On almost-analytic functions, tensors and invariant subspaces

Introduction.In previous papers [ 2 ], [6 I, 1 * we introduced the notion of Φ-tensors as a generalization of analytic tensors.The main purpose of this paper is to introduce the notion of almost-analytic function and almost-analytic submanifold and to define an operator acting on tensor fields defined on an invariant subspace.In § 1 and § 2 we define almost-analytic functions and almost-analytic submanifolds in an almost-complex space.A new Φ-operator on an invariant subspace of the manifold admitting a tensor fields of type (1, 1) is defined in § 4, after preliminary facts are given in § 3. Some formulas about the Φ-operator are given in § 5 and we devote § 6 to Kahlerian spaces.The results of all sections except § 6 are independent from connection or metric. Almost-analytic functions.Consider an ^-dimensional differentiable manifold admitting a tensor field whose components with respect to local coordinates {x λ } are <p/ 2) .A tensor field 7V* } = T μg ... μi λp ' Λ| is called to be pure in \, μj or briefly to be p(X i9 μ 5 ) if it commutes with φ λ μ in λ i? μ Ί i. e. it holds that m <r T (λ) _ λ t rp λ p ...σ...λi ψβj 1 μ Q ...σ...μ 1 ~ ψ σ ± ( μ ) Aαalogously T (μ ( ; λ) is called to be φ(\ i9 λj) or p(μ i9 μ 3 ) if ψσ J-(μ) ψσ Ί (μ) , or ψβi * μ Q ,..σ...μ j ...μ ι ψμ } <L μ Q ...μ t .. .σ... μ,