Volume 54 , issue 3$-$4 ( 2002 ) | back |

5th International Symposium on Mathematical Analysis and Its Applications (MAA5) | 53--56 |

Ascent, descent, quasi-nilpotent part and analytic core of operators | 57--70 |

**Abstract**

This paper concerns a localized version of the single valued extension property of a bounded operator $T\in L(X)$, where $X$ is a Banach space, at a point $\lambda_0 \in \Bbb C$. We shall relate this property to the ascent and the descent of $\lambda_0 I-T$, as well as to some spectral subspaces as the quasi-nilpotent part and the analytic core of $\lambda_0 I- T$. We shall also describe all these notions in the setting of an abstract shift condition, and in particular for weighted right shift operators on $\ell^p (\Bbb N)$, where $1\leq p< \infty$.

**Keywords:** Single valued extension property,
quasi-nilpotent part and analytic core, property (Q), weighted right shift
operators.

**MSC:** 47A10, 47A11, 47A53, 47A55

Weyl's theorem for a generalized derivation and an elementary operator | 71--81 |

**Abstract**

For $a,b\in B(H)$, $B(H)$ the algebra of operators on a complex infinite dimensional Hilbert space $H$, the generalized derivation $\delta_{ab}\in B(B(H))$ and the elementary operator $\triangle_{ab}\in B(B(H))$ are defined by $\delta_{ab}(x)=ax-xb$ and $\triangle_{ab}(x)=axb-x$. Let $d_{ab}=\delta_{ab}$ or $\triangle_{ab}$. It is proved that if $a,b^*$ are hyponormal, then $f(d_{ab})$ satisfies (generalized) Weyl's theorem for each function $f$ analytic on a neighbourhood of $\sigma(d_{ab})$.

**Keywords:** Weyl's theorem, generalized derivation, elementary operator

**MSC:** 47B47, 47B20, 47A53

On Davis-Kahan-Weinberger extension theorem | 83--88 |

**Abstract**

If $R=\bmatrix H\\ B\endbmatrix$, where $H=H^*$, we find a pseudo-inverse form of all solutions $W=W^*$, such that $\|A\|=\|R\|$, where $A=\bmatrix H&B^*\\ B& W\endbmatrix$ and $\|H\|\leq\|R\|$. In this paper we extend well-known results in a finite dimensional setting, proved by Dao-Sheng Zheng [15]. Thus, a pseudo inverse form of solutions of the Davis-Kahan-Weinberger theorem is established.

**Keywords:** Davis-Kahan-Weinberger theorem, Moore-Penrose inverse.

**MSC:** 47A05, 47A20, 15A09

A multivalued fixed point theorem in ultrametric spaces | 89--91 |

**Abstract**

The purpose of this paper is to prove that a class of generalized contractive multivalued mappings on a spherically complete ultrametric space has a fixed point.

**Keywords:** Ultrametric space, spherically complete, fixed point, multivalued
mappings.

**MSC:** 47H10

Mixed norm spaces of difference sequences and matrix transformations | 95--112 |

**Abstract**

In this paper, we generalise the definition of mixed norm spaces, define mixed norm spaces of difference sequences, determine their $\beta$-duals, and characterise matrix transformations on them. We obtain many known results as special cases.

**Keywords:** Mixed norm spaces, difference sequences, matrix transformations.

**MSC:** 40H05, 46A45

On the three-space-problem for dF spaces and their duals | 111--115 |

**Abstract**

It is shown that dF spaces of K. Brauner behave more regularly than DF spaces in connection with the three-space-problem. In particular, this problem has a positive answer in the class of Fréchet spaces for the property of being the strong dual of a barrelled dF space. Thus, a partial positive answer to a question of D. Vogt is obtained.

