Abstract
In this paper, the boundedness problem of factorizable four-dimensional matrices on the space of double sequences is investigated. As an application of our results, the lower bounds and the operator norms of four-dimensional Cesàro matrix and four-dimensional Copson matrix are obtained, which provide an extension of Hardy’s discrete inequality and Copson’s discrete inequality to double series, respectively. Finally, we present complementary results for the operator norm and lower bound of the four-dimensional Hausdorff matrices.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
By \(\Omega \), we denote the space of all real or complex valued double sequences which is the vector space with coordinatewise addition and scalar multiplication. The space \(\mathcal {L}_p\) of double sequences [3] is defined by
where \(1\le p<\infty ,\) which is a complete space with the norm
For more information on the normed spaces of double sequences and domain of triangle matrices in normed/paranormed sequence spaces, and the matrix transformations and summability theory, we refer the readers to the textbook [1] and the recent papers [2, 4, 7,8,9, 14, 17] and [21,22,23,24,25,26,27,28,29].
Let X and Y be two double sequence spaces and \(\mathsf {H}=(h_{nmjk})\) be a four-dimensional infinite matrix of real or complex numbers. Then, we say that \(\mathsf {H}\) defines a matrix mapping from X into Y, and we denote it by writing \(\mathsf {H}:X\rightarrow Y,\) if for every double sequence \(x=(x_{n,m})\in X\) the double sequence \(\mathsf {H}x=\{(\mathsf {H}x)_{n,m}\}\), the \(\mathsf {H}\)-transform of x, exists and is in Y, where
The purpose of this paper is to establish the lower bound and the operator norm of factorizable four-dimensional matrices as operators on the double sequence space \(\mathcal {L}_p\). For \(p\in \mathbb {R} \backslash \{0\}\), the lower bound involved here is the number \(L_{p}(\mathsf {H})\), which is defined as the supremum of those \(\ell ,\) obeying the following inequality
where \(x\ge 0,~x\in \mathcal {L}_p\) and \(\mathsf {H}=(h_{nmjk})\) is a nonnegative four-dimensional matrix. Also, we consider the upper bound \(k>0,\) of the form
for all nonnegative sequences x. The constant k is not depending on x. We seek the smallest possible value of k and denote the best upper bound by \(\left\| {\mathsf {H}} \right\| _{\mathcal {L}_p} \) as the operator norm of \(\mathsf {H}\) on \(\mathcal {L}_p .\) When we deal with two-dimensional matrices, we use, respectively, the notation \( L_p(\mathsf {A})\) and \(\left\| {\mathsf {\mathsf {A}}} \right\| _{\ell _p} \) for the lower bound and the operator norm of the matrix \(\mathsf {A}\) on \(\ell _p\), where \(\ell _p\) is the space of all real or complex \(p-\)absolutely summable sequences.
2 Main Results
Our main results are as follows.
Theorem 2.1
Let \(\mathsf {A}=(a_{nj})\) and \(\mathsf {B}=(b_{mk})\) be two nonnegative infinite matrices with the lower bounds \(L_p(\mathsf {A})\) and \(L_p(\mathsf {B})\), respectively. Let \(\mathsf {H}=(a_{nj}b_{mk})\) be the four-dimensional matrix constructed from \(\mathsf {A}\) and \(\mathsf {B}.\) Then,
Proof
Let \(x=(x_{jk})\) be a nonnegative double sequence in \({\mathcal {L}_p}\). Then,
This implies that
In order to see that one even has equality, look at double sequences of the form
Then, one has that
Now let \(\alpha > {L_p}\left( \mathsf {A}\right) \) and \(\beta > {L_p}\left( \mathsf {B}\right) \). Then, there exist nonnegative sequences \((\varsigma _j)\) and \(({\eta _k})\) such that
and
Then,
This implies that
Consequently,
\(\square \)
Theorem 2.2
Let \(\mathsf {A}=(a_{nj})\) and \(\mathsf {B}=(b_{mk})\) be two nonnegative infinite matrices with the norms \({\left\| \mathsf {A} \right\| _{{\ell _p}}}\) and \({\left\| \mathsf {B} \right\| _{{\ell _p}}}\), respectively. Let \(\mathsf {H}=(a_{nj}b_{mk})\) be the four-dimensional matrix constructed from \(\mathsf {A}\) and \(\mathsf {B}.\) Then,
Proof
The proof can be easily adapted from the one of Theorem 2.1 and so is omitted. \(\square \)
To provide some applications of the above Theorems, we refer the readers to the next two sections.
