Papers by Dr. Bernard Mutuku Nzimbi
Far East Journal of Mathematical Sciences, 2021
International Journal of Statistics and Applied Mathematics, 2021
In this paper, we give a condition in which a paranormal operator T has invariant subspaces. We a... more In this paper, we give a condition in which a paranormal operator T has invariant subspaces. We also show that a hyponormal operator T has an invariant subspace if it is complex symmetric.
International Journal of Mathematics Trends and Technology
Journal of Advances in Mathematics and Computer Science
In this paper, we characterize Murray-von Neumann equivalent projections. We also investigate and... more In this paper, we characterize Murray-von Neumann equivalent projections. We also investigate and compare the relationship between the Murray von Neumann relation and other equivalence relations on the set of orthogonal projections in the von Neumann algebra .
The study of operators having some special spectral properties like Weyl's theorem, Browder&#... more The study of operators having some special spectral properties like Weyl's theorem, Browder's theorem and the SVEP has been of important interest for some time now. The SVEP is very useful in the study of the local spectral theory. In this paper, we explore the single-valued extension property (SVEP) for some operators on Hilbert spaces. We characterize operators with or without SVEP at zero and those where Weyl's and Browder's theorems hold. It is shown that if a Fredholm operator has no SVEP at zero, then zero is an accumulation point of the spectrum of the operator. It is also shown that quasi similar Fredholm operators have equal Weyl spectrum.
In this paper we investigate unitary quasi-equivalence of operators in Hilbert spaces. We charact... more In this paper we investigate unitary quasi-equivalence of operators in Hilbert spaces. We characterize operators that are unitarily quasi-equivalent. We also investigate equivalence relations closely related to unitary quasi-equivalence. We give and prove conditions under which unitary quasi-equivalence coincides with other operator equivalence relations.
In this paper we investigate -isometries and related classes of operators. We will also introduce... more In this paper we investigate -isometries and related classes of operators. We will also introduce and study the notions of similitudes, self-similarity and -metric equivalence relations of operators. It will be shown that self-similarity implies -metric equivalence of operators. We will characterize -isometric and -unitary and prove that quasisimilar -isometries are unitarily equivalent.
Mathematics and Computer Science, 2018
In this paper we introduce the notion of Quasi-similarity of bounded linear operators in Hilbert ... more In this paper we introduce the notion of Quasi-similarity of bounded linear operators in Hilbert Spaces. We do so by defining a quasi-affinity from one Hilbert Space H to K. Some results on quasi-affinities are also discussed. It has already been shown that on a finite dimensional Hilbert Space, quasi similarity is an equivalence relation that is; it is reflexive, symmetric and also transitive. Using the definition of commutants of two operators, we give an alternative result to show that quasi similarity is an equivalence relation on an infinite dimensional Hilbert Space. Finally, we establish the relationship between quasi similarity and almost similarity equivalence relations in Hilbert Spaces using hermitian and normal operators.
We consider the almost similarity property which is a new class in operator theory and was first ... more We consider the almost similarity property which is a new class in operator theory and was first introduced by A. A. S. Jibril. We establish that almost similarity is an equivalence relation. Some results on almost similarity and isometries, compact operators, hermitian, normal and projection operator are also shown. By characterization of unitary equivalence operators in terms of almost similarity we prove that operators that are similar are almost similar. We also claim that quasi-similarity implies almost similarity under certain conditions (i.e. if the quasiaffinities are assumed to be unitary operators). Furthermore, a condition under which almost similarity of operators implies similarity is investigated. Lastly, we show that two bounded linear operators of a Banach algebra on a Hilbert space are both completely nonunitary if they are contractions which are almost similar to each other.
Similarity and unitary equivalence can be shown to be of equivalence relations. We discuss a resu... more Similarity and unitary equivalence can be shown to be of equivalence relations. We discuss a result showing that two similar operators have equal spectra (i.e. point and approximate point spectrum). More so, unitary equivalence results for invariant subspaces and normal operators are proved. For similar normal operators, we state the Fuglede – Putnam –Rosenblum theorem that makes proofs for similar normal operators more simplified. It is also noted that direct sums and summands are preserved under unitary equivalence. Furthermore, we show that the natural concept of equivalence between Hilbert Space operators is unitary equivalence which is stronger than similarity. By introducing the notion of quasisimilarity of operators which is the same as similarity in finite dimensional spaces, but in infinite dimensional spaces, it is a much weaker relation, we further show that quasisimilarity is an equivalence relation. We also link invariant subspaces and hyperinvariant subspaces with quasisimilarity where it is seen that similarity preserves nontrivial invariant subspaces while quasisimilarity preserves nontrivial hyperinvariant subspaces. 152 Mathematics Subject Classification: 47A10; 47A15; 47B15
Far East Journal of Mathematical Sciences
Far East Journal of Mathematical Sciences
ABSTRACT
We study properties of W 2 -recurrent LP-Sasakian manifolds and prove symmetry and skew-symmetry ... more We study properties of W 2 -recurrent LP-Sasakian manifolds and prove symmetry and skew-symmetry properties of the W 2 -curvature tensor.
In this paper we introduce the notions of an A-self-adjoint, an A-skew-adjoint linear operator, w... more In this paper we introduce the notions of an A-self-adjoint, an A-skew-adjoint linear operator, where A is a self-adjoint and invertible operator and related classes of operators, which generalize some known classes of operators. We investigate some properties of these operators and show that these operators share some properties with some known classes of operators. We prove some results on some equivalence of these operators and investigate conditions under which these operators are self-adjoint, unitary, skew-adjoint, normal, hyponormal, quasinormal or binormal. We also attempt to locate the spectra of such operators.
Mathematics and Computer Science, 2017
In this paper, we survey various results concerning n-involution operators and k-potent operators... more In this paper, we survey various results concerning n-involution operators and k-potent operators in Hilbert spaces. We gain insight by studying the operator equation n T I = , with , 1 k T I k n ≠ ≤ − where , n k ∈ ℕ. We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.
Uploads
Papers by Dr. Bernard Mutuku Nzimbi