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)
References
[1] Aluthge, A: On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory 13 (1990), 307–315.10.1007/BF01199886Suche in Google Scholar
[2] Aluthge, A: Some generalized theorems on p-hyponormal operators, Integral Equations Operator Theory 24 (1996), 497–501.10.1007/BF01191623Suche in Google Scholar
[3] Beck, W. A.—Putnam, C. R.: A note on normal operators and their adjoints, J. London Math. Soc. 31 (1956), 213–216.10.1112/jlms/s1-31.2.213Suche in Google Scholar
[4] Berberian, S. K.: Some conditions on an operator implying normality, Math. Ann. 184 (1970), 188–192.10.1007/BF01351562Suche in Google Scholar
[5] Campbell, S. L.: Linear operators for which T∗T and TT∗ commute, Proc. Am. Math. Soc. 34 (1972), 177–180.10.2307/2037922Suche in Google Scholar
[6] Campbell, S. L.: Linear operators for which T∗T and TT∗ commute II, Pacific. J. Math. 53 (1980), 355–361.10.2140/pjm.1974.53.355Suche in Google Scholar
[7] Campbell, S. L.: Linear operators for which T∗T and TT∗ commute III, Pacific. J. Math. 91(1) (1980), 39–45.10.2140/pjm.1980.91.39Suche in Google Scholar
[8] Conway, J. B.: The Theory of Subnormal Operators. Math. Surveys Monographs, Vol. 36, Amer. Math. Soc., Providence, 1991.10.1090/surv/036Suche in Google Scholar
[9] Cvetković-Ilić, D. S.—Stanković, H.: On normal complements, J. Math. Anal. Appl. 535(2) (2024), Art. ID 128216.10.1016/j.jmaa.2024.128216Suche in Google Scholar
[10] Embry, M. R.: Conditions implying normality in Hilbert space, Pacific. J. Math. 18 (1966), 457–460.10.2140/pjm.1966.18.457Suche in Google Scholar
[11] Embry, M. R.: Similarities involving normal operators on Hilbert space, Pacific. J. Math. 35(2) (1970), 331–336.10.2140/pjm.1970.35.331Suche in Google Scholar
[12] Fong, C. K.—Istrătescu, V. I.: Some characterizations of Hermitian operators and related classes of operators, Proc. Amer. Math. Soc. 76 (1979), 107–112.10.1090/S0002-9939-1979-0534398-XSuche in Google Scholar
[13] Fong, C. K.—Tsui, S. K.: A note on positive operators, J. Operator Theory 5(1) (1981), 73–76.Suche in Google Scholar
[14] Furuta, T.: Invitation to Linear Operators. From Matrices to Bounded Linear Operators on a Hilbert Space, Taylor and Francis, London and New York, 2001.10.1201/b16820Suche in Google Scholar
[15] Heinz, E.: Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415–438.10.1007/BF02054965Suche in Google Scholar
[16] Kittaneh, F.: Some characterizations of self-adjoint operators, Acta Sci. Math. 47 (1984), 441–444.Suche in Google Scholar
[17] Ko, E.—Lee, M.-J.: Remarks on n-power quasinormal operators, Filomat 37(11) (2023), 3371–3381.10.2298/FIL2311371KSuche in Google Scholar
[18] Löwner, K.: Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177–216.10.1007/BF01170633Suche in Google Scholar
[19] Mbekhta, M.—Suciu, L.: Partial isometries and the conjecture of C. K. Fong and S. K. Tsui, J. Math. Anal. Appl. 437 (2016), 431–444.10.1016/j.jmaa.2015.12.057Suche in Google Scholar
[20] Mccarthy, C. A.: Cρ, Israel J. Math. 5 (1967), 249–271.10.1007/BF02771613Suche in Google Scholar
[21] Mortad, M. H.: Similarities involving unbounded normal operators, Tsukuba J. Math. 34(1) (2010), 129–136.10.21099/tkbjm/1283967412Suche in Google Scholar
[22] Mortad, M. H.: A contribution to the Fong-Tsui conjecture related to self-adjoint operators, arXiv:1208. 4346v1.Suche in Google Scholar
[23] Mortad, M. H.: An Operator Theory Problem Book, World Scientific Publishing Co., 2018.10.1142/10884Suche in Google Scholar
[24] Mortad, M. H.: Counterexamples in Operator Theory, Birkhäuser/Springer Cham, 2022.10.1007/978-3-030-97814-3Suche in Google Scholar
[25] Sid Ahmed, O. A. M.: On the class of n-power quasinormal operators on the Hilbert space, Bull. Math. Anal. Appl. 3(2) (2011), 213–228.Suche in Google Scholar
[26] Stampfli, J. G.: Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458.10.2140/pjm.1962.12.1453Suche in Google Scholar
[27] Stanković, H.: Converse of Fuglede Theorem, Operators and Matrices 17(3) (2023), 705–714.10.7153/oam-2023-17-46Suche in Google Scholar
[28] Stanković, H.: Subnormal n-th roots of matricially and spherically quasinormal pairs, Filomat 37(16) (2023), 5325–5331.10.2298/FIL2316325SSuche in Google Scholar
[29] Stanković, H.: Conditions implying self-adjointness and normality of operators, Complex Analysis Operator Theory 18 (2024), Art. No. 149.10.1007/s11785-024-01596-0Suche in Google Scholar
[30] Whitley, R. A note on hyponormal operators, Proc. Amer. Math. Soc. 49(2) (1975), 399–400.10.1090/S0002-9939-1975-0365216-7Suche in Google Scholar
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Artikel in diesem Heft
- Copies of monomorphic structures
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- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
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- Remarks on some one-ended groups
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- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
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- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
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