**Keywords:** Three-space-problem, DF space, dF space, dual Fréchet space.

**MSC:** 46A03, 46A04

Integrated semigroups of unbounded linear operators and $C_0$ semigroups on subspaces | 117--124 |

**Abstract**

We give conditions under which an infinitesimal generator of an integrated semigroups of unbounded linear operator becomes an infinitesimal generator of a perturbated semigroup of bounded linear operators. Also, we analyze when a linear operator in a Banach space is an infinitesimal generator of an integrated semigroups of unbounded linear operators.

**Keywords:** $C_{0}$-semigroups, integrated
semigroups, infinitesimal generator, perturbed operator.

**MSC:** 47D06, 47D40

An $L_p$ estimate for the difference of derivatives of spectral expansions arising by one-dimensional Schrödinger operators | 125--131 |

**Abstract**

We prove the estimate $$ \|\sigma_{\mu}^{\prime}(x,f)- \tilde\sigma_{\mu}^{\prime}(x,f)\|_{L_p(G)}\le C\|f\|_{BV(G)}\cdot\mu^{1-1/p}, $$ where $2\le p<+\infty$, and $\sigma_{\mu}(x,f),\tilde \sigma_{\mu}(x,f)$ are the partial sums of spectral expansions of a function $f(x)\in BV(G)$, corresponding to arbitrary non-negative self-adjoint extensions of the operators $\Cal Lu=-u^{\prime\prime}+q(x)u$, $\tilde{\Cal L}u=-u^{\prime\prime}+\tilde q(x)u$ $(x\in G)$ respectively; the operators are defined on an arbitrary bounded interval $G\subset \Bbb R$.

**Keywords:** Spectral expansions, self-adjoint extension, Schrödinger
operator.

**MSC:** 34L10, 47E05

Distributions generated by boundary values of functions of the Nevanlinna class $N$ | 133--138 |

**Abstract**

In this work necesarry and sufficient conditions are given for a regular distribution in $D'$ to be distribution generated by the boundary function of some function from the Nevanlinna class $N$.

**Keywords:** Distribution, boundary value of function, Nevanlinna space.

**MSC:** 46F20, 30E25, 32A35

Displacement structure of generalized inverse $A_{T,S}^{(1,2)}$ | 139--144 |

**Abstract**

It is well known that matrices with a $UV$-displacement structure possess generalized inverse with a $VU$-displacement structure. Estimation for the displacement rank of $A_{T,S}^{(1,2)}U-VA_{T,S}^{(1,2)}$ are presented, where $A_{T,S}^{(1,2)}$ is the $(1,2)$-inverse of $A$ with prescribed range $T$ and null space $S$. We extend the results due to G. Heinig and F. Hellinger, Wei and Ng, Cai and Wei for the Moore-Penorse inverse, group inverse and weighted Moore-Penrose inverse, respectively.

**Keywords:** Displacement, $(1,2)$-inverse, structured matrix.

**MSC:** 15A09, 65F20

Unbounded solutions to some systems of conservation laws---split delta shock waves | 145--149 |

**Abstract**

The solution concept is based on splitting of delta measures along regular curves in $\Bbb R^2$. Now, their product with piecewise smooth functions with discontinuities along such curves makes sense. The differentiation is defined by their mapping into the usual Radon measure space (naturally embedded into the space of Schwartz distributions).

**Keywords:** Systems of conservation laws, delta shocks, generalized solutions.

**MSC:** 35L65, 35D05, 40F10

Generalized fractional calculus with application in mechanics | 151--158 |

**Abstract**

In the studies concerned with the stability of the viscoelastic rod of fractional type it is shown that many problems are described by stability of coupled systems of differential equations with fractional derivatives. Here, we are dealing with analysis of such systems in the space of distributions. The main result of this work is stated in Theorem~5.

**MSC:** 26A33, 34D99, 46F99, 73H99

The invariant subspace lattice of an algebraic operator | 159--162 |

**Abstract**

The main object in this work is to analyze the invariant subspace lattice of an algebraic operator.

**MSC:** 47A15

A modification of revised simplex method | 163--169 |

**Abstract**

We introduce a modification of the revised simplex method, which accelerates the process of finding the first basic feasible solution in the two phases revised simplex method. We report computational results on numerical examples from Netlib test set.

