On Fong-Tsui conjecture and binormality of operators
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Hranislav Stanković
Abstract
In [Fong, C. K.—Tsui, S. K.: A note on positive operators, J. Operator Theory 5(1) (1981), 73–76], the authors conjectured that T ∈ 𝔅(ℋ) is self-adjoint if |T| ≤ |Re T|. After more than 40 years, the validity of the conjecture is still unknown. In this paper, we demonstrate, amongst other results, that the conjecture is true for binormal operators which are either injective, 2-quasinormal, or Im T is compact.
Funding statement: This work was supported by by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia [Grant Number: 451-03-137/2025-03/200102].
Acknowledgement
The author wishes to express sincere gratitude to the anonymous referee for his/her valuable comments and suggestions, which have enhanced the clarity and overall quality of the manuscript.
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(Communicated by Emanuel Chetcuti)
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