on thissituation, we propose a strategy of a gauge-invariant treatments of $l=0,1$mode perturbations through the decomposition of the metric perturbations bysingular harmonic functions at once and the regularization of this singularitythrough the imposition of the boundary conditions to the Einstein equations.Following this proposal, we derive the linearized Einstein equations for anymodes of $l\geq 0$ in a gauge-invariant manner. We discuss the solutions to theodd-mode perturbation equations in the linearized Einstein equations and showthat these perturbations include the Kerr parameter perturbation in theseodd-mode perturbation, which is physically reasonable.

Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : -- Formulation and odd-mode perturbations