Quasiaffine Inverse and Moore-Penrose Inverse of Operators in Hilbert Spaces
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Date
2024-05
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Kenyatta University
Abstract
The study of inverses of operators by the concept of the Moore-Penrose Inverse and the quasiaffine inverse started in 1920 and 1985 respectively. Precisely, Moore (1920) and Penrose (1955) independently gave conditions satisfied by the MPI. Hongke and Chuan (1985) and Khalagai (1996) studied invertibility of normal (sub-normal) operators and one-sided invertible operators respectively, by the concept of the quasiaffine inverse. Since then, several researchers have contributed to these areas. Particularly, it has been shown that the Moore-Penrose inverse of an operator A with closed range satisfies the following conditions:AA^+ A=A,〖 A〗^+ AA^+=〖A 〗^+, 〖(AA〗^+ )^*=AA^+ and 〖(A〗^+ A)^*=A^+ A. If A is a quasiaffine inverse of B then both A and B are quasiaffinities and if A is an EP operator then 〖Ran(A)=Ran(A〗^*). It is also known that the Fuglede-Putnam Theorems and Fuglede-Putnam type commutativity theorems hold for normal operators and EP operators under some conditions. On quasiaffine inverses, this thesis establishes the uniqueness of the quasiaffine inverse of A given AXB=X and BYA=Y as well as establishing that B=A^(-1) under given conditions. The invertibility of quasinormal partial isometry with dense range as well as results on invertibility of operators A and B satisfying the equations AX=XB or BY=YA or both is shown under some given conditions. On MPI, the invertibility of an EP operator in terms of its Moore-Penrose inverse is established. In particular, the case where the Moore-Penrose inverse of an EP operator turns to be its usual inverse under some given conditions is shown. The Moore-Penrose inverse of a perturbed operator A+B with closed range, where A is expressible as a product of two operatorsP,Q∊B_C (H) with closed ranges and B a bounded operator satisfying some given conditions is exhibited as well as the relation between the ranges and null spaces of these operators. Moreover, this thesis establishes that Fuglede-Putnam-type results hold for EP operators, injective operators and operators with dense range satisfying some commutativity conditions involving operators 〖 AA〗^*, A^* A, 〖A^* A〗^+, BB^*,B^* B and 〖B^* B〗^+.
The study of inverses of operators by the concept of the Moore-Penrose Inverse and the quasiaffine inverse started in 1920 and 1985 respectively. Precisely, Moore (1920) and Penrose (1955) independently gave conditions satisfied by the MPI. Hongke and Chuan (1985) and Khalagai (1996) studied invertibility of normal (sub-normal) operators and one-sided invertible operators respectively, by the concept of the quasiaffine inverse. Since then, several researchers have contributed to these areas. Particularly, it has been shown that the Moore-Penrose inverse of an operator A with closed range satisfies the following conditions:AA^+ A=A,〖 A〗^+ AA^+=〖A 〗^+, 〖(AA〗^+ )^*=AA^+ and 〖(A〗^+ A)^*=A^+ A. If A is a quasiaffine inverse of B then both A and B are quasiaffinities and if A is an EP operator then 〖Ran(A)=Ran(A〗^*). It is also known that the Fuglede-Putnam Theorems and Fuglede-Putnam type commutativity theorems hold for normal operators and EP operators under some conditions. On quasiaffine inverses, this thesis establishes the uniqueness of the quasiaffine inverse of A given AXB=X and BYA=Y as well as establishing that B=A^(-1) under given conditions. The invertibility of quasinormal partial isometry with dense range as well as results on invertibility of operators A and B satisfying the equations AX=XB or BY=YA or both is shown under some given conditions. On MPI, the invertibility of an EP operator in terms of its Moore-Penrose inverse is established. In particular, the case where the Moore-Penrose inverse of an EP operator turns to be its usual inverse under some given conditions is shown. The Moore-Penrose inverse of a perturbed operator A+B with closed range, where A is expressible as a product of two operatorsP,Q∊B_C (H) with closed ranges and B a bounded operator satisfying some given conditions is exhibited as well as the relation between the ranges and null spaces of these operators. Moreover, this thesis establishes that Fuglede-Putnam-type results hold for EP operators, injective operators and operators with dense range satisfying some commutativity conditions involving operators 〖 AA〗^*, A^* A, 〖A^* A〗^+, BB^*,B^* B and 〖B^* B〗^+.
Description
A Thesis Submitted in Fulfilment of the Requirements for the Award of the Degree of Doctor of Philosophy in Pure Mathematics in the School of Pure and Applied Sciences of Kenyatta University May 2024
Supervisors:
1. Mutie Kavila
2. Jairus Khalagai