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.
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.
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}.
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.
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.
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}.
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.
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).
Authors: Anatolij Plichko
Let be a probability measure on a separable Banach space
. A subset
is
-continuous if
. In the paper the
-continuity and uniform
-continuity of convex bodies in
,
especially of balls and half-spaces, is considered. The
-continuity is
interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer
to a question of F. Tops{\o}e is given.
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).