**Keywords:** Linear programming, revised simplex method, MATHEMATICA.

**MSC:** 90C05

Mean value theorems in $q$-calculus | 171--178 |

**Abstract**

In this paper, some properties of continuous functions in $q$-analysis are investigated. The behavior of $q$-derivative in a neighborhood of a local extreme point is described. Two theorems are proved which are $q$-analogons of the fundamental theorems of the differential calculus. Also, two $q$-integral mean value theorems are proved and applied to estimating remainder term in $q$-Taylor formula. Finally, the previous results are used in considering some new iterative methods for equation solving.

**Keywords:** $q$-calculus.

**MSC:** 26A99

A note on limiting behavior of solutions of one-dimensional nonlinear stochastic wave equation with functionals of the white noise as initial data | 179--186 |

**Abstract**

The limitting behaviour of solutions to stochastic wave equations with singularities represented by stochastic terms is considered. In cases when the initial data are the white noise process or its functionals it is proved that the triviality effect appears.

**Keywords:** Nonlinear stochastic wave equation,
generalized stochastic processes, generalized solutions.

**MSC:** 35R60, 60H15

A system of partial differential equations with fractional derivatives | 187--194 |

**Abstract**

A subspace $\Cal D'_{\ast}(P)$ of the space of distributions has been analized and the Laplace transform of their elements applied to solve, in a prescribed domain, the system of linear partial differential equations with fractional derivatives, as well. To this system belongs a mathematical model of a visco elastic rod submited to the axial force of the form $F(t)=B+A\theta (t-t_0)$, where $A$ and $B$ are constants and $\theta$ is the Heaviside functions.

**MSC:** 46F99, 26A33, 73C99

On holomorphically projective mappings of generalized Kählerian spaces | 195--202 |

**Abstract**

In this work we define generalized Kählerian spaces and for them consider holomorphically projective mappings with an invariant complex structure. Also we consider equitorsion holomorphically projective mappings and for them we find some invariant geometric objects.

**Keywords:** Generalized Kählerian space,
holomorphically projective mappings, equitorsion holomorphically
projective mappings, holomorphically projective parameter,
holomorphically projective tensor.

**MSC:** 53C25, 53A45, 53B05

Transformations of dual problem and decreasing dimensions in linear programming | 203--210 |

**Abstract**

We investigate behavior of the potential function in a modification of the Mehrotra's primal-dual algorithm. This modification reduces dimensions of the problem and eliminates need for the finite termination algorithm. Numerical results on some examples from the Netlib test set are provided. We also regard problems about applying a stabilization procedure proposed by Kovačević-Vujčić and Ašić in the Mehrotra's primal dual interior-point algorithm for linear programming. Transformations of the dual problem required for the application of the stabilization procedure are considered.

**Keywords:** Primal-dual interior point methods, potential function.

**MSC:** 90C05

Integral transforms and summation of some Schlömilch series | 211--218 |

**Abstract**

In this paper we present a survey of the results given in the papers [11, 12, 13, 14]. Connections between integral transforms and some Schlömilch series have also been considered. These series are represented in terms of the Riemann zeta and related functions of reciprocal powers and can be brought in so called closed form in certain cases, which means that the infinite series are represented by finite sums. As applications of our results, recursive relations of some related functions and the sums of new Schlömilch series involving the Neumann, MacDonald, Struve or Bessel functions are given as well.

**Keywords:** Bessel functions, the Riemann zeta and related functions.

**MSC:** 33C10, 11M06, 65B10

Infinitesimal deformations of curvature tensors at non-symmetric affine connection space | 219--226 |

**Abstract**

At the present work we consider infinitesimal deformations of geometric objects, especially of curvature tensors, at a space $L_N$ of non-symmetric affine connection.

**Keywords:** Infinitesimal deformation, non-symmetric affine connection,
Lie derivative, geometric object, curvature tensor.

**MSC:** 53C25, 53A45, 53B05