**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.

2 years ago

**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.

2 years ago

**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.

——-

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}.

2 years ago

**Authors:** E. Andruchow, E. Chiumiento, M. E. Di Iorio y Lucero

Let be a positive injective operator in a Hilbert space (\h, <,>), and
denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace
is called A-compatible if there exists a closed complement for
, which is orthogonal to with respect to the inner product [,].
Equivalently, if there exists a necessarily unique idempotent operator
such that , which is symmetric for this inner product. The
compatible Grassmannian is the set of all A-compatible subspaces of
. By parametrizing it via the one to one correspondence , this set is shown to be a differentiable submanifold of the Banach space
of all operators in 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 which preserve the form [,]. Each connected
component in of a compatible subspace of finite dimension, turns
out to be a symplectic leaf in a Banach Lie-Poisson space. For , in the presence of a fixed [,]-orthogonal decomposition of ,
, 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.

2 years ago

**Authors:** I. Beltita, M. Mantoiu

To a continuous action of a vector group on a -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 -algebra of
compact operators in a separable Hilbert space and we indicate some
applications.

2 years ago

**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}.

2 years ago

**Authors:** Jean-Christophe Bourin, Eun-Young Lee, Minghua Lin

Let be a positive semi-definite matrix partitioned in
Hermitian blocks, , . 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.

2 years ago

**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).

2 years ago

**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).

2 years ago