Generalized Hausdorff Matrices Research Articles

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Lower bounds for matrices and its applications in frame theory

Lower bounds of non-negative triangular matrices and generalized Hausdorff matrices on the Taylor sequence space are considered. Some estimates are found for their lower bounds that depend on the ℓ1-norm of the columns of the Taylor matrix. Further, we show that similar estimates are obtained if we consider such matrices on the domain space of an arbitrary summability matrix Λ in ℓp, in which again the ℓ1-norm of the columns of the matrix Λ appears. As an application of such estimates to frame theory, we present the concept of Λ-frames for a separable Hilbert space H, as a special case of E-frames which were recently introduced in Talebi and Dehghan (2015). We study some properties of Λ-frames and specially Taylor frames. We characterize all Taylor orthonormal bases, Taylor Riesz bases and Taylor frames starting with an arbitrary orthonormal basis for H. Finally, we characterize all dual Taylor frames for a given Taylor frame.

Lower Bound for Matrices on the Fibonacci Sequence Spaces and its Applications in Frame Theory

Let \(A=(a_{n,k})_{n,k\ge 0}\) be a non-negative matrix. Denote by \(L_{\ell _p,F_q}(A)\) the supremum of those \(\ell ,\) satisfying the following inequality: $$\begin{aligned} {\left\{ {\sum \limits _{n = 0}^\infty {{{\left( {\sum \limits _{k = 0}^n {\frac{{f_k^2}}{{{f_n}{f_{n + 1}}}}\sum \limits _{i = 0}^\infty {{a_{k,i}}{x_i}} } } \right) }^q}} } \right\} ^{1/q}} \ge \ell {\left( {\sum \limits _{n = 0}^\infty {x_n^p} } \right) ^{1/p}}, \end{aligned}$$ where \(x\ge 0\), \(x\in \ell _p\) and \(\{f_n\}_{n=0}^\infty \) is the Fibonacci numbers sequence. In this paper, first we introduce the Fibonacci weighted sequence space, \(F_{w,p}~(0< p < 1)\), of non-absolute type which is the p-normed space and is linearly isomorphic to the space \(\ell _p(w)\), where the weight sequence \(w=\{w_n\}_{n=0}^\infty \) is a increasing, non-negative sequence of real numbers. Then we focus on the evaluation of \(L_{\ell _p,F_q}(A^t)\) for a lower triangular matrix A, where \(0<q\le p<1\). A similar result is also established for \(L_{\ell _p,F_q}(H_\mu ^\alpha )\) where \(H^\alpha _\mu \) is the generalized Hausdorff matrix, \(0<q\le p\le 1\) and \(\alpha \ge 0.\) In each case, a lower estimate is obtained which is related to the Fibonacci numbers. As an application of such estimate in frame theory, we present the concept of Fibonacci frames for a separable Hilbert space \(\mathcal {H},\) as a special case of E-frames which were recently introduced by the authors in Talebi and Dehghan (Banach J Math Anal 9(3):43–74, 2015). We study some properties of Fibonacci frames and characterize all Fibonacci orthonormal bases, Fibonacci Riesz bases and Fibonacci frames starting with an arbitrary orthonormal basis for \(\mathcal {H}\). Finally, we characterize all dual Fibonacci frames for a given Fibonacci frame.

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

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 ${(H_\mu ^\alpha )^t}$.

On double Hausdorff summability method

Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savas and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savas and Sevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices. MSC:40F05, 40G05.

The below boundedness of matrix operators on the special Lorentz sequence spaces

Let A=(an,k)n,k⩾0 be a non-negative matrix. Denote by Lℓp(w),weakℓq(A) the supremum of those L, satisfying the following inequality:supB{1(#B)1−1q∑n∈B(∑k=0∞an,kxk)}⩾L{∑n=0∞wnxnp}1p, where x={xk}k=0∞∈ℓp(w), x⩾0, w={wn}k=0∞ is a non-negative decreasing sequence and the supremum is taken over all nonempty subset B of non-negative integers with finite cardinal. In this paper we focus on the evaluation of Lℓp(w),weakℓq(At) for a lower triangular matrix A, where 0<q⩽p<1. A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for Lℓp,weakℓq(Hμα) where Hμα is the generalized Hausdorff matrix, 0<q⩽p<1 and α>0. A similar result is also established for the case in which (Hμα)t is replaced by Hμα. In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices and some special generalized Hausdorff matrices such as generalized Cesàro, generalized Hölder, generalized Gamma and generalized Euler matrices. Our results provide some analogue to those given by R. Lashkaripour, G. Talebi, Lower bound for matrix operators on the Euler weighted sequence space ew,pθ (0<p<1), Rend. Circ. Mat. Palermo 61 (1) (2012) 1–12; and C.-P. Chen, K.-Z. Wang, Operator norms and lower bounds of generalized Hausdorff matrices, Linear Multilinear Algebra 59 (3) (2011) 321–337.