3 Extension of Hardy’s and Copson’s Inequalities
In this section, using Theorems 2.1 and 2.2, we are going to provide an extension of Hardy’s and Copson’s inequalities to double series. First, consider the Hardy’s inequality [13]
where \(x_k\ge 0\) for all \(k\in \mathbb {N}\) and the constant \(\left( \frac{p}{p-1}\right) ^p\) is the best possible. Inequality (3.1) can be rewritten as \({\left\| C(1)\right\| _{\ell _p} } = \frac{p}{p-1},\) where \(C(1)=(c_{nk})_{n,k\ge 0}\) is the Cesàro matrix of order 1, defined by
Now, consider the four-dimensional Cesàro matrix of order 1 and 1, \(\mathsf {C(1,1)}=\left( h_{nmjk}\right) \) defined by
Obviously, this matrix can be factorized as \(h_{nmjk}=c_{nj}c_{mk}\) where \({c_{nj}}\) and \({c_{mk}}\) are defined by (3.2). Applying Theorem 2.2, we have
This leads us to the following generalization of Hardy’s inequality.
Corollary 3.1
Let \({1< p < \infty } \) and \(x=\left( x_{nk}\right) \) be nonnegative sequence of real numbers in \({\mathcal {L}_p}\). Then,
The constant \({\left( {\frac{p}{{p - 1}}} \right) ^{2p}}\) in (3.3) is the best possible.
Inequality (3.3) is an extension of Hardy’s inequality to double series. It can be extended to multiple series [16].
Next, consider the Copson’s inequality [11] [see also ([13], Theorem 344)]
where \(x_k\ge 0\) for all \(k\in \mathbb {N}\). The inequality switch order when \(p>1\) and the constant \(p^p\) is the best possible. Again, inequality (3.4) can be rewritten as \(L_p\left( C(1)^t\right) =p,\) where \(C(1)^t\) denotes the transpose of C(1), and C(1) is the Cesàro matrix of order 1, defined by (3.2). The transpose of the Cesàro matrix is called Copson matrix. Now, consider the four-dimensional Copson matrix \(\mathsf {{C}}^{\mathsf {t}}({\mathsf {1,1}})=\left( h_{nmjk}\right) \) defined by
Since the four-dimensional Copson matrix \({\mathsf {C}}^{\mathsf {t}}({\mathsf {1,1}})\) can be factorized as \({h_{nmjk}} = {c^t_{nj}}{c^t_{mk}}\), where \({c_{nj}}\) and \({c_{mk}}\) are defined by (3.2), applying Theorem 2.1 we have
whenever \(0<p\le 1.\) Further, since \({\left\| C(1)^t\right\| _{\ell _p} } = p,\) by Theorem 2.2, we deduce that
for \(1<p<\infty .\) These lead us to the following generalization of Copson’s inequality.
Corollary 3.2
Let \(x=\left( x_{nk}\right) \) be nonnegative sequence of real numbers. Then
The inequality switch order when \(p>1\) and the constant \(p^{2p}\) is the best possible.
Inequality (3.5) is an extension of Copson’s inequality to double series. It can be extended to multiple series [18].
4 Complimentary Results for Four-Dimensional Hausdorff Matrices
Let \(\hbox {d}\mu \) and \(\hbox {d}\lambda \) be two Borel probability measures on [0,1] and \(\mathsf {H}_{\mu \times \lambda }=(h_{nmjk})\) be the four-dimensional Hausdorff matrix defined by [15]
for all \(n,m,j,k\in \mathbb {N}\). Clearly, we have
where
and
The four-dimensional Hausdorff matrix contains some famous classes of matrices. These classes are as follows:
- 1.
The choices \(\hbox {d}\mu (\alpha ) = \eta (1 - \alpha )^{\eta - 1} \hbox {d}\alpha \) and \(\hbox {d}\lambda (\beta ) = \gamma (1 - \beta )^{\gamma - 1} \hbox {d}\beta \) give the four-dimensional Cesàro matrix of order \(\eta \) and \(\gamma \) which is denoted by \(\mathsf {C}(\eta , \gamma )\).
- 2.
The choices \(\hbox {d}\mu (\alpha ) =\) point evaluation at \(\alpha =\eta \) and \(\hbox {d}\lambda (\beta ) =\) point evaluation at \(\beta =\gamma \) give the four-dimensional Euler matrix of order \(\eta \) and \(\gamma \) which is denoted by \(\mathsf {E}(\eta , \gamma )\),
- 3.
The choices \( \hbox {d}\mu (\alpha ) = \left| {\log \alpha } \right| ^{\eta - 1} /\Gamma (\eta )\hbox {d}\alpha \) and \(\hbox {d}\lambda (\beta ) = \left| {\log \beta } \right| ^{\gamma - 1} /\Gamma (\gamma )\hbox {d}\beta \) give the four-dimensional Hölder matrix of order \(\eta \) and \(\gamma \), which is denoted by \(\mathsf {H}(\eta , \gamma )\).
