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1
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Singular integral models for p-hyponormal operators and the Riemann-Hilbert problem

100%
Chō M., Huruya T., Itoh M.
Studia Mathematica
|
1998
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tom 130
|
nr 3
213-221
EN
The purpose of this paper is to give singular integral models for p-hyponormal operators and apply them to the Riemann-Hilbert problem.
2
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Extensions of convex functionals on convex cones

100%
Ignaczak E., Paszkiewicz A.
Applicationes Mathematicae
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1998-1999
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tom 25
|
nr 3
381-386
EN
We prove that under some topological assumptions (e.g. if M has nonempty interior in X), a convex cone M in a linear topological space X is a linear subspace if and only if each convex functional on M has a convex extension on the whole space X.
3
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Unbounded Hermitian operators and relative reproducing kernel Hilbert space

100%
Jorgensen P.
Open Mathematics
|
2010
|
tom 8
|
nr 3
569-596
EN
We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.
4
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Gradient method for non-injective operators in Hilbert space with application to Neumann problems

100%
Karátson J.
Applicationes Mathematicae
|
1999
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tom 26
|
nr 3
333-346
EN
The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.
5
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A new view of some operators and their properties in terms of the Non-Newtonian Calculus

100%
Ünlüyol E., Salas S., Iscan I.
Topological Algebra and its Applications
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2017
|
tom 5
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nr 1
49-54
EN
Differentiation and integration are basic operations of calculus and analysis. Indeed, they are in- finitesimal versions of substraction and addition operations on numbers, respectively. From 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of substraction and addition into division and multiplication, respectively and thus established a new calculus, called Non-Newtonian Calculus. So in this paper, it is investigated to a new view of some operators and their properties in terms of Non-Newtonian Calculus. Then we compare with the Newtonian and Non-Newtonian Calculus.
6
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Operator fractional-linear transformations: convexity and compactness of image; applications

100%
Khatskevich V., V. Shul'Man
Studia Mathematica
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1995
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tom 116
|
nr 2
189-195
EN
The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space $ℒ(X_1,X_2)$ of all linear bounded operators acting from $X_1$ into $X_2$, where $X_1, X_2$ are Banach spaces. We show that in the case of Hilbert spaces $X_1, X_2$ the image F(ℬ) of any (open or closed) ball ℬ ⊂ D(F) is convex, and if ℬ is closed, then F(ℬ) is compact in the weak operator topology (w.o.t.) (Theorem 1.2). These results extend the corresponding results on compactness obtained in [3], [4] under some additional restrictions imposed on F. We also establish that the convexity of the image of f.-l.t. is a characteristic property of Hilbert spaces, that is, if for the f.-l.t. $F:K → (I+K)^{-1}$ the image $F(𝘒)$ of the open unit ball 𝘒 of the space ℒ(X) is convex, then X is a Hilbert space (Theorem 1.3). In Section 2 we apply the compactness of F(𝘒̅) for the closed unit operator ball 𝘒̅ to the study of the behavior of solutions to evolution problems in a Hilbert space ℋ. Namely, we establish the exponential dichotomy of solutions for the so-called hyperbolic case (such that the evolution operator is invertible). This is an extension of Theorem 1.1 of [5], where the corresponding assertion was established for the particular case of a Pontryagin space ℋ.
7
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A Simple Proof of the Polar Decomposition Theorem

100%
Wójcik P.
Annales Mathematicae Silesianae
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2017
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tom 31
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nr 1
165-171
EN
In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.
8
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Schauder's fixed-point theorem in approximate controllability problems

88%
Babiarz A., Klamka J., Niezabitowski M.
International Journal of Applied Mathematics and Computer Science
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2016
|
tom 26
|
nr 2
263-275
EN
The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder's fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.
9
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Measure valued solutions for stochastic evolution equations on Hilbert space and their feedback control

88%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2005
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tom 25
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nr 1
129-157
EN
In this paper, we consider a class of semilinear stochastic evolution equations on Hilbert space driven by a stochastic vector measure. The nonlinear terms are assumed to be merely continuous and bounded on bounded sets. We prove the existence of measure valued solutions generalizing some earlier results of the author. As a corollary, an existence result of a measure solution for a forward Kolmogorov equation with unbounded operator valued coefficients is obtained. The main result is further extended to cover Borel measurable drift and diffusion which are assumed to be bounded on bounded sets. Also we consider control problems for these systems and present several results on the existence of optimal feedback controls.
10
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Existence of solutions for second order stochastic differential inclusions in Hilbert spaces

88%
Balasubramaniam P., Ntouyas S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 2
365-384
EN
In this paper, sufficient conditions are given for the existence of solutions for a class of second order stochastic differential inclusions in Hilbert space with the help of Leray-Schauder Nonlinear Alternative.
11
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Hyperspaces of Peano continua of euclidean spaces

63%
Gladdines H., van Mill J.
Fundamenta Mathematicae
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1993
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tom 142
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nr 2
173-188
EN
If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L(ℝ^n)$ is homeomorphic to $B^∞$, where B denotes the pseudo-boundary of the Hilbert cube Q.
12
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Topological dual of $B_∞(I,𝓛 ₁(X,Y))$ with application to stochastic systems on Hilbert space

63%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2009
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tom 29
|
nr 1
67-90
EN
In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as the structural control, is to be chosen so as to minimize fluctuation (volatility). Both existence of optimal policy and necessary conditions of optimality are presented including a conceptual algorithm.
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