We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by “componental operators” acting on the Hilbert spaces that form the product space. We attribute operatorial properties to the componental operators rather than to the full operators. The operatorial properties that we discuss include nonexpansivity, firm nonexpansivity, relaxed firm nonexpansivity, averagedness, being a cutter, quasi-nonexpansivity, strong quasi-nonexpansivity, strict quasi-nonexpansivity and contraction. Some relationships between operators whose componental operators have such properties and operators that have these properties on the product space are studied. This enables also to define componental fixed point sets and to study their properties. For componental contractions we offer a variant of the Banach fixed point theorem. Our motivation comes from the desire to extend a fully-simultaneous method that takes into account sparsity of the linear system in order to accelerate convergence. This was originally applicable to the linear case only and gives rise to an iterative process that uses different componental operators during iterations.
|Number of pages||17|
|Journal||Numerical Functional Analysis and Optimization|
|State||Published - 2021|
Bibliographical noteFunding Information:
The work of Yair Censor is supported by the ISF-NSFC joint research program Grant No. 2874/19. We thank Eliahu Levy and Daniel Reem for some fruitful discussions on this topic.
© 2021 Taylor & Francis Group, LLC.
- Componental operators
- Hilbert spaces
- fixed point
- nonexpansive operator
- quasi-nonexpansive operator
ASJC Scopus subject areas
- Signal Processing
- Computer Science Applications
- Control and Optimization