الملخص الإنجليزي
Many of the problems encountered in several branches of mathematics,
engineering, and physics require the existence and uniqueness of solutions of
operator equations of the form
Tx = b,
where b is a given vector in some Hilbert space X and T is a densely defined
linear operator on X. It is obvious that the above equation has a unique
solution if and only if the equation Tx = 0 has only the trivial solution x = 0.
It is also well-known that if T has a closed range, then there exists a solution
of Tx = b if and only if
⟨y, b⟩ = 0,
where ⟨·, ·⟩ denotes the inner product on X and y is any solution of the adjoint
equation
T
∗
y = 0.
This project will focus on the perturbation of closed range operators T on
a Hilbert space X with emphasis on the stability of the closedness of the range
under relatively bounded perturbations. The effect of these perturbations on
the nullity and the deficiency of T in addition to the solutions of the operator
equation above will also be considered. The theory of the minimum modulus
of T and that of its generalized inverse are expected to play a significant role
in this study.
