In this note, we introduce a notion of the J-kernel of a bounded linear operator on a Krein space and study the J-Fredholm theory for Krein space operators. Using J-Fredholm theory, we discuss and study the J-essential pseudospectra of bounded linear operators on Krein spaces, and some of their properties are studied. Furthermore, we discuss the structure of the J-essential pseudospectra of infinite-dimensional Hamiltonian operators.
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Dhahri et al. (2024) studied this question.
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