# functional analysis from the arXiv

curated by isomorphismes

### A note on the continuous self-maps of the ladder system space.

Authors: Claudia Correa, Daniel V. Tausk

We give a partial characterization of the continuous self-maps of the ladder system space K_S. Our results show that K_S is highly nonrigid. We also discuss reasonable notions of “few operators” for spaces C(K) with scattered K and we show that C(K_S) does not have few operators for such notions.

1 year ago

### Discrete-Time Path Distributions on Hilbert Space.

Authors: Mathieu Beau (STP-DIAS), T. C. Dorlas (STP-DIAS)

We construct a path distribution representing the kinetic part of the Feynman path integral at discrete times similar to that defined by Thomas [1], but on a Hilbert space of paths rather than a nuclear sequence space. We also consider different boundary conditions and show that the discrete-time Feynman path integral is well-defined for suitably smooth potentials.

1 year ago

### On a connection between a generalised modulus of smoothness of order~$r$ and the best approximation by algebraic polynomials.

Authors: Mikhail K. Potapov, Faton M. Berisha

In this paper an asymmetrical operator of generalised translation is introduced, the generalised modulus of smoothness is defined by its means and the direct and inverse theorems in approximation theory are proved for that modulus.

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V danno\v{i} rabote vvoditsya nesimmetrichny\v{i} operator obobshchennogo sdviga, s ego pomoshchyu opredelyaetsya obobshchenny\v{i} modul’ gladkosti i dlya nego dokazyvaetsya pryamaya i obratnaya teoremy teorii priblizheni\v{i}.

1 year ago

### The compatible Grassmannian.

Let $\large \dpi{100} \bg_white A$ be a positive injective operator in a Hilbert space (\h, &lt;,&gt;), and denote by [,] the inner product defined by A: [f,g]=&lt;Af,g&gt;. A closed subspace $\large \dpi{100} \bg_white \s \subset \h$ is called A-compatible if there exists a closed complement for $\large \dpi{100} \bg_white \s$, which is orthogonal to $\large \dpi{100} \bg_white \s$ with respect to the inner product [,]. Equivalently, if there exists a necessarily unique idempotent operator $\large \dpi{100} \bg_white Q_\s$ such that $\large \dpi{100} \bg_white R(Q_\s)=\s$, which is symmetric for this inner product. The compatible Grassmannian $\large \dpi{100} \bg_white Gr_A$ is the set of all A-compatible subspaces of $\large \dpi{100} \bg_white \h$. By parametrizing it via the one to one correspondence $\large \dpi{100} \bg_white \s\leftrightarrow Q_\s$, this set is shown to be a differentiable submanifold of the Banach space of all operators in $\large \dpi{100} \bg_white \h$ which are symmetric with respect to the form [,]. A Banach-Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in $\large \dpi{100} \bg_white \h$ which preserve the form [,]. Each connected component in $\large \dpi{100} \bg_white Gr_A$ of a compatible subspace $\large \dpi{100} \bg_white \s$ of finite dimension, turns out to be a symplectic leaf in a Banach Lie-Poisson space. For $\large \dpi{100} \bg_white 1\le p \le \infty$, in the presence of a fixed [,]-orthogonal decomposition of $\large \dpi{100} \bg_white \h$, $\large \dpi{100} \bg_white \h=\s_0\dot{+} \n_0$, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p-Schatten operators which are symmetric for the form [,]. It carries the locally transitive action of the Banach-Lie group of invertible operators which preserve [,], and are of the form G=1+K, with K in p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.

1 year ago

### Rieffel deformation and twisted crossed products.

Authors: I. Beltita, M. Mantoiu

To a continuous action of a vector group on a $\large \dpi{100} \bg_white C^*$-algebra, twisted by the imaginary exponential of a symplectic form, one associates a Rieffel deformed algebra as well as a twisted crossed product. We show that the second one is isomorphic to the tensor product of the first one with the $\large \dpi{100} \bg_white C^*$-algebra of compact operators in a separable Hilbert space and we indicate some applications.

1 year ago

### Differentiable mappings on products with different degrees of differentiability in the two factors.

Authors: Hamza Alzaareer, Alexander Schmeding

We develop differential calculus of C^{r,s}-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class C^{r,s}.

1 year ago

### Positive definite matrices with Hermitian blocks and their partial traces.

Let $\large \dpi{100} \bg_white H$ be a positive semi-definite matrix partitioned in $\large \dpi{100} \bg_white \beta\times \beta$ Hermitian blocks, $\large \dpi{100} \bg_white H=[A_{s,t}]$, $\large \dpi{100} \bg_white 1\le s,t,\le \beta$. Then, for all symmetric norms, |H| \le |\sum_{s=1}^{\beta} A_{s,s}|. The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.

1 year ago

### Shannon's sampling theorem in a distributional setting.

Authors: Amol Sasane

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-\pi,\pi) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(\pi (t -n)))/(\pi (t-n)) (t in R). In this article we prove a generalization of this result under the assumption that F is a compactly supported distribution with its support contained in (-\pi,\pi).

1 year ago

### On uniform continuity of convex bodies with respect to measures in Banach spaces.

Authors: Anatolij Plichko

Let $\large \dpi{100} \bg_white \mu$ be a probability measure on a separable Banach space $\large \dpi{100} \bg_white X$. A subset $\large \dpi{100} \bg_white U\subset X$ is $\large \dpi{100} \bg_white \mu$-continuous if $\large \dpi{100} \bg_white \mu(\partial U)=0$. In the paper the $\large \dpi{100} \bg_white \mu$-continuity and uniform $\large \dpi{100} \bg_white \mu$-continuity of convex bodies in $\large \dpi{100} \bg_white X$, especially of balls and half-spaces, is considered. The $\large \dpi{100} \bg_white \mu$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Tops{\o}e is given.

1 year ago

### Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients.

Authors: Omar Mellah (LMRS, LMPA), Paul Raynaud De Fitte (LMRS)

We show that, contrarily to what is claimed in some papers, the nontrivial solutions of some stochastic differential equations with almost periodic coefficients are never mean square almost periodic (but they can be almost periodic in distribution).

1 year ago