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Numerical Radius Inequalities for Nonlinear Operators in Hilbert Spaces

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Abstract

In this paper, the numerical radius of nonlinear operators in Hilbert spaces is studied. First, the relationship between the spectral radius and the numerical radius of nonlinear operators is given. Then, the famous inequality \(\frac{1}{2}\Vert T\Vert \le w(T)\le \Vert T\Vert \) and inclusion \(\sigma (A^{-1}B)\subseteq \frac{\overline{W(B)}}{\overline{W(A)}}\) of bounded linear operators are generalized to the case of certain nonlinear operators, where \(w(\cdot )\), \(\Vert \cdot \Vert \) and \(\sigma (\cdot )\) are the numerical radius, the usual operator norm and the spectrum, respectively. Finally, some numerical radius inequalities for nonlinear operator matrices are obtained.

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Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions and comments which have greatly improved this paper. This work is supported by the NNSF of China (Grant Nos. 11561048, 11761029), and the NSF of Inner Mongolia (Grant Nos. 2019MS01019, 2020ZD01).

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  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, People’s Republic of China

    Xiaomei Dong & Deyu Wu

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  1. Xiaomei Dong
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  2. Deyu Wu
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Correspondence to Deyu Wu.

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Dong, X., Wu, D. Numerical Radius Inequalities for Nonlinear Operators in Hilbert Spaces. Mediterr. J. Math. 19, 214 (2022). https://doi.org/10.1007/s00009-022-02117-z

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