The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty }\left(G\right)$ and $C^{\infty }\left(K\right)$ of infinitely differentiable functions where G is an arbitrary domain in ${ℝ}^p$, p≥1, while K is a compact set in ${ℝ}^p$ with non-void interior K̇ such that $\overline{K}̇=K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G\subseteq {ℂ}^p$ are also investigated.

Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator $C_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \left(tz\right)\right)g\left(tz\right)dt/t$, f ∈ H(B). The boundedness and compactness of $C_{\phi }^g$ from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied

We show that in the space C[-1,1] there exists an orthogonal algebraic polynomial basis with optimal growth of degrees of the polynomials.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for others give...

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ${\omega }_{\omega }$ and with the third derived set $K^{\left(3\right)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c₀\left({\omega }_{\omega }\right)$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

Define $h^{\infty }\left(E\right)$ as the subspace of $C^{\infty }(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-\infty }(E*)$ consisting of all harmonic E*-valued functions g such that ${(1-|x\left|\right)}^{m}f$ is bounded for some m>0. Then the dual $h^{\infty }(E*)$ is represented by $h^{-\infty }(E*)$ through $⟨f,g{⟩}_0=li{m}_{r\to 1}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in h^{-\infty }(E*),g\in h^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.