It is well known that if φ(t) ≡ t, then the system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is a basis in some Lebesgue space $L_p$. The aim of this short note is to show that the answer to this question is negative.

If $E=e_i$ and $F=f_i$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i,j}$ with norm $\left|\right|a_{i,j}{\left|\right|}_{E\left(F\right)}=\left|\right|{\sum }_k\left|\right|{\sum }_l{a}_{k,l}{f}_l\left|\right|e_k\left|\right|$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_p$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_p$ (1 ≤ p < +∞).

Let $Q={\left(q_n\right)}_{n=1}^{\infty }$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

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 ${\ell }_p$.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

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,...

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

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 $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between...

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

Let ${\sigma }_A\left(n\right)=\left|(a,a^{\text{'}})\in A²:a+a^{\text{'}}=n\right|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, ${\sigma }_A\left(n\right)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which ${\sigma }_A\left(n\right)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and ${\sigma }_A\left(n̅\right)\le 768$ for all n̅ ∈ Zₘ.

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................

The aim of this paper is to show that for every Banach space $(X,\parallel ·\parallel )$ containing asymptotically isometric copy of the space $c_0$ there is a bounded, closed and convex set $C\subset X$ with the Chebyshev radius $r\left(C\right)=1$ such that for every $k\ge 1$ there exists a $k$-contractive mapping $T:C\to C$ with $\parallel x-Tx\parallel >1-1/k$ for any $x\in C$.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

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 $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.