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The convergence of the sequence $x_{n+1}=A{x}_n+b$ to ${(l-A)}^{-1}b$ (where $x_n,b$ are vectors in Banach space, $A$ is a bounded linear operator with bounded ${(l-A)}^{-1}$) can be accelerated by constructing certain linear combinations of several ordinary successive approximations. A sufficient condition is that the spectrum of $A$ decompose into a finite set and a subset of a sufficiently small neighborhood of zero (e. g., $A$ is compact).

In this paper a natural generalization of the gradual power method, known in matrix algebra, is given. It is an iterative process whose result is a basis of an invariant subspace of a given bounded linear operator $A$, and a matrix operator induced on this subspace. This process is shown to be contractive in certain metric, and can be used e.g. for compact operators.