Research Article
Year 2025,
Volume: 54 Issue: 2, 516 - 528, 28.04.2025
Abstract
In this paper, we introduce a new class of operators on a complex Hilbert space $\mathcal{H}$ which is called polynomially accretive operators, and thereby extending the notion of accretive and $n$-real power positive operators. We give several properties of the newly introduced class, and generalize some results for accretive operators. We also prove that every $2$-normal and $(2k+1)$-real power positive operator, for some $k\in\mathbb{N}$, must be $n$-normal for all $n\geq2$. Finally, we give sufficient conditions for the normality in the preceding implication.
Keywords
polynomially accretive operators accretive operators n–real power positive operators Re–nnd matrices polynomially normal operators
References
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- [8] M. Ch, J. E. Lee, K. Tanahashi and A. Uchiyama, Remarks on n–normal operators, Filomat 32(15), 5441-5451, 2018.
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- [12] B. Fuglede, A Commutativity Theorem for Normal Operators, Proc. Nati. Acad. Sci. 36, 35–40, 1950.
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- [14] E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123, 415– 438, 1951.
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- [16] F. Kittaneh, On the structure of polynomially normal operators, Bull. Austral. Math Soc. 30, 11-18, 1984.
- [17] K. Löwner, Uber monotone Matrixfunktionen, Math. Z. 38, 177–216, 1934.
- [18] E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc. 26, 394–395, 1920.
- [19] M. H. Mortad, An operator theory problem book, World Scientific Publishing Co. Pte. Ltd., 2018.
- [20] M. H. Mortad, Counterexamples in operator theory, Birkhäuser/Springer, Cham, 2022.
- [21] M. Z. Nashed, Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim. 9(3-4), 261–325, 1987.
- [22] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406–413, 1955.
- [23] R. Penrose, On best approximate solutions of linear matrix equations, Proc. Cambridge Philos. Soc. 52, 17–19, 1956.
- [24] H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34, 653–664, 1971.
- [25] W. Rudin, Functional Analysis, McGraw-Hill, International Editions, 1991.
- [26] M. Uchiyama, Powers and commutativity of selfadjoint operators, Math. Ann. 300(4), 643–647, 1994.
- [27] L. Wu and B. Cain. The Re-nonnegative definite solutions to the matrix inverse problem, Linear Algebra Appl. 236, 137–146, 1996.
- [28] H. Zhang, Y. Dou and W. Yu, Real positive solutions of operator equations $AX=C$ and $XB=D$, AIMS Mathematics 8(7), 15214–15231, 2023.
Year 2025,
Volume: 54 Issue: 2, 516 - 528, 28.04.2025
Abstract
References
- [1] S. A. Aluzuraiqi and A. B. Patel, On of n–normal operators, General Math Notes. 1, 61–73, 2010.
- [2] W. N. Anderson and E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28, 60–71, 1975.
- [3] A. Benali, On the class of n–real power positive operators on a Hilbert space, Filomat 10(2), 23–31, 2018.
- [4] A. Bjerhammer, Application of the calculus of matrices to the method of least squares with special reference to geodetic calculations, Kungl. Tekn. Hogsk. Hand. Stockholm 49, 1–86, 1951.
- [5] A. Bjerhammer, Rectangular reciprocal matrices with special reference to geodetic calculations, Bull. Geodesique 52, 188–220, 1951.
- [6] F. Bouchelaghem and M. Benharrat, The Moore–Penrose Inverse of Accretive Operators with Application to Quadratic Operator Pencils, Filomat 36(7), 2475–2491, 2022.
- [7] M. Ch and B. Načevska, Spectral properties of n–normal operators, Filomat 32(14), 5063-5069, 2018.
- [8] M. Ch, J. E. Lee, K. Tanahashi and A. Uchiyama, Remarks on n–normal operators, Filomat 32(15), 5441-5451, 2018.
- [9] D. S. Cvetković–Ilić, Re-nnd solutions of the matrix equation $AXB=C$, J. Australian Math. Soc. 84, 63–72, 2008.
- [10] D. S. Djordjević, M. Ch and D. Mosić, Polynomially normal operators, Ann. Funct. Anal. 11, 493–504, 2020.
- [11] R. G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert spaces, Proc. Amer. Math. Soc. 17(2), 413–416, 1966.
- [12] B. Fuglede, A Commutativity Theorem for Normal Operators, Proc. Nati. Acad. Sci. 36, 35–40, 1950.
- [13] M. Guesba and M. Nadir, On operators for which $T^2\geq -T^{*2}$, Aust. J. Math. Anal. Appl. 13(1), 1–5, 2016.
- [14] E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123, 415– 438, 1951.
- [15] A. A. S. Jibril, On n-Power Normal Operators, Arab. J. Sci. Eng. 33(24), 247–251, 2008.
- [16] F. Kittaneh, On the structure of polynomially normal operators, Bull. Austral. Math Soc. 30, 11-18, 1984.
- [17] K. Löwner, Uber monotone Matrixfunktionen, Math. Z. 38, 177–216, 1934.
- [18] E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc. 26, 394–395, 1920.
- [19] M. H. Mortad, An operator theory problem book, World Scientific Publishing Co. Pte. Ltd., 2018.
- [20] M. H. Mortad, Counterexamples in operator theory, Birkhäuser/Springer, Cham, 2022.
- [21] M. Z. Nashed, Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim. 9(3-4), 261–325, 1987.
- [22] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406–413, 1955.
- [23] R. Penrose, On best approximate solutions of linear matrix equations, Proc. Cambridge Philos. Soc. 52, 17–19, 1956.
- [24] H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34, 653–664, 1971.
- [25] W. Rudin, Functional Analysis, McGraw-Hill, International Editions, 1991.
- [26] M. Uchiyama, Powers and commutativity of selfadjoint operators, Math. Ann. 300(4), 643–647, 1994.
- [27] L. Wu and B. Cain. The Re-nonnegative definite solutions to the matrix inverse problem, Linear Algebra Appl. 236, 137–146, 1996.
- [28] H. Zhang, Y. Dou and W. Yu, Real positive solutions of operator equations $AX=C$ and $XB=D$, AIMS Mathematics 8(7), 15214–15231, 2023.
There are 28 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 27, 2024 |
| Publication Date | April 28, 2025 |
| Submission Date | January 23, 2024 |
| Acceptance Date | June 1, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 2 |