Mathematics

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Browsing Mathematics by Subject "Hilbert spaces"

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    On browder spectrum and quasisimilarity of some classes of operators in hilbert spaces
    (Chuka University, 2024) Kamau Faith Wairimu
    The spectrum of operators and quasisimilarity on classes of operators in Hilbert Spaces has been extensively studied. It has been shown that quasisimilarity on m-hyponormal, w hyponormal and (p,k) quasihyponormal operators have equal spectrum, equal Essential Spectrum and equal Weyl Spectrum. However, the consideration of equality of the Browder spectrum for such operators together with quasisimilarity has not been done. Therefore, this study considered the properties of Browder spectrum and quasisimilarity on classes of operators such as m-hyponormal, w-hyponormal and (p,k) quasihyponormal operators and established that quasisimilarity preserves normality, hyponormality, m-hyponormality, w-hyponormality, log-hyonormality and (p,k)-quasihyponormality. We also determined the conditions under which quasisimilarity of such operators yields equal Browder spectrum, that, the operators must be quasisimilar and intersected with biquasitriangular operators. In addition, the study determined the relationship of Browder spectrum and Numerical range and established the spectral theory of Browder spectrum. Finally, the study established an equivalent relation of Browder spectrum and Single-valued extension property. In achieving this, the properties of normal operators, hyponormal operators, biquasitriangular and spectral properties of classes of operators together with properties of Single-Valued Extension Property (SVEP) and Browder’s theorem were considered. The results that have been established on other equivalence relation on these classes of operators on almost similarity and unitary equivalence were used in comparison. The findings on the Browder spectrum and quasisimilarity of these classes of operators will be useful in other branches of Mathematics and Physics such as differential operators and algebra. The study of the spectrum of operators will be important for ones applying these operators in spectral analysis, algebra, fluid mechanics, quantum mechanics and differential operators for the calculations of wave function and formulation of theory.
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    On orthogonality and micro transitivity characterization of Hilbert spaces
    (Chuka University, 2024) Mugure Damaris Njeri
    Characterization of a transitive separable Banach spaces as Hilbert spaces has been an open area of research. It has been shown that separable Banach spaces which are transitive, almost transitive, convex transitive and micro transitive together with isometries of various characteristics such as unitary, reflection, differentiable properties are Hilbert spaces. It has also been shown that a separable real Banach space which is almost transitive with vector orthogonalities of dimension greater than three is a Hilbert space. However, properties such as micro transitive together with vector orthogonalities for n-dimension have essential property that can be utilized to characterize Banach spaces as Hilbert spaces. Additionaly, by this characterization, properties of matrix numerical range and numerical radius can also be determined. Therefore, by utilizing micro transitivity and Isosceles vector (I-vector), Pythagorean vector (P-vector) and Isosceles Pythagorean vector (IP-vector) in the unit sphere of separable Banach space this research determined that an n-dimension separable Banach spaces is a Hilbert space. In addition, by the use of properties of numerical range in general Banach space the study also established properties of matrix numerical range and numerical radius in separable transitive Banach space. The findings of this study will find use in algebra and differential operators for the purpose of calculation of wave functions and formulation of theory. In addition, the findings of the study will find use in spectral analysis of functions for the study of vibrations and interfacial waves stability analysis.