arXiv subjects
Publications and source records attributed to Beata Randrianantoanina.
We prove that if $X$ is a real Banach space, with $\dim X\geq 3$, which contains a subspace of codimension 1 which is 1-complemented in $X$ and whose group of isometries is almost transitive then $X$ is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.
Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $ε> 0$ there exists $x \in X$ with $x^2= χ_A, \int x d μ= 0$ and $\| Tx \| < ε$. In particular all compact operators are narrow. We prove that if $X$ is a Lorentz function space $L_{w,p}$ on [0,1] with $p>2$, then there exists a constant $k_X>1$ so that for every narrow projection $P$ on $L_{w,p}$ $\| Id - P \| \geq k_X. $ This generalizes earlier results on $L_p$ and partially answers a question of E. M. Semenov. Moreover we prove that every rearrangement-invariant function space $X$ with an absolutely continuous norm contains a complemented subspace isomorphic to $X$ which is the range of a narrow projection and a non-narrow projection, which gives a negative answer to a question of A.Plichko and M.Popov.
This is the survey of results about norm one projections and 1-complemented subspaces in Köthe function spaces and Banach sequence spaces. The historical development of the theory is presented from the 1930's to the newest ideas. Proofs of the main results are outlined. Open problems are also discussed. Every effort has been made to include as complete a bibliography as possible.
We show that injective isometries in Orlicz space $L_M$ have to preserve disjointness, provided that Orlicz function $M$ satisfies $Δ_2$-condition, has a continuous second derivative $M''$, satisfies another ``smoothness type'' condition and either $\lim_{t\to0} M''(t) = \infty$ or $M''(0) = 0$ and $M''(t)>0$ for all $t>0$. The fact that surjective isometries of any rearrangement-invariant function space have to preserve disjointness has been determined before. However dropping the assumption of surjectivity invalidates the general method. In this paper we use a differential technique.
We show that Rudin-Plotkin isometry extension theorem in $L_p$ implies that when $X$ and $Y$ are isometric subspaces of $L_p$ and $p$ is not an even integer, $1 \leq p < \infty$, then $X$ is complemented in $L_p$ if and only if $Y$ is; moreover the constants of complementation of $X$ and $Y$ are equal. We provide examples demonstrating that this fact fails when $p$ is an even integer larger than 2.
Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.
We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.
Let $X_0, X_1, ..., X_k$ with $k \in \IN\cup\{\infty\}$ be sequence spaces $($finite or infinite dimensional$)$ over $\IC$ or $\IR$ with absolute norms $N_i$ for $i = 0, ..., k$, $($i.e., with 1-unconditional bases$)$ such that $\dim X_0 = k$. Define an absolute norm on the cross product space $($also known as the $X_0$ 1-unconditional sum$)$ $X_1 \times ... \times X_k$ by $$ N(x_1, ..., x_k) = N_0(N_1(x_1), ..., N_k(x_k)) \quad \hbox{for all} \quad (x_1, ..., x_k) \in X_1 \times ... \times X_k. $$ We show that every sequence space with an absolute norm has an intrinsic cross product structure of this form. The result is used to prove a characterization of isometries of complex cross product spaces that covers all the existing results. We demonstrate by examples and the theory of finite reflection groups that it is impossible to extend the complex result to the real case. Nevertheless, some new isometry theorems are obtained for real cross product spaces.
Suppose that a real nonatomic function space on $[0,1]$ is equipped with two re\-arran\-ge\-ment-invariant norms $\|\cdot\|$ and $|||\cdot|||$. We study the question whether or not the fact that $(X,\|\cdot\|)$ is isometric to $(X,|||\cdot|||)$ implies that $\|f\|= |||f|||$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, resp. Lorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spaces where it fails. We provide a complete description of Orlicz functions $φ\neqψ$ with the property that $L_φ$ and $L_ψ$ are isometric.
Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize surjective isometries of $X(H).$ We prove that if $T$ is such an isometry then there exist Borel maps $a:\Om\to\bbK$ and $σ:\Om\lra\Om$ and a strongly measurable operator map $S$ of $\Om$ into $\calB(H)$ so that for almost all $\om$ $S(\om)$ is a surjective isometry of $H$ and for any $f\in X(H)$ $$Tf(\om)=a(\om)S(\om)(f(σ(\om))) \text{ a.e.}$$ As a consequence we obtain a new proof of characterization of surjective isometries in complex rearrangement-invariant function spaces.
We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.
We prove that there is no 1-complemented subspace of finite codimension in separable rearrangament-invariant function spaces.
We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(σ(s))$ for a Borel function $a$ and an invertible Borel map $σ:[0,1] \to [0,1].$ If $X$ is not equal to $L_p$, up to renorming, for some $1\le p\le \infty$ then in addition $|a|=1$ a.e. and $σ$ must be measure-preserving.