We define embedding of an $n$-dimensional normed space into $L_{-p},\ 0<p<n$ by extending analytically with respect to $p$ the corresponding property of the classical $L_p$-spaces. The well-known connection between embeddings into $L_p$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in $L_{-p}$ if and only if $\|x\|^{-p}$ is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg's problems by proving that the spaces $\ell_q^n,\ 2<q\le \infty$ embed in $L_{-p}$ if and only if $p\in [n-3,n).$ We show that the technique of embedding in $L_{-p}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $p\in [n-3,n),$ $\Bbb E(\max_{i=1,...,n} |X_i|^{-p}) \ge \Bbb E(\max_{i=1,...,n} |Y_i|^{-p}),$ where $X_1,...,X_n$ and $Y_1,...,Y_n$ are jointly $q$-stable symmetric random variables, $0<q\le 2,$ so that, for some $k\in \Bbb N,\ 1\le k <n,$ the vectors $(X_1,...,X_k)$ and $(X_{k+1},...,X_n)$ have the same distributions as $(Y_1,...,Y_k)$ and $(Y_{k+1},...,Y_n),$ respectively, but $Y_i$ and $Y_j$ are independent for every choice of $1\le i\le k,\ k+1\le j\le n.$
A definition of frames in Krein spaces is proposed which extends the concept of $J$-frames defined by J.I. Giribet et al., J. Math. Anal. Appl. ${\textbf{393}}$ (2012), 122-137. The principal difference consists in the fact that a $J$-frame is related to maximal definite subspaces $\mathcal{M}_\pm$ which are not assumed to be uniformly definite. The latter allows one to extend the collection of $J$-frames. In particular, some complete $J$-orthogonal sequences and $J$-orthogonal Schauder bases can be interpreted as $J$-frames.
Extensions of dual definite subspaces to dual maximal definite ones are described. The concepts of dual quasi maximal subspaces and quasi basis are introduced and studied. The obtained results are applied to the classification of C-symmetries.
We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.
We introduce bi-parametric fractional integrals of the Erdelyi-Kober type that generalize known Garding-Gindikin constructions associated to the cone of positive definite matrices. It is proved that the Radon transform, which maps a zonal function on the Grassmann manifold $G_{n,m}$ of $m$-dimensional linear subspaces of $R^n$ into a function on the similar manifold $G_{n,k}$, $ 1\leq m<k \leq n-1$, is represented as analytic continuation of the corresponding Erdelyi-Kober integral. This result shows that different Grinberg-Rubin's formulas for such transforms [GR] have, in fact, a common stru…
To describe a set of functions, which forms a reflexive subspace B of the classical Banach space L a special function that characterizes their average integral growth is introduced. It is shown that this function essentially depends on the geometry of B. By the way, one question of la Vallee Poussin is answered. Also a short proof of the known result about the existence of an uncomplemented subspace isomorphic to the Hilbert space in every Lebesgue - Riesz space Lp (1<p<2) is obtained.
For a locally compact group $G$, the first-named author considered the closed subspace $a_0(G)$ which is generated by the pure positive definite functions. In many cases $a_0(G)$ is itself an algebra. We illustrate using Heisenburg groups and the $2\times 2$ real special linear group, that this is not the case in general. We examine the structures of the algebras thereby created and examine properties related to amenability.
For a $\mathcal{C}^2$-smooth function on a finite-dimensional space, a necessary condition for its quasiconvexity is the positive semidefiniteness of its Hessian matrix on the subspace orthogonal to its gradient, whereas a sufficient condition for its strict pseudoconvexity is the positive definiteness of its Hessian matrix on the subspace orthogonal to its gradient. Our aim in this paper is to extend those conditions for $\mathcal{C}^{1,1}$-smooth functions by using the Fréchet and Mordukhovich second-order subdifferentials.
We show that the class of subspaces of c_0 is stable under Lipschitz isomorphisms. The main corollary is that any Banach space which is Lipschitz isomorphic to c_0 is linearly isomorphic to c_0.
Let $(Ω,Σ,μ)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(μ)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have $$ \inf_{|λ|=1} \|f-λg\|\leq C\||f|-|g|\|. $$ In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $λ$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this article, we construct various subspaces doing stable phase retrieva…
In this paper, we study random sampling on reproducing kernel space $V$, which is a range of an idempotent integral operator. Under certain decay condition on the integral kernel, we show that any element in $V$ can be approximated by an element in a finite-dimensional subspace of $V$. Moreover, we prove with overwhelming probability that random points uniformly distributed over a cube $C$ is stable sample for the set of functions concentrated on $C$
Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm…
The diametral dimension, $Δ(E)$, and the approximate diametral dimension, $δ(E)$ of an element $E$ of a large class of nuclear Fréchet spaces are set theoretically between the corresponding invariant of power series spaces $Λ_{1}(\varepsilon)$ and $Λ_{\infty}(\varepsilon)$ for some exponent sequence $\varepsilon$. Aytuna et al., \cite{AKT2}, proved that $E$ contains a complemented subspace which is isomorphic to $Λ_{\infty}(\varepsilon)$ provided $Δ(E)=Δ( Λ_{\infty}(\varepsilon))$ and $\varepsilon$ is stable. In this article, we will consider the other extreme case and we proved that in this l…
We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in R^n$ the random variables $\sum a_iX_i$ and $γ(a) Y$ are identically distributed, where $γ:R^n\to [0,\infty)$ is called the standard of $X.$ An old problem is to characterize those functions $γ$ that can appear as the standard of an $n$-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in $L_0.$ This result is almost optimal, as the norm of any finite dimensional subspace of $L_p$ with $p…
Phase retrieval is known to always be unstable when using a frame or continuous frame for an infinite dimensional Hilbert space. We consider a generalization of phase retrieval to the setting of subspaces of $L_2$ which coincides with using a continuous frame for phase retrieval when the subspace is the range of the analysis operator of a continuous frame. We then prove that there do exist infinite dimensional subspaces of $L_2$ where phase retrieval is stable. That is, we give a method for constructing an infinite dimensional subspace $Y\subseteq L_2$ such that there exists $C\geq 1$ so that …
In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions ψon the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) of bilinear forms on a real vector space V. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of ψto the strip { z \in C\: 0 \leq Im z \leq b} with a coupling condition ψ(ib + t) = \oline{ψ(t)} on the boundary. Our first main result consists of a characterization of thes…
A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a strongly stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix …
We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace). We give the complete description of this class for infinite symmetric group ${\frak S}_{\Bbb N}$. It happened the family of the stable representations of ${\frak S}_{\Bbb N}$ coincides with the set of representations of the components (left or right) of the admissible representations of the double-group in the sense of Ol'shanski. In particular all these are …
