## Annals of Functional Analysis

### Approximation of lower bound for matrix operators on the weighted sequence space $C_{p}^r(w)(p \in (0,1))$

#### Abstract

Let $A=(a_{n,k})_{n,k\geq 0}$ be a non-negative matrix. We denote by $L_{\ell_p(w),C_{q}^r(w)}(A)$ the supremum of those $\ell,$ satisfying the following inequality: ${\left( {\sum\limits_{n = 0}^\infty {{w_n}{{\left( {\frac{1}{{{{\left( {1 + r} \right)}^n}}}\sum\limits_{k = n}^\infty {\frac{{{{\left( {1 + r} \right)}^k}}}{{1 + k}}\sum\limits_{j = 0}^\infty {{a_{k,j}}{x_j}} } } \right)}^q}} } \right)^{1/q}} \ge \ell {\left( {\sum\limits_{n = 0}^\infty {{w_n}x_n^p} } \right)^{1/p}},$ where $x\geq 0$, $x\in \ell_p(w)$, $r\in (0,1)$, $q\le p$ are numbers in $(0,1)$ and $\left(w_n \right)_{n=0}^\infty$ is a non-negative and non-increasing sequence of real numbers. In this paper, first we introduce the weighted sequence space $C_{p}^r(w)~(p \in (0,1))$ of non-absolute type which is a $p$-normed space and is isometrically isomorphic to the space $\ell_p(w)$. Then we focus on the evaluation of $L_{\ell_p(w),C_{q}^r(w)}(A^t)$ for a lower triangular matrix $A$, where $q\le p$ are in $(0,1)$. A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for $L_{\ell_p,C_{q}^r}(H_\mu ^\alpha )$ where $H^\alpha_\mu$ is the generalized Hausdorff matrix, $q\le p\le1$ are positive numbers and $\alpha\geq 0.$ A similar result is also established for the case in which $H^\alpha_\mu$ is replaced by ${(H_\mu ^\alpha )^t}$.

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 3 (2015), 203-215.

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.afa/1429286043

Digital Object Identifier
doi:10.15352/afa/06-3-17

Mathematical Reviews number (MathSciNet)
MR3336916

Zentralblatt MATH identifier
06441319

#### Citation

Talebi, Gholamreza; Salmei, Hossein. Approximation of lower bound for matrix operators on the weighted sequence space $C_{p}^r(w)(p \in (0,1))$. Ann. Funct. Anal. 6 (2015), no. 3, 203--215. doi:10.15352/afa/06-3-17. https://projecteuclid.org/euclid.afa/1429286043

#### References

• B. Altay, F. Basàr and M. Mursaleen, On the Euler sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$ I, Information Sci. 176 (2006), no. 10, 1450–1462.
• G. Bennett, Factorizing the classical inequalities, Mem. Amer. Math. Soc. 120 (1996), no. 576, 1–130.
• G. Bennett, Inequalities complimentary to Hardy, Quart. J. Math. Oxford 49 (1998), no. 4, 395-432.
• C.P. Chen and K.Z. Wang, Lower bounds of Copson type for the transposes of lower triangular matrieces, J. Math. Anal. Appl. 341 (2008), no. 2, 1284–1294.
• C.P. Chen and K.Z. Wang, Operator norms and lower bounds of generalized Hausdorff matrices, Linear Multilinear Algebra 59 (2011), no. 3, 321–337.
• E.T. Copson, Note on series of positive terms, J. London Math. Soc. s1-3 (1928), no. 1, 49–51.
• M.A. Dehghan, G. Talebi, A.R. Shojaeifard, Lower bounds for generalized Hausdorff matrices and lower triangular matrices on the block weighted sequence space $\ell_p(w,F)$, Linear Multilinear Algebra 62 (2014), no. 1, 126–138.
• D. Foroutannia, Upper bound and lower bound for matrix operators on weighted sequence space, Doctoral Dissertation, Zahedan, 2007.
• R. Lashkaripour and G. Talebi, Lower bound for matrix operators on the Euler weighted sequence space $e_{w,p}^\theta(0<p<1)$, Rend. Circ. Mat. Palermo 61 (2012), no. 1, 1–12.
• G. Talebi and M.A. Dehghan, The below boundedness of matrix operators on the special Lorentz sequence spaces, Linear Algebra Appl. 439 (2013), no. 8, 2411–2421.