Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2013
Volume
Vol. 33
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 33 (2013)
Projects
Pages
Articles
A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes
(2013) Alpay, Daniel; Kipnis, Alon
Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.
On the decomposition of families of quasinormal operators
(2013) Burdak, Zbigniew
The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decomposition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of commuting quasinormal operators is shown. Examples are given.
Existence results for Dirichlet problems with degenerated p-Laplacian
(2013) Cavalheiro, Albo Carlos
In this article, we prove the existence of entropy solutions for the Dirichlet problem $(P)\left\{ \begin{array}{ll} & -{\rm div}[{\omega}(x){\vert{\nabla}u\vert}^{p-2}{\nabla}u]= f(x) - {\rm div}(G(x)),\ \ {\rm in} \ \ {\Omega} \\ & u(x)=0, \ \ {\rm in} \ \ {\partial\Omega} \end{array} \right.$
where $\Omega$ is a bounded open set of $\mathbb{R}^N$ $(N \geq 2)$, $f \in L^1(\Omega)$ and $G/\omega \in [L^p(\Omega,\omega)]^N$.
Periodic solutions in multivariate invariance arguments
(2013) Chudziak, Jacek; Wójcik, Sebastian
Inspired by the recent results of A.E. Abbas we determine continuous multivariate utility functions invariant with respect to a wide family of transformations related to the shift transformations.
Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials
(2013) Eckhardt, Jonathan; Gesztesy, Fritz; Nichols, Roger; Teschl, Gerald
We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type $\begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*}$ where the coefficients $p,q,r,s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p \neq 0$, $r \gt 0% a.e. on $(a,b)$, and $p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)$, and $f$ is supposed to satisfy $\begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*}$ In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H_{loc}^{-1}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{max}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{min}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{min}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{min}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{min}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).