Some results about the Schroeder-Bernstein Property for separable Banach spaces

On Bernstein classes of quasianalytic maps

We study the structure of families of well approximable elements of tensor products of Banach spaces including analogs of the classical quasianalytic classes in the sense of Bernstein and Beurling. As in the case of quasianalytic functions, we prove for members of these families variants of the Mazurkiewicz and Markushevich theorems and in some particular cases, if such elements are Banach-valued continuous maps on a compact metric space, estimate massivity of their graphs and level sets.

Lifshitz-Krein trace formula for Hirsch functiuonal calculus on Banach spaces

We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kre\uın type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.

Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus

We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove a generalization of the important formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.

Sampling formulas for one-parameter groups of operators in Banach spaces

We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_σ(D),\>\>σ>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_σ(D)$ to known sampling results for regular functions of exponential type $σ$.

Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors

Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on connection between the degree of smoothness of vector $x\in X$ with respect to operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for operator A, and the k-module of continuity are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtai…

Bernstein's Lethargy Theorem in Frechet Spaces

In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Fréchet spaces. Let $X$ be an infinite-dimensional Fréchet space and let $\mathcal{V}=\{V_n\}$ be a nested sequence of subspaces of $ X$ such that $ \bar{V_n} \subseteq V_{n+1}$ for any $ n \in \mathbb{N}$ and $ X=\bar{\bigcup_{n=1}^{\infty}V_n}.$ Let $ e_n$ be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on $\sup\{\{dist}(x, V_n)\}$, we prove that there exists $ x \in X$ and $ n_o \in \mathbb{N}$ such that $$ \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n $$ for any $ …

Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance $ρ(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \leq ρ(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weak…

Tail spaces estimates on Hamming cube and Bernstein--Markov inequality

This note contains some estimates for tail spaces on Hamming cube. We use analytic paraproduct operator for that purpose. We also show several types of Bernstein--Markov inequalities for Banach space valued functions on Hamming cube. Here the novelty is in getting rid of some irritating logarithms and in proving Bernstein--Markov inequalities for $|\nabla f|_X$ rather than for $Δ^{1/2} f$ for $X$ valued polynomials on Hamming cube.

On exposed functions in Bernstein spaces of functions of exponential type

For $σ>0$, the Bernstein space \ $B^1_σ$ consists of those $L^1(R)$\ functions whose Fourier transforms are supported by $[-σ,σ]$. Since $B^1_σ$ is separable and dual to some Banach space, the closed unit ball $D(B^1_σ)$ of $B^1_σ$\ has sufficiently large sets of both exposed and strongly exposed points. Moreover, $D(B^1_σ)$ coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples and obtain as corollaries the relations between the sets of exposed, strongly exposed, weak$^{\ast}$ exposed, and weak$^{\ast}…

多维从属半群的解析性准则

Criteria for Analyticity of Multidimensionally Subordinate Semigroups

该研究将一维Bernstein函数相关的半群解析性结果推广到多维情形，给出了多维从属半群满足对应解析性质的充分条件，拓展了算子半群领域的相关理论。

Some $s$-numbers of an integral operator of Hardy type in Banach function spaces

Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\,\,\,x\in(a,b)\,\,(-\infty<a<b<+\infty) $$ and mapping a Banach function space $E$ to itself. We investigate some geometrical properties of $E$ for which $$ C_{1}\int_{a}^{b}u(x)v(x)dx \leq\limsup\limits_{n\rightarrow\infty}ns_{n}(T) \leq \limsup\limits_{n\rightarrow\infty}ns_{n}(T)\leq C_{2}\int_{a}^{b}u(x)v(x)dx $$ under appropriate conditions on $u$ and $v.$ The constants $C_{1},C_{2}>0$ depend only on the s…

s-数之间的不等式

Inequalities between s-numbers

本文将Hilbert空间算子的奇异数推广为Banach空间的s-数，并给出最小与最大s-数之间界限的初等证明。

On subordination of holomorphic semigroups

We prove that for any Bernstein function $ψ$ the operator $-ψ(A)$ generates a holomorphic $C_0$-semigroup $(e^{-tψ(A)})_{t \ge 0}$ on a Banach space, whenever $-A$ does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that $(e^{-tψ(A)})_{t \ge 0}$ is holomorphic in the holomorphy sector of $(e^{-tA})_{t \ge 0},$ and if $(e^{-tA})_{t \ge 0}$ is sectorially bounded in this sector then $(e^{-tψ(A)})_{t \ge 0}$ has the same property. We also obtain new sufficient conditions on $ψ$ in order that…