For a bounded linear operator T in a Banach space the Ritt resolvent condition (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, . This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.
It is shown that an operator with the properties mentioned in the title does exist in the space , 1 ≤ p ≤ ∞. The maximal sector for the extended resolvent condition can be prescribed a priori jointly with the corresponding order of the exponential growth of the resolvent in the complementary sector.
We characterize the Banach space operators T whose arithmetic means form a precompact set in the operator norm topology. This occurs if and only if the sequence is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.
We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...
For the Abel equation on a real-analytic manifold a dynamical criterion of solvability in real-analytic functions is proved.
The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.
A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Kolmogorov.
The polynomial functions on a projective space over a field = ℝ, ℂ or ℍ come from the corresponding sphere via the Hopf fibration. The main theorem states that every polynomial function ϕ(x) of degree d is a linear combination of “elementary” functions .
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