We investigate almost ball remotal and ball remotal subspaces of Banach spaces. Several subspaces of the classical Banach spaces are identified having these properties. Some stability results for these properties are also proved.

In the normed space of bounded operators between a pair of normed spaces, the set of operators which are "bounded below" forms the interior of the set of one-one operators. This note is concerned with the extension of this observation to certain spaces of pairs of operators.

A linear functional F on a Banach algebra A is almost multiplicative if |F(ab) - F(a)F(b)| ≤ δ∥a∥ · ∥b∥ for a,b ∈ A, for a small constant δ. An algebra is called functionally stable or f-stable if any almost multiplicative functional is close to a multiplicative one. The question whether an algebra is f-stable can be interpreted as a question whether A lacks an almost corona, that is, a set of almost multiplicative functionals far from the set of multiplicative functionals. In this paper we discuss...

Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by ${\sigma }_{ϵ}\left(a\right):=\lambda \in ℂ:\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|\ge 1/ϵ$ with the convention that $\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|=\infty$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $\varphi \left(a\right)\in {\sigma }_{\delta }\left(a\right)$ for all a in A. (2) If ϕ is linear and $\varphi \left(a\right)\in {\sigma }_{ϵ}\left(a\right)$ for all a in A,...

The paper aims to study systems of linear ordinary differential equations in the context of an algebra of almost periodic generalized ultradistributions. Conditions on the existence of generalized solutions are given.

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set $X$ in a metric space, it is proved the existence of an almost periodic sequence ${\left\{{\psi }_k\right\}}_{k\in \mathbb{Z}}$ such that $\{{\psi }_k;k\in \mathbb{Z}\}=X$ and ${\psi }_k={\psi }_{k+lq\left(k\right)}$, $l\in \mathbb{Z}$ for all $k$ and some $q\left(k\right)\in \mathbb{N}$ which depends on $k$.

The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,\mu )$ by multiples of projections or unitary operators are examined.

We investigate Hyers-Ulam stability of non-surjective ε-isomeries of Banach spaces. We also pose and discuss an open problem.

Dunford-Pettis type properties are studied in individual Banach spaces as well as in spaces of operators. Bibasic sequences are used to characterize Banach spaces which fail to have the Dunford-Pettis property. The question of whether a space of operators has a Dunford-Pettis property when the dual of the domain and the codomain have the respective property is studied. The notion of an almost weakly compact operator plays a consistent and important role in this study.