Advances in Operator Theory
- Adv. Oper. Theory
- Volume 3, Number 3 (2018), 473-490.
Perturbation of minimum attaining operators
We prove that the minimum attaining property of a bounded linear operator on a Hilbert space $H$ whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on $H$. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.
Adv. Oper. Theory Volume 3, Number 3 (2018), 473-490.
Received: 10 August 2017
Accepted: 20 December 2017
First available in Project Euclid: 7 February 2018
Permanent link to this document
Digital Object Identifier
Primary: 47B07: Operators defined by compactness properties
Secondary: 47A10: Spectrum, resolvent 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47B65: Positive operators and order-bounded operators
Ganesh, Jadav; Ramesh, Golla; Sukumar, Daniel. Perturbation of minimum attaining operators. Adv. Oper. Theory 3 (2018), no. 3, 473--490. doi:10.15352/aot.1708-1215. https://projecteuclid.org/euclid.aot/1518016380