Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces via Two Tominaga’s Results

Authors

DOI:

https://doi.org/10.63286/jima.2025.03

Keywords:

Tensorial product, Hadamard product, selfadjoint operators, convex functions

Abstract

In this paper we provide several generalizations for tensorial and Hadamard products of positive linear operators on complex Hilbert spaces of the celebrated scalar inequalities due to Tominaga. They give both multiplicative and additive reverses of Young’s inequality for positive operators in terms of Specht’s ratio and logarithmic mean.

References

[1] H. Budak, T. Tunc, and M. Z. Sarikaya, Fractional Hermite–Hadamard-type inequalities for interval-valued functions, Proceedings of the American Mathematical Society, 148(2), 2020, 705–718.

DOI: 10.1090/proc/14741

[2] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra and its Applications, 26, 1979, 203–241.

DOI: 10.1016/0024-3795(79)90179-4

[3] H. Araki and F. Hansen, Jensen’s operator inequality for functions of several variables, Proceedings of the American Mathematical Society, 128(7), 2000, 2075–2084.

DOI: 10.1090/S0002-9939-00-05371-5

[4] J. S. Aujila and H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Mathematica japonicae, 42(2), 1995, 265–272.

[5] S. S. Dragomir, A trapezoid type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces, Istanbul Journal of Mathematics, 1(2), 2023, 48–56.

DOI: 10.26650/ijmath.2023.00006

[6] A. Korányi, On some classes of analytic functions of several variables, Transactions of the American Mathematical Society, 101(3), 1961, 520–554.

DOI: 10.1090/S0002-9947-1961-0136765-6

[7] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Mathematica japonicae, 41(3), 1995, 531–535.

[8] T. Furuta, J. Mičić Hot, J. Pečarić and Y. Seo, Mond-Pečarić Method in Operator Inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

[9] K. Kitamura and Y. Seo, Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scientiae Mathematicae, 1(2), 1998, 237–241.

[10] W. Specht, Zer Theorie der elementaren, Mittel, Mathematische Zeitschrift, 74, 1960, 91–98.

[11] V. Stojiljković, Twice differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces, Electronic Journal of Mathematical Analysis and Applications, 11(2), 2023, 1–15.

[12] V. Stojiljković and S. S. Dragomir, Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces, Gulf Journal of Mathematics, 15(2), 2023, 40–55.

DOI: 10.56947/gjom.v15i2.1247

[13] M. Tominaga, Specht’s ratio in the Young inequality, Scientiae Mathematicae Japonicae Online, 55, 2002, 583–588.

[14] S. Wada, On some refinement of the Cauchy-Schwarz Inequality, Linear Algebra and its Applications, 420, 2007, 433–440.

DOI: 10.1016/j.laa.2006.07.019

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Published

14.04.2025

How to Cite

Dragomir, S. S. (2025). Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces via Two Tominaga’s Results. Journal of Inequalities and Mathematical Analysis, 1(1), 28–46. https://doi.org/10.63286/jima.2025.03

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Articles