Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

Series expansion of the Gamma function and its reciprocal

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

On linear functional equations modulo $${\mathbb {Z}}$$

In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\rightarrow {\mathbb {R}}$$ f i : V → R , $$g_i:V\rightarrow {\mathbb {Z}}$$ g i : V → Z such that $$\varphi _i=f_i+g_i$$ φ i = f i + g i , $$(i=0,\dots ,n+1)$$ ( i = 0 , ⋯ , n + 1 ) and $$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ ∑ i = 0 n + 1 f i ( r i x + q i y ) = 0 for all $$x, y\in V$$ x , y ∈ V .

Linear functional equations and their solutions in generalized Orlicz spaces

Assume that $$\Omega \subset \mathbb {R}^k$$ Ω ⊂ R k is an open set, V is a real separable Banach space and $$f_1,\ldots ,f_N :\Omega \rightarrow \Omega $$ f 1 , … , f N : Ω → Ω , $$g_1,\ldots , g_N:\Omega \rightarrow \mathbb {R}$$ g 1 , … , g N : Ω → R , $$h_0:\Omega \rightarrow V$$ h 0 : Ω → V are given functions. We are interested in the existence and uniqueness of solutions $$\varphi :\Omega \rightarrow V$$ φ : Ω → V of the linear equation $$\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$ φ = ∑ k = 1 N g k · ( φ ∘ f k ) + h 0 in generalized Orlicz spaces.

Linear Li–Yorke Chaos in a Finite-Dimensional Space with Weak Topology

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted.

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x , u , D u + a 0 x , u = f x , u , D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N , a 0 : Ω × ℝ ⟶ ℝ , and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering “≤,” which are called barrier solutions.