We prove that, for any $p\in (1,\infty )$, the second-order Cesàro sequence space ${Ces}^2\left(p\right)$ has the $\left(\beta \right)$-property and the $k$-NUC property for $k\ge 2$. In addition, we show that ${Ces}^2\left(p\right)$ has the Kadec–Klee, rotundity, and uniform convexity properties. For any positive integer $k$, we also investigate the uniform Opial and $\left(L\right)$ properties of the sequence space. We also establish that ${Ces}^2\left(p\right)$ is reflexive and has the fixed-point property. Finally, we calculate the packing constant $\left(C\right)$ of the space.

We introduce the concept of the strong Gelfand–Phillips (GP) property in Banach lattices, and we characterize Banach lattices with the strong GP property. Next, by introducing the class of almost limited completely continuous operators from an arbitrary Banach lattice $E$ to another $F$, we give some properties of them related to some well-known classes of operators and, especially, to the strong GP property of the Banach lattice $E$.

We give a systematic account of the various pictures of $KK$-theory for real $C^{\ast }$-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of $KK$-theory, and we use CRT-structures to prove that a natural transformation $F\left(A\right)\to G\left(A\right)$ between homotopy equivalent, stable, half-exact functors defined on real $C^{\ast }$-algebras is an isomorphism, provided it is an isomorphism on the smaller class of $C^{\ast }$-algebras. Finally, we develop $E$-theory for real $C^{\ast }$-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.

Given an increasing sequence $(X_n{)}_{n\in \omega }$ of quasi-uniform spaces and paratopological groups, we study the topology of the direct limits $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ and $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ of the sequence $(X_n{)}_{n\in \omega }$ in the categories of quasi-uniform spaces and paratopological groups, respectively. First, we prove that the quasi-uniformity of the quasi-uniform direct limit $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ is generated by some special family of quasi-pseudometrics. Then we discuss some properties of the direct limits $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$. Finally, we give an explicit description of the topology of the direct limit $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ under certain conditions on the sequence of paratopological groups $(X_n{)}_{n\in \omega }$. Moreover, some questions about direct limits of $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ and $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ are posed.

Let $E_i$ be Banach spaces, and let $X_{E_i}$ be Banach spaces continuously contained in the spaces of $E_i$-valued sequences $\left(\hat{x}\right(j){)}_j\in E_i^{\mathbb{N}}$, for $i=1,2,3$. Given a bounded bilinear map $B:E_1\times E_2\to E_3$, we define $(X_{E_2},X_{E_3}{)}_B$, the space of $B$-multipliers between $X_{E_2}$ and $X_{E_3}$, to be the set of sequences $({\lambda }_j{)}_j\in E_1^{\mathbb{N}}$ such that $\left(B\right({\lambda }_j,\hat{x}\left(j\right)){)}_j\in X_{E_3}$ for all $\left(\hat{x}\right(j){)}_j\in X_{E_2}$, and we define the Hadamard projective tensor product $X_{E_1}{⊛}_B{X}_{E_2}$ as consisting of those elements in $E_3^{\mathbb{N}}$ that can be represented as ${\sum }_n{\sum }_{j}B\left({\hat{x}}_n\right(j),{\hat{y}}_n(j\left)\right)$, where $(x_n{)}_n\in X_{E_1}$, $(y_n{)}_n\in X_{E_2}$, and ${\sum }_n\Vert x_n{\Vert }_{X_{E_1}}\Vert y_n{\Vert }_{X_{E_2}}<\infty$.

We will analyze some properties of these two spaces, relate them, and compute the Hadamard tensor products and the spaces of vector-valued multipliers in several cases, getting applications in the particular case where $E=\mathcal{L}(E_1,E_2)$ and $B(T,x)=T\left(x\right)$.

The aim of this paper is to describe the operators between spaces of continuously differentiable functions whose adjoint preserves extreme points. It is important to mention that no condition regarding injectivity or surjectivity of the operators is assumed. Previously known results characterizing surjective isometries can be immediately derived from such descriptions.

We take advantage of the recent developments in the isomorphic classification of the infinite matrix spaces of mixed norms ${\ell }_q\left({\ell }_p\right)$ for the whole range of values $0<p,q\le \infty$ to give a unified approach to the classification of Besov spaces over Euclidean spaces. In particular, we show that different Besov spaces with generalized smoothness ${\mathring{\mathrm{B}}}_{p,q}^w\left({\mathbb{R}}^d\right)$ over the Euclidean space ${\mathbb{R}}^d$ are isomorphic if and only if the indices $p$ and $q$ match.

In this paper, we give the generalizations of Jessen’s inequality, Hölder’s inequality, the converse Jessen inequality, and the Jessen–Mercer inequality for superquadratic functions of several variables. We also give applications of the obtained inequalities to time-scale integrals.

Suppose that $A$ and $B$ are positive operators on an ordered Banach space with a normal and generating cone satisfying $0\le AB\le BA$. It is known that, in this case, we have

$$r\left(AB\right)\le r\left(A\right)r\left(B\right)\mathrm{and}r(A+B)\le r\left(A\right)+r\left(B\right).$$ In this article we consider less restrictive assumptions on $A$ and $B$ which give us the same inequalities as above. Moreover, we also prove these inequalities in a more general setting of ordered algebras, and we consider related results in the Banach algebra setting. We apply our results to the essential spectral radius $r_{\mathrm{ess}}$ of AM-compact operators and prove the equality $r_{\mathrm{ess}}(AB+BA)=2r_{\mathrm{ess}}\left(AB\right)$ under reasonable assumptions.

The boundary value problems for linear and nonlinear regular degenerate abstract differential equations are studied. The equations have the principal variable coefficients and a small parameter. The linear problem is considered on a parameter-dependent domain (i.e., on a moving domain). The maximal regularity properties of linear problems and the optimal regularity of the nonlinear problem are obtained. In application, the well-posedness of the Cauchy problem for degenerate parabolic equations and boundary value problems for degenerate anisotropic differential equations are established.

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

The study of the generalization of the Bernstein and Bernstein–Durrmeyer polynomials attached to a continuous function $f:[0,1]\to \mathbb{R}$ and based on the Polya distribution were considered in some recent publications. The aim of the present note is to study the approximation properties of the complex version of these Durrmeyer-type operators, attached to analytic functions in a disk $D_R=\{z\in \mathbb{C};|z|<R\}$ with $R>1$.