# Functional Analysis

## New submissions

[ total of 18 entries: 1-18 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Integral comparisons of nonnegative positive definite functions on locally compact abelian groups
Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)

In this paper, we discuss the following general questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying that $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized and what is the best value?
First we discuss the problem in locally compact Abelian groups and then apply the results to the case when $\mu, \nu$ are the traces of the usual Lebesgue measure over centered and arbitrary intervals, respectively. This special case was earlier investigated by Shapiro, Montgomery, Hal\'asz and Logan. It is a close companion of the more familiar problem of Wiener, as well.

[2]
Title: Pseudo-differential operators with nonlinear quantizing functions
Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Operator Algebras (math.OA)

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi,$$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and $\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $\tau$ and here we investigate the corresponding calculus in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.

[3]
Title: Generalization of a Real-Analysis Result to a Class of Topological Vector Spaces
Authors: Leonard T. Huang
Subjects: Functional Analysis (math.FA); Probability (math.PR)

In this paper, we generalize an elementary real-analysis result to a class of topological vector spaces. We also give an example of a topological vector space to which the result cannot be generalized.

[4]
Title: Generators of semigroups on Banach spaces inducing holomorphic semiflows
Comments: 14 pages, Accepted, Israel Journal of Mathematics 2018
Subjects: Functional Analysis (math.FA)

Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.

[5]
Title: New characterizations of operator monotone functions
Comments: Linear Algebra and Its Applications, 2018
Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)

If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one of these inequalities for all positive operators $A$ and $B$ and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone.
In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

[6]
Title: New Approach To Fixed Point Theorems
Subjects: Functional Analysis (math.FA)

In this article, we discuss a new version of metric fixed point theory especially of Banach Contraction Principle, Ran-Reurings Theorem and others.

[7]
Title: Fixed Point Theorems In Ordered Partial b-Metric Spaces With New Setting
Subjects: Functional Analysis (math.FA)

The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the existence of fixed point can be proved in incomplete metric spaces with non-contraction map on it. We have reported an example in support our result.

[8]
Title: Weighted composition operator on quaternionic Fock space
Subjects: Functional Analysis (math.FA)

In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such that this (right)-anti-linear weighted composition operator is a (right)-conjugation. Specially, we present equivalent conditions ensuring weighted composition operators which are conjugate $\mathcal{C}_{a,b,c}-$commuting or complex $\mathcal{C}_{a,b,c}-$ symmetric on $\mf$, which generalized the classical results on $\mathcal{F}^2(\mathbb{C}).$ At last part of the paper, we exhibit the closed expression for the kernel function of $\mf.$

[9]
Title: Lossless Analog Compression
Subjects: Functional Analysis (math.FA); Information Theory (cs.IT)

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where the description complexity K(x) is given by the infimum of the lower modified Minkowski dimensions over all support sets U of x. We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution---with respect to the s-dimensional Hausdorff measure---supported on countable unions of s-dimensional differentiable manifolds. Countable unions of differentiable manifolds include essentially all signal models used in compressed sensing theory, in spectrum-blind sampling, and in the matrix completion problem. Specifically, we prove that, for almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n>s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n<s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n greater than or equal to s being necessary for recovery with probability of error smaller than one. The central conceptual tool in the development of our theory is geometric measure theory.

[10]
Title: Row-finite systems of ordinary differential equations in a scale of Banach spaces
Subjects: Functional Analysis (math.FA)

We study an infinite system of first-order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite lifetime and prove that they live in a scale of increasing Banach spaces. For this, we approximate our system by finite systems and obtain uniform estimates of the corresponding solutions using the version of Ovsyannikov's method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.

### Cross-lists for Tue, 20 Mar 18

[11]  arXiv:1803.06803 (cross-list from math.CA) [pdf, ps, other]
Title: Hadamard powers of some positive matrices
Authors: Tanvi Jain
Journal-ref: Linear Algebra and its Applications, 528, (2017) 147-158
Subjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)

Positivity properties of the Hadamard powers of the matrix $\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers $x_1,\ldots,x_n$ and the matrix $\begin{bmatrix}|\cos((i-j)\pi/n)|\end{bmatrix}$ are studied. In particular, it is shown that $\begin{bmatrix}(1+x_ix_j)^r\end{bmatrix}$ is not positive semidefinite for any positive real number $r<n-2$ that is not an integer, and $\begin{bmatrix}|\cos((i-j)\pi/n)|^r\end{bmatrix}$ is positive semidefinite for every odd integer $n\ge 3$ and $n-3\le r<n-2.$

[12]  arXiv:1803.06970 (cross-list from math.DG) [pdf, ps, other]
Title: Higher symmetries of symplectic Dirac operator
Comments: Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Functional Analysis (math.FA); Representation Theory (math.RT); Symplectic Geometry (math.SG)

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

[13]  arXiv:1803.07005 (cross-list from math.AP) [pdf, ps, other]
Title: Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
Authors: Jonas M. Tölle
Subjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA); Probability (math.PR)

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with $C^{1}$-vector field coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^o-formula.

### Replacements for Tue, 20 Mar 18

[14]  arXiv:1704.01597 (replaced) [pdf, ps, other]
Title: Fourier Series of Gegenbauer-Sobolev Polynomials
Journal-ref: SIGMA 14 (2018), 024, 11 pages
Subjects: Functional Analysis (math.FA)
[15]  arXiv:1710.10823 (replaced) [pdf, ps, other]
Title: Some remarks on the notions of boundary systems and boundary triple(t)s
Subjects: Functional Analysis (math.FA)
[16]  arXiv:1803.05742 (replaced) [pdf, ps, other]
Title: On the piecewise pseudo almost periodic solution of nondensely impulsive integro-differential systems with infinite delay
Subjects: Functional Analysis (math.FA)
[17]  arXiv:1706.03844 (replaced) [pdf, ps, other]
Title: Interpolation and Fatou-Zygmund property for completely Sidon subsets of discrete groups (New title: Completely Sidon sets in discrete groups)
Authors: Gilles Pisier
Comments: This new version contains a significant addition, namely the operator space version of a result of Varopoulos, showing that if the closed span of a subset of G in C*(G) is completely isomorphic to $\ell_1$ (by an arbitrary isomorphism) or if the dual operator space is exact then the set is completely Sidon. v3: more polished, longer and more detailed version with a new title
Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
[18]  arXiv:1803.03606 (replaced) [pdf, ps, other]
Title: A Simple proof of Johnson-Lindenstrauss extension
Authors: Manor Mendel
Comments: 2 pages. Elimination of typos
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
[ total of 18 entries: 1-18 ]
[ showing up to 2000 entries per page: fewer | more ]

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