Upper and Lower Bounds for Products of Multiplication Operator and Hausdorff Matrix on Block Weighted Sequence Spaces (original) (raw)
Lower bound and upper bound of operators on block weighted sequence spaces
Czechoslovak Mathematical Journal, 2012
Let A = (a n,k ) n,k 1 be a non-negative matrix. Denote by L v,p,q,F (A) the supremum of those L that satisfy the inequality where x 0 and x ∈ ℓp(v, F ) and also v = (vn) ∞ n=1 is an increasing, non-negative sequence of real numbers. If p = q, we use L v,p,F (A) instead of L v,p,p,F (A). In this paper we obtain a Hardy type formula for L v,p,q,F (Hµ), where Hµ is a Hausdorff matrix and 0 < q p 1. Another purpose of this paper is to establish a lower bound for A N M W v,p,F , where A N M W is the Nörlund matrix associated with the sequence W = {wn} ∞ n=1 and 1 < p < ∞. Our results generalize some works of Bennett, Jameson and present authors.
Lower Bounds for Matrices on Weighted Sequence Spaces
2007
This paper is concerned with the problem of finding a lower bound for certain matrix operators such as Hausdorff and Hilbert matrices on sequence spaces lp(w) and Lorentz sequence spaces d(w,p), which is recently considered in [7,8], similar to [13] considered by J. Pecaric, I. Peric and R. Roki. Also, this study is an extension of some works which are studied before in [1,2,7,8].
Lower bound for the norm of lower triangular matrices on block weighted sequence spaces
Journal of Mathematical Inequalities, 2011
Let 1 < p < ∞ and A = (a n,k ) n,k 1 be a non-negative matrix. Denote by A w,p,F , the infimum of those U satisfying the following inequality: where x 0 and x ∈ l p (w,I) and also w = (w n ) ∞ n=1 is a decreasing, non-negative sequence of real numbers. The purpose of this paper is to give a lower bound for A w,p,F , where A is a lower triangular matrix. In particular, we apply our results to Weighted mean matrices and Nörlund matrices which recently considered in on the usual sequence spaces. Our results generalize some work of Jameson, Lashkaripour, Frotannia and Chen in .
Linear Algebra and its Applications, 2012
AMS classification: 15A18 15A45 15A60 15B48 47B65 Keywords: Hadamard-Schur product Spectral radius Non-negative matrices Positive operators Generalized spectral radius Joint spectral radius Maximum circuit geometric mean Max algebra Matrix inequality Recently, Audenaert (2010) [2], Horn and Zhang (2010) [15], Huang (2011) [16] and Schep (2011) proved inequalities between the spectral radius ρ of Hadamard product (denoted by •) of finite and infinite non-negative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequalities even in the case of the usual spectral radius of non-negative matrices. In particular, we prove that
Norm and Lower Bounds of Operators on Weighted Sequence Spaces
Matematicki Vesnik
This paper is concerned with the problem of finding the upper and lower bounds of matrix operators from weighted sequence spaces lp(v;I) into lp(v;F). We consider certain matrix operators such as Cesaro, Copson and Hilbert which were recently considered in (7, 8, 11, 13) on the usual weighted sequence spaces lp(v).
Lobachevskii Journal of Mathematics, 2009
The main goal of the present study is to give some estimations for upper bound and lower bound of some matrix operators on weighted sequence spaces d(w, p) and l p (w). We considered this problem for certain matrix operators such as N¨orlund, Weighted mean, Ceas`aro and Copson matrices, which is recently considered in [7-13]. Also, this study is an extension of [2]. Moreover, we present the exact solution of the lower and upper estimates for some particular matrix operators.
Banach Journal of Mathematical Analysis, 2016
Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang ( ) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of n × n non-negative matrices.
Some inequalities involving upper bounds for some matrix operators I
Czechoslovak Mathematical Journal, 2007
In this paper we consider the problem of finding upper bounds of certain matrix operators such as Hausdorff, Nörlund matrix, weighted mean and summability on sequence spaces lp(w) and Lorentz sequence spaces d(w, p), which was recently considered in [9] and [10] and similarly to [14] by Josip Pecaric, Ivan Peric and Rajko Roki. Also, this study is an extension of some works by G. Bennett on lp spaces, see [1] and .
Lower Bounds of Copson Type for Hausdorff Matrices on Weighted Sequence Spaces
journal of sciences islamic republic of iran, 2011
Let H = , , 0 () nk n k h ≥ be a non-negative matrix. Denote by ( ) ,, , w p q LH µ where 0. qp −∞ < ≤ < In particular, we apply our results to the Cesaro matrices, Holder matrices and Gamma matrices. Our results also generalize some works due to R. Lashkaripour and D. Foroutannia (6). Moreover, in this study we extend some results mentioned in (3) and (4).
Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means
Computers & Mathematics with Applications, 2010
In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on those spaces.
A new sequence space and norm of certain matrix operators on this space
Communications in Mathematical Analysis, 2016
In the present paper, we introduce the sequence space [{l_p}(E,Delta) = left{ x = (x_n)_{n = 1}^infty : sum_{n = 1}^infty left| sum_{j in {E_n}} x_j - sum_{j in E_{n + 1}} x_jright| ^p < infty right},] where E=(En)E=(E_n)E=(En) is a partition of finite subsets of the positive integers and pge1pge 1pge1. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of finding the norm of certain matrix operators from lpl_plp into $ l_p(E,Delta)$, and apply our results to Copson and Hilbert matrices.
Upper lp-estimates in vector sequence spaces, with some applications
Mathematical Proceedings of the Cambridge Philosophical Society, 1993
In , Partington proved that if A is a Banach sequence space with a monotone basis having the Banach-Saks property, and (X n ) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space S A X n has this same property. In addition, Partington gave an example showing that if A and each X n have the weak Banach-Saks property, then S A X n need not have the weak Banach-Saks property.
Applications of Measure of Noncompactness in Matrix Operators on Some Sequence Spaces
Abstract and Applied Analysis, 2012
We determine the conditions for some matrix transformations fromn(ϕ), where the sequence spacen(ϕ), which is related to theℓpspaces, was introduced by Sargent (1960). We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.