On browder spectrum and quasisimilarity of some classes of operators in hilbert spaces

Abstract

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.