Necessary conditions for the exact observability of systems on Hilbert spaces

2008 ◽  
Vol 57 (3) ◽  
pp. 222-227 ◽  
Author(s):  
Gen Qi Xu ◽  
Chao Liu ◽  
Siu Pang Yung
Keyword(s):  
Hilbert Spaces ◽  
Necessary Conditions ◽  
Exact Observability

scholarly journals Approximative K-atomic decompositions and frames in Banach spaces

2018 ◽  
Vol 26 (1/2) ◽  
pp. 153-166
Author(s):  
Shah Jahan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic Decomposition ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Special Kind ◽  
Frame Theory ◽  
Bessel Sequence ◽  
Atomic Decompositions ◽  
Counter Example

L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative d-frame and approximative d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative d-Bessel sequence and approximative d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

scholarly journals Equivalence and stability of random fixed point iterative procedures

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Ismat Beg ◽  
Mujahid Abbas
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
Random Operator ◽  
Random Operators ◽  
Random Fixed Point ◽  
The Stability ◽  
Iterative Procedures ◽  
Measurable Mappings

We generate a sequence of measurable mappings iteratively and study necessary conditions for its strong convergence to a random fixed point of strongly pseudocontractive random operator. We establish the weak convergence of an implicit random iterative procedure to common random fixed point of a finite family of nonexpansive random operators in Hilbert spaces. We prove the equivalence between the convergence of random Ishikawa and random Mann iterative schemes for contraction random operator and strongly pseudocontractive random operator. We also examine the stability of random fixed point iterative procedures for the random operators satisfying certain contractive conditions in the context of metric spaces.

scholarly journals New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 951
Author(s):  
Yuankang Fu ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Upper And Lower Bounds ◽  
Uniformly Convex ◽  
Equilateral Triangles ◽  
Geometric Constant ◽  
Geometric Constants

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

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