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Dissertation
Fixed-point and implicit/inverse function theorems for free noncommutative functions
Doctor of Philosophy (Ph.D.), Drexel University
01 Jun 2015
DOI:
https://doi.org/10.17918/etd-6318
Abstract
We establish a fixed-point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices. The conclusions are stronger if such a mapping is a free noncommutative function, i.e., if it respects matrix similarities. As a special case, we obtain the corresponding version of the Banach Contraction Mapping Theorem. This result is then applied to prove the existence and uniqueness of a solution of the initial value problem for ODEs in noncommutative spaces. As a by-product of the ideas developed in this paper, we establish a noncommutative version of the principle of nested closed sets. We prove the implicit function theorem and the inverse function theorem in two different settings: for free noncommutative functions over operator spaces and for free noncommutative functions on the set of nilpotent matrices.
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Details
- Title
- Fixed-point and implicit/inverse function theorems for free noncommutative functions
- Creators
- Gulnara K. Abduvalieva - DU
- Contributors
- Dmitry S. Kaliuzhnyi-Verbovetskyi (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 6318; 991014632561104721