7 research outputs found
An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality
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 \A_i,\B_i,\X_i\in\bh (), , with
and and
are non-negative functions on which are continuous such that
for all , then \begin{equation*}
w^{2r}\bra{\sum_{i=1}^{n}\X_i\A_i^m\B_i}\leq
\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\norm{\sum_{i=1}^{n}\frac{1}{p}S_{i,j}^{pr}+\frac{1}{q}T_{i,j}^{qr}}-r_0\inf_{\norm{x}=1}\rho(\xi),
\end{equation*}
where ,
S_{i,j}=\X_i\phi^2\bra{\abs{\A_i^{j*}}}\X_i^*,
T_{i,j}=\bra{\A_i^{m-j}\B_i}^*\psi^2\bra{\abs{\A_i^j}}\A_i^{m-j}\B_i and
\rho(x)=\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}\bra{\seq{S_{i,j}^r\xi,\xi}^{\frac{p}{2}}-\seq{T_{i,j}^r\xi,\xi}^{\frac{q}{2}}}^2.Comment: No comment
An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality
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 (), , p, q > 1 with and and are non-negative functions on which are continuous such that for all , 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 , , 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. </p
New norm equalities and inequalities for operator matrices
Abstract We prove new inequalities for general 2 ร 2 operator matrices. These inequalities, which are based on classical convexity inequalities, generalize earlier inequalities for sums of operators. Some other related results are also presented. Also, we prove a numerical radius equality for a 5 ร 5 tridiagonal operator matrix
General Power Inequalities for Generalized Euclidean Operator Radius
In this paper, we establish generalizations and refinements for some results including upper bounds for the general Euclidean operator radius and the numerical radius
Regarding the Ideal Convergence of Triple Sequences in Random 2-Normed Spaces
In our ongoing study, we explore the concepts of I3-Cauchy and I3-Cauchy for triple sequences in the context of random 2-normed spaces (RTNS). Moreover, we introduce and analyze the notions of I3-convergence, I3-convergence, I3-limit points, and I3-cluster points for random 2-normed triple sequences. Significantly, we establish a notable finding that elucidates the connection between I3-convergence and I3-convergence within the framework of random 2-normed spaces, highlighting their interrelation. Additionally, we provide an illuminating example that demonstrates how I3-convergence in a random 2-normed space might not necessarily imply I3-convergence. Our observations underscore the importance of condition (AP3) when examining summability using ideals. Furthermore, we thoroughly investigate the relationship between the properties (AP) and (AP3), illustrating through an example how the latter represents a less strict condition compared to the former
General numerical radius inequalities for matrices of operators
Let Ai โ B(H), (i = 1, 2, ..., n), and T=[0โฏ0A1โฎโฐA200โฐโฐโฎAn0โฏ0]
T = \left[ {\matrix{
0 & \cdots & 0 & {A_1 } \cr
\vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu
\raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {A_2 } & 0 \cr
0 & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu
\raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu
\raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \cr
{A_n } & 0 & \cdots & 0 \cr
} } \right]
. In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials