Related papers: A Schauder basis for $L_2$ consisting of non-negat…

We prove the following. For any complex valued $L^p$-function $b(x)$, $2 \leq p < \infty$ or $L^\infty$-function with the norm $\| b | L^{\infty}\| < 1$, the spectrum of a perturbed harmonic oscillator operator $L = -d^2/dx^2 + x^2 + b(x)$…

In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in…

Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…

Necessary and sufficient conditions for the equivalence of the Rademacher chaos to the canonical basis of l_2 and also for the complementability of the corresponding generated subspace are derived. In particular, we obtain the…

We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C>0$ with the following property: for any $A\in\mathcal{L}(X)$ and convex compact…

Let $(\mathbb{X},d,\mu)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$…

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{\alpha_k\}_{k=1}^\infty$ such that…

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative we define the generalized fractional Littlewood-Paley $g$-function for semigroups acting on $L^p$-spaces of functions with values in…

Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of…

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they posses a generalised version of Riesz--Fischer property, that embeddings between them are always…

We consider self-similar measures $\mu $ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2\pi nx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(\mu) $. Depending on the fractal dimension…

If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\overset\phi\longrightarrow E$ into a Banach space $E$…

Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $\chi_{\lambda}$ corresponding to a…

If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $\alpha $ the dual of $X$ contains a subspace whose…

We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every…

We introduce the so-called Bishop-Phelps-Bollob\'as property for positive functionals, a particular case of the Bishop-Phelps-Bollob\'as property for positive operators. First we show a version of the Bishop-Phelps-Bollob\'as theorem for…

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.$$ We show that $L_1(L_p)$ $(1 < p < \infty)$ is primary,…

We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the…

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise…

Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a…