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An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

  • Received: 20 July 2023 Revised: 28 August 2023 Accepted: 10 September 2023 Published: 15 September 2023
  • MSC : 47A12, 47A30, 47B15, 47A63

  • We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if $ {\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H}) $ ($ i = 1, 2, \cdots, n $), $ m\in \mathbb N $, $ p, q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $ and $ \phi $ and $ \psi $ are non-negative functions on $ [0, \infty) $ which are continuous such that $ \phi(t)\psi(t) = t $ for all $ t \in [0, \infty) $, then

    $ \begin{equation*} w^{2r}\left({\sum\limits_{i = 1}^{n} {\bf X}_i {\bf A}_i^m {\bf B}_i}\right)\leq \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\left\Vert{\sum\limits_{i = 1}^{n}\frac{1}{p}S_{i, j}^{pr}+\frac{1}{q}T_{i, j}^{qr}}\right\Vert-r_0\inf\limits_{\left\Vert{\xi}\right\Vert = 1}\rho(\xi), \end{equation*} $

    where $ r_0 = \min\{\frac{1}{p}, \frac{1}{q}\} $, $ S_{i, j} = {\bf X}_i\phi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^* $, $ T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\psi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i $ and

    $ \rho(\xi) = \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left({\left<{S_{i, j}^r\xi, \xi}\right>^{\frac{p}{2}}-\left<{T_{i, j}^r\xi, \xi}\right>^{\frac{q}{2}}}\right)^2. $

    Citation: Mohammad H. M. Rashid, Feras Bani-Ahmad. An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality[J]. AIMS Mathematics, 2023, 8(11): 26384-26405. doi: 10.3934/math.20231347

    Related Papers:

  • We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if $ {\bf A}_i, {\bf B}_i, {\bf X}_i\in \mathcal{B}(\mathcal{H}) $ ($ i = 1, 2, \cdots, n $), $ m\in \mathbb N $, $ p, q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $ and $ \phi $ and $ \psi $ are non-negative functions on $ [0, \infty) $ which are continuous such that $ \phi(t)\psi(t) = t $ for all $ t \in [0, \infty) $, then

    $ \begin{equation*} w^{2r}\left({\sum\limits_{i = 1}^{n} {\bf X}_i {\bf A}_i^m {\bf B}_i}\right)\leq \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\left\Vert{\sum\limits_{i = 1}^{n}\frac{1}{p}S_{i, j}^{pr}+\frac{1}{q}T_{i, j}^{qr}}\right\Vert-r_0\inf\limits_{\left\Vert{\xi}\right\Vert = 1}\rho(\xi), \end{equation*} $

    where $ r_0 = \min\{\frac{1}{p}, \frac{1}{q}\} $, $ S_{i, j} = {\bf X}_i\phi^2\left({\left\vert{ {\bf A}_i^{j*}}\right\vert}\right) {\bf X}_i^* $, $ T_{i, j} = \left({ {\bf A}_i^{m-j} {\bf B}_i}\right)^*\psi^2\left({\left\vert{ {\bf A}_i^j}\right\vert}\right) {\bf A}_i^{m-j} {\bf B}_i $ and

    $ \rho(\xi) = \frac{n^{2r-1}}{m}\sum\limits_{j = 1}^{m}\sum\limits_{i = 1}^{n}\left({\left<{S_{i, j}^r\xi, \xi}\right>^{\frac{p}{2}}-\left<{T_{i, j}^r\xi, \xi}\right>^{\frac{q}{2}}}\right)^2. $



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