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.

We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L¹+L^{\infty }.$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L¹+L^{\infty }$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L¹+L^{\infty }$ is empty.

We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N₀,N₁)}_{\theta ,q}$ and ${(X₀,X₁)}_{\theta ,q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_i$ is the normed space N with the norm inherited from $X_i$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{\theta }=S-2^{\theta }I$ (S...

A complete description of the real interpolation space $L={(L_{p₀}\left(\omega ₀\right),...,L_{pₙ}\left(\omega ₙ\right))}_{\theta ⃗,q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces ${\Omega }_i$ (i ∈ I) such that L is an $l_q$ sum of the restrictions of L to ${\Omega }_i$, and L on each ${\Omega }_i$ is a result of interpolation of just two weighted $L_p$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X*,Y^{\perp })$ has the λ-bounded approximation property. Then there exists a net $\left(S_{\alpha }\right)$ of finite-rank operators on X such that $S_{\alpha }\left(Y\right)\subset Y$ and $\left|\right|S_{\alpha }\left|\right|\le \lambda$ for all α, and $\left(S_{\alpha }\right)$ and $\left(S{*}_{\alpha }\right)$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

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...

If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;L^p\left(C\right),L^{\infty }\left(C\right))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;L^p\left(C\right)\cap QC,L^{\infty }\left(C\right)\cap QC)$. We apply this result to identify the interpolation space ${(L^{p₀,q₀}\left(C\right)\cap QC,L^{p₁,q₁}\left(C\right)\cap QC)}_{\theta ,q}$.

This paper deals with Besov spaces of logarithmic smoothness $B_{p,r}^{0,b}$ formed by periodic functions. We study embeddings of $B_{p,r}^{0,b}$ into Lorentz-Zygmund spaces $L_{p,q}{\left(logL\right)}_{\beta }$. Our techniques rely on the approximation structure of $B_{p,r}^{0,b}$, Nikol’skiĭ type inequalities, extrapolation properties of $L_{p,q}{\left(logL\right)}_{\beta }$ and interpolation.

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...

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.

Let d be a positive integer and α a real algebraic number of degree d + 1. Set $\alpha ̲:=(\alpha ,\alpha ²,...,{\alpha }^d)$. It is well-known that $c\left(\alpha ̲\right):=limin{f}_{q\to \infty }{q}^{1/d}·\left|\right|q\alpha ̲\left|\right|>0$, where ||·|| denotes the distance to the nearest integer. Furthermore, $c\left(\alpha ̲\right)n^{-1/d}\le c\left(n\alpha ̲\right)\le nc\left(\alpha ̲\right)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c\left(n\alpha ̲\right)\le C{n}^{-1/d}$ for any integer n ≥ 1.

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.

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

Let $\theta \in (0,1)$, $\lambda \in [0,1)$ and $p,p_0,p_1\in (1,\infty ]$ be such that $(1-\theta )/p_0+\theta /p_1=1/p$, and let $\varphi ,{\varphi }_0,{\varphi }_1$ be some admissible functions such that $\varphi ,{\varphi }_0^{p/p_0}$ and ${\varphi }_1^{p/p_1}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $\langle L_{{\varphi }_0}^{p_0),\lambda }\left(𝒳\right),L_{{\varphi }_1}^{p_1),\lambda }\left(𝒳\right),\theta \rangle$ of two generalized grand Morrey spaces on a quasi-metric measure space $𝒳$ is the generalized grand Morrey space $L_{\varphi }^{p),\lambda }\left(𝒳\right)$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.

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,...

The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $Ce{s}_p\left(I\right)$ is an interpolation space between $Ce{s}_{p₀}\left(I\right)$ and $Ce{s}_{p₁}\left(I\right)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $Ce{s}_p[0,1]$ is not an interpolation space between Ces₁[0,1] and $Ce{s}_{\infty }[0,1]$.

A computationally simple method for generating reduced-order models that minimise the $L_2$ norm of the approximation error while preserving a number of second-order information indices as well as the steady-state value of the step response, is presented. The method exploits the energy-conservation property peculiar to the Routh reduction method and the interpolation property of the $L_2$-optimal approximation. Two examples taken from the relevant literature show that the suggested techniques...

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$, $1\le p<\infty$.

We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$, and those defined by the dual property, the sequentially Right${}^{*}$ Banach spaces of order $p$ for $1\le p\le \infty$. These classes of Banach spaces are characterized by the notions of $L_p$-limited sets in the corresponding dual space and $R_p^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T:X\to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$-algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C\left(K\right)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Let X be an analytic set defined by polynomials whose coefficients $a₁,...,a_s$ are holomorphic functions. We formulate conditions on sequences $a_{1,\nu },...,a_{s,\nu }$ of holomorphic functions converging locally uniformly to $a₁,...,a_s$, respectively, such that the sequence $X_{\nu }$ of sets obtained by replacing $a_j$’s by $a_{j,\nu }$’s in the polynomials converges to X.

Let $(A_0,A_1)$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{\beta }$, we show that $A^{\theta }=A_{\theta }$ for every $\theta \in (0,1)$. We also deal with assumptions on ${\overline{A}}^{\beta }$, the closure of $A^{\beta }$ in the dual of ${(A_0^{*},A_1^{*})}_{\beta }$.

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...