In this article, we study the reduced minimum modulus of the Drazin inverse of an operator on a Hilbert space and give lower and upper bounds of the reduced minimum modulus of an operator and its Drazin inverse, respectively. Using these results, we obtain a characterization of the continuity of Drazin inverses of operators on a Hilbert space.
Let P and Q be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if P + Q − I is invertible, then P and Q are homotopic with˜s(P, Q) ≤ 2. In this paper, we have given a necessary and sufficient condition that˜s(P, Q) ≤ 2, where˜s(P, Q)… (More)
Let P, Q be two linear idempotents on a Banach space. We show that the closeness of the range and complementarity of the kernel (range) of linear combinations of P and Q are independent of the choice of coefficients. This generalizes known results and shows that many spectral properties do not depend on the coefficients. The non-singularity of the… (More)