Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and ${\ell }^{\infty }$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then ${\ell }^{\infty }$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X{\otimes }_{\gamma }Y*$. Applications to embeddings of c₀ in various spaces of operators are given.

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:{[0,1]}^m\to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:{[0,1]}^m\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^m\to ℝ$ such that the sets of...

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

We give a full solution of the following problems concerning the spaces $M_{\phi }\left(X⃗\right)$: (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{\phi }\left(X⃗\right)\ne M_{\psi }\left(X⃗\right)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{\phi }\left(X⃗\right)⊊M_{\psi }\left(X⃗\right)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{\phi }\left(X⃗\right)$-space for some (unknown beforehand) Banach couple X⃗.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

A separable Banach space X contains ${\ell }_1$ isomorphically if and only if X has a bounded fundamental total $w{c}_0*$-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w{c}_0*$-biorthogonal system.

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\overline{linf\left(\right[0,1\left]\right)}$ contains an order isomorphic copy of D(0,1), (2) $\overline{linf\left(Q\right)}$ contains an isomorphic copy of C([0,1]), (3) $\overline{linf\left(\right[0,1\left]\right)}/\overline{linf\left(Q\right)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that ${𝒳}^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then ${𝒳}^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

We construct, under Axiom ♢, a family ${\left(C\left(K_{\xi }\right)\right)}_{\xi <2^{\left(2^{\omega }\right)}}$ of indecomposable Banach spaces with few operators such that every operator from $C\left(K_{\xi }\right)$ into $C\left(K_{\eta }\right)$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{\omega }$ instead of $2^{\left(2^{\omega }\right)}$.

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ\left(E\right)=\left(E\right)+ℂi{d}_E$. Still, ℬ(ℓ²) is...

Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and $T:\overline{span}\left(f\left(X\right)\right)\to X$ be the Figiel operator with $T\circ f=I{d}_X$ and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when $\overline{span}\left(f\left(X\right)\right)$ is weakly nearly strictly convex.

The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g\left(R_F\right)\int f_x\left(R_F\right)d\mu \left(x\right)=h\left(R_F\right)$. (1) They involve functions of the kind $X\ni x\mapsto f_x\left(R_F\right)\in B\left(F\right)$, where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

In this note we survey the partial results needed to show the following general theorem: $l_p\left(l_q\right):1\le p,q\le +\infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d̂({\overline{co}}^{w*}\left(K\right),X)\le 2d̂(K,X)$ and, if K ∩ X is w*-dense in K, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (iii) if X has a 1-symmetric basis, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$.

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{\alpha }:\alpha \in \subset A$ there exist α ≠ β ∈ such that $a_{\alpha }\in a_{\beta }{A}^{}$ is isomorphic to ${⨁}_{i=1}^r(ℂ\left[X\right]/X^{d_i}ℂ\left[X\right])\oplus E$, where $d₁,...,d_r\in \mathbb{N}$, and E is either $Xℂ\left[X\right]/X^{d₀}ℂ\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${⨁}_{i=1}^rℂ\left[X\right]/X^{d_i}ℂ\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

Let ${\beta \left(n\right)}_{n=0}^{\infty }$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space ${\ell }^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }f̂\left(n\right)zⁿ$ such that ${\sum }_{n=0}^{\infty }{\left|f̂\left(n\right)\right|}^p{\left|\beta \left(n\right)\right|}^p<\infty$. We give some sufficient conditions for the multiplication operator, $M_z$, to be unicellular on the Banach space ${\ell }^p\left(\beta \right)$. This generalizes the main results obtained by Lu Fang [1].

The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $\left(\mathrm{L}\right)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\left(\mathrm{L}\right)$ sets. As a sequence characterization of such operators, we see that an operator $T:X\to E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $\left|T\right(x_n\left)\right|\to 0$ for $\sigma (E,E^{\text{'}})$ for every...

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p\left(G\right)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O{A}_p\left(G\right)$ of the classical Figà-Talamanca-Herz algebras $A_p\left(G\right)$. If p ∈ (1,∞) is arbitrary, then $A_p\left(G\right)\subset O{A}_p\left(G\right)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $O{A}_p\left(G\right)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O{A}_q\left(G\right)\subset O{A}_p\left(G\right)$ completely contractively...

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

We provide a complex version of a theorem due to Bednar and Lacey characterizing real $L_1$-preduals. Hence we prove a characterization of complex $L_1$-preduals via a complex barycentric mapping.

For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K\subset {\sum }_{n=1}^{\infty }\alpha ₙxₙ:\alpha ₙ\in Ball\left(l_{p^{\text{'}}}\right)$, where p’ = p/(p-1) and $xₙ\in l_p^s\left(X\right)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $xₙ\in l_p^w\left(X\right)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_p$ in a particular manner. The normed operator ideals $(K_p,{\kappa }_p)$ of p-compact operators and $(W_p,{\omega }_p)$ of weakly p-compact operators, arising from these factorizations,...

In this paper we study existence and uniqueness of solutions for the Hammerstein equation $$u\left(x\right)=v\left(x\right)+\lambda {\int }_{I_a^b}K(x,y)f(y,u\left(y\right))dy$$ in the space of function of bounded total $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, where $\lambda \in ℝ$, $K:I_a^b\times I_a^b⟶ℝ$ and $f:I_a^b\times ℝ⟶ℝ$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C\left(2^{}\times [0,\alpha ]\right)$ of all real continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological product of the Cantor cubes $2^{}$ with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently,...

Let A be a unital Banach function algebra with character space ${\Phi }_A$. For $x\in {\Phi }_A$, let $M_x$ and $J_x$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_x$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y\in {\Phi }_A:M_x\supseteq J_y$ has an infinite connected subset. Various related results are given.

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ${}_{\nu }\left(X\right)$ and ${}_{\nu }\left(X\right)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ${\omega }_{\omega }$ and with the third derived set $K^{\left(3\right)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c₀\left({\omega }_{\omega }\right)$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

There exists a universal control sequence ${p̅\left(m\right)}_{m=1}^{\infty }$ of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, $x={\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}*\left(x\right)x_{\pi \left(n\right)}$ where π(n) is a permutation of n which depends on x but is uniformly controlled by ${p̅\left(m\right)}_{m=1}^{\infty }$, that is, $n_{n=1}^m\subseteq \pi {\left(n\right)}_{n=1}^{p̅\left(m\right)}\subseteq n_{n=1}^{p̅(m+1)}$ for each m.

We prove that the quasi-Banach spaces ${\ell }_p\left(c₀\right)$ and ${\ell }_p\left(\ell ₂\right)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $\ell ₁\left(c₀\right)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X,|\left|·\right|{|}_X)$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${\left|\right|f\left(·\right)\left|\right|}_X$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_u(={f\in E\left(X\right):\left|\right|f\left(·\right)\left|\right|}_X\le u)$ stand for the order interval in E(X). For a real Banach space $(Y,|\left|·\right|{|}_Y)$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...