Absolute Tauberian constants for H-J Hausdorff matrices

We obtain absolute Tauberian constants for the H-J generalized Hausdorff transformations. As corollaries we obtain the analogous results for the E-J generalized Hausdorff matrices, and the results of Sherif for regular Hausdorff matrices.

Lower bounds for generalized Hausdorff matrices and lower triangular matrices on the block weighted sequence space ℓp(w, F)

Let and be a non-negative matrix. Denote by , the supremum of those , satisfying the following inequalitywhere is a partition of positive integers, , is a non-negative sequence in and is a monotone and non-negative sequence of real number. In this paper a Hardy-type formula is obtained for , where is the generalized Hausdorff matrix, and Another purpose of this paper is to establish a general upper estimate for the exact value of , for which recently a lower estimate was established in Lashkaripour and Talebi [Lashkaripour R, Talebi G. Bull. Iran. Math. Soc. 2011;37:115–126], where is a non-negative lower triangular matrix and . We also derive the corresponding result for with In particular, we apply our results to summability matrices, weighted mean matrices, Nörlund matrices. Our results also generalize some results in Chen and Wang [Chen C-P, Wang K-Z. Linear Multilinear Algebra, March 2011;59:321–337] and Lashkaripour and Talebi [Lashkaripour R, Talebi G. Czech. Math. J. 2012; 62: 293–04.].

Operator norms and lower bounds of generalized Hausdorff matrices

Let A = (a n,k ) n,k≥0 be a non-negative matrix. Denote by L p,q (A) the supremum of those L satisfying the following inequality: The purpose of this article is to establish a Bennett-type formula for and a Hardy-type formula for and , where is a generalized Hausdorff matrix and 0 < p ≤ 1. Similar results are also established for and for other ranges of p and q. Our results extend [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 (2007), pp. 208–217] and [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 (2007) pp. 563–573] from to with α ≥ 0 and completely solve the value problem of , , and for α ∈ ℕ ∪ {0}.

Some E‐J Generalized Hausdorff Matrices Not of Type M

We show that there exists a regular E‐J generalized Hausdorff matrix which has no zero elements on the main diagonal and which is not of type M and establish several other related theorems.

Size of Convergence Domains for Generalized Hausdorff Prime Matrices

We show that there exit E-J generalized Hausdorff matrices and unbounded sequences such that each matrix has convergence domain .

TypeMfor E–J generalized quasi-Hausdorff matrices

In this paper, we determine sufficient condition for the transpose of an E–J generalized Hausdorff matrix to be of type M .

The question of equivalence for generalized Hausdorff matrices

In the present paper we investigate when Hausdorff matrices and generalized Hausdorff matrices, with the same mass function, are equivalent, as bounded operators on c and ℓ p .

Generalized Hausdorff and Weighted Mean Matrices as Operators on lp

Two theorems are proved. Theorem 1 establishes sufficient conditions for a generalized Hausdorff matrix H(λ, α) either to be in B(lp) or not to be in B(lp). Theorem 2 shows, inter alia, that if 1 ≤ p < ∞, an > 0, An ≔ a0 + a1 + ··· + an, and An/nan → c > 0, then the weighted mean matrix Ma with weights an is in B(lp) if and only if c < p. There are two examples about cases when the conditions of the theorems are not satisfied. A short proof of the fact that weighted mean matrices are special generalized Hausdorff matrices is also given.

The point spectra for certain classes of operators in B(? p )

In a recent paper [5] Maddox determined the point spectrum of the Cesaro matrices of order α>0, considered as series-to-series operators. In this paper we obtain the point spectrum for the Endl generalized Hausdorff matrices, the Hausdorff matrices, and for a wide class of generalized Hausdorff matrices, which include the Endl and ordinary Hausdorff matrices as special cases. We also obtain point spectrum results for a wide class of weighted mean matrices. Since the Cesaro matrices are examples of Hausdorff matrices, our results contain the corresponding theorems of Maddox as special cases.

Lp-Norms of Some Generalized Hausdorff Matrices: Corrections

<i>l^p</i>-Norms of Some Generalized Hausdorff Matrices: Corrections

Eigenvalues for generalized Hausdorff matrices

We introduce a new class of generalized Hausdorff matrices and show that their eigenvalues and corresponding eigenspaces can be obtained very easily. We also consider these matrices as operators on l p (1≤p≤∞).

Lp-Norms of Some Generalized Hausdorff Matrices

AbstractIn a recent paper D. Borwein [Math. Z. 183(1983), 483- 487] obtained an upper bound for the lp -norms of some generalized Hausdorff matrices, where the sequence (λn) satisfies the condition λn+1 ≦ λn + c, for some positive c. In this paper we obtained the lp -norms of these generalized Hausdorff matrices for which the mass functions are totally monotone.

Hardy's Inequality in Orlicz-type sequence spaces, for operators related to generalized Hausdorff matrices

Hardy's Inequality in Orlicz-type sequence spaces, for operators related to generalized Hausdorff matrices

Transformations of certain sequences of random variables by generalized Hausdorff matrices

Transformations of certain sequences of random variables by generalized Hausdorff matrices