Abstract In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, show that if there exists relatively compact set K (in weak operator topology) such $0$ is an element its closure but it not norm-closed convex hull, then can guarantee does attain norm. This ...

For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function $\rightsquigarrow$ denotes appropriate type weak convergence. Here $Z$ a local $L^1$-type space, either space $\...

Let $${{\mathcal {F}}}_n$$ be the Farey series of order $$n\ge 1$$ . We obtain sharp effective estimates sums $$\begin{aligned} F_n(f)= \sum _{\frac{{\kappa }}{{\lambda }}\in {{\mathcal {F}}}_n} f\Big (\frac{{\kappa }}\Big ),{\qquad }F_{n,{\sigma }}(f)= \frac{1}{{\kappa }^{\sigma }{\lambda }}f\Big }0< {\sigma }\le 1, \end{aligned}$$ for 1-periodic functions f satisfying weak local regularity as...

Abstract Let $k\geq 2$ be an integer. We prove that the 2-automatic sequence of odious numbers $\mathcal {O}$ is a k -additive uniqueness set for multiplicative functions: if function f satisfies multivariate Cauchy’s functional equation $f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$ arbitrary $x_1,\ldots ,x_k\in \mathcal , then identity $f(n)=n$ all $n\in \mathbb {N}$ .

Let G be a graph embedded in surface and let the set of all faces G. For given subset $$\mathcal F$$ even (faces bounded by an cycle), resonance with respect to $${\mathcal {F}}$$ , denoted $$R(G;\mathcal F)$$ is such that its vertex perfect matchings two vertices $$M_1$$ $$M_2$$ are adjacent if only symmetric difference $$M_1\oplus M_2$$ cycle bounding some face . It has been shown plane eleme...