We determine the asymptotic behavior of the entropy numbers of diagonal operators D: lp → lq, (xk) → (skxk), 0 < p,q ≤ ∞, under mild regularity and decay conditions on the generating sequence (σk). Our results extend the known estimates for polynomial and logarithmic diagonals (σk). Moreover, we also consider some exotic intermediate examples like (σk)=exp(-√log k).

It is shown that for every k ∈ ℕ and every spreading sequence eₙₙ that generates a uniformly convex Banach space E, there exists a uniformly convex Banach space $X_{k+1}$ admitting eₙₙ as a k+1-iterated spreading model, but not as a k-iterated one.

It is proved that every operator from a weak*-closed subspace of ${\ell }_1$ into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to an operator from ${\ell }_1$ to C(K).

A class of closed, bounded, convex sets in the Banach space $c_0$ is shown to be a complete PCA set.

The exponential stability property of an evolutionary process is characterized in terms of the existence of some functionals on certain function spaces. Thus are generalized some well-known results obtained by Datko, Rolewicz, Littman and Van Neerven.

It is proved that if X is a rotund Banach space and M is a closed and proximinal subspace of X, then the quotient space X/M is also rotund. It is also shown that if Φ does not satisfy the δ₂-condition, then $h{⁰}_{\Phi }$ is not proximinal in $l{⁰}_{\Phi }$ and the quotient space $l{⁰}_{\Phi }/h{⁰}_{\Phi }$ is not rotund (even if $l{⁰}_{\Phi }$ is rotund). Weakly nearly uniform convexity and weakly uniform Kadec-Klee property are introduced and it is proved that a Banach space X is weakly nearly uniformly convex if and only if it is reflexive and it has the weakly...

Let L be a normal Banach sequence space such that every element in L is the limit of its sections and let E = ind En be a separated inductive limit of the locally convex spaces. Then ind L(En) is a topological subspace of L(E).

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ${\ell }^{q̅}\left(L_v^{p̅}\right)$ into ${\ell }^q\left(L_u^p\right)$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form ${\ell }^q\left(L_w^p\right)$, 1 < p,q < ∞ , q ≠ p, $w\in A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form $(X,L_{\infty }\left(w\right))$. As an application we extend Ovchinnikov’s interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented....

It is shown that $l_2^n$ imbeds isometrically into $l_4^{n^2+1}$ provided that n is a prime power plus one, in the complex case. This and similar imbeddings are constructed using elementary techniques from number theory, combinatorics and coding theory. The imbeddings are related to existence of certain cubature formulas in numerical analysis.

It is shown that the weak $L^p$ spaces ${\ell }^{p,\infty },L^{p,\infty }[0,1]$, and $L^{p,\infty }[0,\infty )$ are isomorphic as Banach spaces.

Estimation of the Jung constants of Orlicz sequence spaces equipped with either the Luxemburg norm or the Orlicz norm is given. The exact values of the Jung constants of a class of reflexive Orlicz sequence spaces are found by using new quantitative indices for 𝓝-functions.