Advances in Operator Theory

Perturbation of minimum attaining operators

Jadav Ganesh, Golla Ramesh, and Daniel Sukumar

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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.

Article information

Adv. Oper. Theory , Number (2018), 18 pages.

Received: 10 August 2017
Accepted: 20 December 2017
First available in Project Euclid: 7 February 2018

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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

minimum modulus spectrum essential spectrum porous set


Ganesh, Jadav; Ramesh, Golla; Sukumar, Daniel. Perturbation of minimum attaining operators. Adv. Oper. Theory, advance publication, 7 February 2018. doi:10.15352/aot.1708-1215.

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