Yuri A. Abramovich, Mikhail Zaidenberg
The following theorem is the main result of this note. Theorem 1. Let $(E, \|\cdot\|_E) $ be a rearrangement invariant Banach function space on the interval $[0, 1]$. If $E$ is isometric to $Ł_p [0, 1]$ for some $1\le p<\infty$, then $E$ coincides with $Ł_p [0, 1]$ and furthermore $\|\cdot\|_E = λ\|\cdot\|_{Ł_p}$, where $λ= \|{\bf 1}\|_E$.
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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 spac…
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.
Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of $σ$-algebras $({\cal F}_n)$ such that the conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and such that the sequence (g_n) is conditionally independent. Then it is known that $\normo{\sum f_k}_p \le C \normo{\sum g_k}_p$, $1 \le p \le \infty$ where the constant C is independent of p. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant…
The space Weak L^1 consists of all measurable functions on [0,1] such that q(f) = sup_{c>0} c λ{t : |f(t)| > c} is finite, where λdenotes Lebesgue measure. Let ρbe the gauge functional of the unit ball {f : q(f) \leq 1} of the quasi- norm q, and let N be the null space of ρ. The normed envelope of Weak L^1, which we denote by W, is the space (Weak L^1/N, ρ). The Banach envelope of Weak L^1, \overline{W}, is the completion of W. We show that \overline{W} is isometrically lattice isomorphic to a sublattice of W. It is also shown that all rearrangement invariant Banach function spaces are isometr…
We investigate the structure of norm-preserving and linear but not necessarily surjective operators on variable-exponent, discrete Lebesgue spaces. A certain class of isometries, novel to this work, are especially considered; this class completely coincides with all isometries when the Lebesgue space is classical, i.e. of a fixed-exponent. For said isometries it is shown that their actions are completely determined by pairs consisting of set-mappings and bounded functions on $\mathbb{N}$. This result recovers the previously-known structure of isometries on fixed-exponent spaces as a special ca…
Passive discrete-time systems with Hilbert spaces as an incoming and outgoing space and a Pontryagin space as a state space are investigated. A geometric characterization when the index of the transfer function coincides with the negative index of the state space is given. In this case, an isometric (co-isometric) system has a product representation corresponding to the left (right) Krein-Langer factorization of the transfer function. A new criterion, based on the inclusion of reproducing kernel spaces, when a product of two isometric (co-isometric) systems preserves controllability (observabi…
We define the notion of isometric envelope of a subspace in a Banach space, and relate it to a) the mean ergodic projection on the space of fixed points of a semigroup of contractions, b) results on Korovkin sets from the 70's, and c) extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $L_p$, $p \notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé B…
On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces
本文研究有限秩 de Branges–Rovnyak 空间中 Dirichlet 型函数空间上的移位算子，证明有理符号与有限原子测度及有限秩亏格算子的等价性，并刻画了多参数情形下扩张移位算子的允许元组条件。
Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, …
Optimal linear approximation and isometric extensions
本文证明仅用n维等距延拓即可达到绝对线性宽度，引入预绝对宽度概念，给出最优延拓并建立光滑函数集的阶估计，发现超小和超高光滑情形下预绝对与绝对线性宽度同阶，中间情形不同。
Let $M$ and $N$ be two unital JB$^*$-algebras and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the sets of all unitaries in $M$ and $N$, respectively. We prove that the following statements are equivalent: $(a)$ $M$ and $N$ are isometrically isomorphic as (complex) Banach spaces; $(b)$ $M$ and $N$ are isometrically isomorphic as real Banach spaces; $(c)$ There exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N).$ We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N…
In this article, using ideas of Liero, Mielke and Savaré in [21], we establish a Kantorovich duality for generalized Wasserstein distances $W_1^{a,b}$ on a generalized Polish metric space, introduced by Picolli and Rossi. As a consequence, we give another proof that $W_1^{a,b}$ coincide with flat metrics which is a main result of [25], and therefore we get a result of independent interest that $\left(\mathcal{M}(X), W^{a,b}_1\right)$ is a geodesic space for every Polish metric space $X$. We also prove that $(\mathcal{M}^G(X),W_p^{a,b})$ is isometric isomorphism to $(\mathcal{M}(X/G),W_p^{a,b})…
Let $ n\geq 2, A=(a_{ij})_{i,j=1}^{n}$ be a real symmetric matrix, $a=(a_i)_{i=1}^{n}\in \Bbb R^n.$ Consider the differential operator $D_A = \sum_{i,j=1}^n a_{ij}{\partial^2 \over \partial x_i \partial x_j}+ \sum_{i=1}^n a_i{\partial \over \partial x_i}.$ Let $E$ be a bounded domain in $\Bbb R^n,$ $p>0.$ Denote by $L_{D_A}^p(E)$ the space of solutions of the equation $D_A f=0$ in the domain $E$ provided with the $L_p$-norm. We prove that, for matrices $A,B,$ vectors $a,b,$ bounded domains $E,F,$ and every $p>0$ which is not an even integer, the space $L_{D_A}^p(E)$ is isometric to a subspace …
On the geometry of $G$-norm
本文研究$G$-范数在算子空间中的几何性质，探讨其数值指标，并刻画相对尖锐算子，证明在有限维希尔伯特空间中，此类算子为部分等距算子。
Characterization of surjective isometries: the real case
该研究刻画了两类空间上的满实线性等距映射形式，确定对应因子的唯一性，还推导了Mityagin问题的交换与非交换等距形式。
Let $X$ be an operator space, let $ϕ$ be a product on $X$, and let $(X,ϕ)$ denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping $ϕ$ for the algebra $(X,ϕ)$ to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher-Ruan-Sinclair that solved the case when the …
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