- 4.
The choices \(\hbox {d}\mu (\alpha ) = \eta \alpha ^{\eta - 1} \hbox {d}\alpha \) and \(\hbox {d}\lambda (\beta ) = \gamma \beta ^{\gamma - 1} \hbox {d}\beta \) give the four-dimensional Gamma matrix of order \(\eta \) and \(\gamma \) which is denoted by \({{\Gamma }}\left( {\eta ,\gamma } \right) \).
The four-dimensional Cesàro, Hölder and Gamma matrices have nonnegative entries whenever \(\eta >0\) and \(\gamma >0,\) and also the four-dimensional Euler matrices, when \(0<\eta <1\) and \(0<\gamma <1\).
The study of the boundedness problem of four-dimensional Hausdorff matrices goes back to the some recent works of the author. For example, it is proved in ([19], Theorem 3.1) that
Further, it is proved in ([20], pp. 7–8) that
and
According to the Hellinger–Toeplitz theorems ([6], Propositions 7.2 and 7.3), (4.1), (4.2) and (4.3), respectively, give
and
and
The proof of (4.1), (4.2) and (4.3), in those papers, is all based on the special version of four-dimensional Euler matrix. On the other hand, the four-dimensional Hausdorff matrix can be factorized as
where \({\mathsf {H}}_{{\mu }} = {( {h_{nj}^{(\mu )} })_{nj}}\) is the classical (two-dimensional) Hausdorff matrix [5] corresponding to the Borel probability measure \(\hbox {d}\mu \) (and similarly for \({\mathsf {H}}_{\lambda }\)). A similar factorization holds for the transpose \(\mathsf {H}_{\mu \times \lambda }^t.\) Therefore, (4.1), (4.2) and (4.3), can be obtained by a different way. In fact they are all special cases of Theorems 2.1 and 2.2, using the classical results on lower bounds and norms of the classical (two-dimensional) Hausdorff matrices in [6, 10, 12]. For example, to achieve (4.4), consider the transpose of the four-dimensional Hausdorff matrices as operators selfmap of the space \({\mathcal {L}_p}.\) Applying Theorem 2.2, we have the following result.
Theorem 4.1
Let p be fixed, \(1<p<\infty \) and \(\mathsf {H}_{\mu \times \lambda }\) be the four-dimensional Hausdorff matrix associated with the measures \(\mathrm{d}\mu \) and \(\mathrm{d}\lambda .\) Then, the transpose \(\mathsf {H}_{\mu \times \lambda }^t\) is bounded on \({\mathcal {L}_p}\) if and only if both \(\int _0^1 {{\alpha ^{\frac{{1 - p}}{p}}}\mathrm{d}\mu ( \alpha )} < \infty \) and \(\int _0^1 {{\beta ^{\frac{{1 - p}}{p}}}\mathrm{d}\lambda \left( \beta \right) } < \infty ,\) and we have
Proof
For the classical (two-dimensional) Hausdorff matrix \(\mathsf {H}_{\mu },\) it is proved by Bennett [5] that \(\mathsf {H}^\mathsf {t}_{\mu }\) is bounded on \(\ell _p\) if and only if \( \int _0^1 {{\alpha ^{\frac{{1 - p}}{p}}}\hbox {d}\mu (\alpha )}<\infty , \) and that \({\left\| {{\mathsf {H}^\mathsf {t}_{\mu }}} \right\| _{{\ell _p}}} = \int _0^1 {{\alpha ^{\frac{{1 - p}}{p}}}\hbox {d}\mu (\alpha )}.\) The result of the theorem is now a consequence of Theorem 2.2. \(\square \)
Let \(\mathsf {E}(\eta ,\gamma )\), \(\mathsf {C}(\eta , \gamma )\), \(\mathsf {H}(\eta , \gamma )\) and \({\Gamma }\left( {\eta ,\gamma } \right) \) be the four-dimensional Euler matrices, Cesàro matrices, Hölder matrices and Gamma matrices, respectively. Applying Theorem 4.1 to these matrices, we have the following corollary.
Corollary 4.2
Let p be fixed, \(1<p<\infty \) and \(\eta ,\gamma >0\). Then,
- 1.
\(\left\| \mathsf {E}^\mathsf {t}(\eta ,\gamma ) \right\| _{\mathcal {L}_p} =\left( \eta \gamma \right) ^{\frac{1-p}{p}}, \eta \le 1, \gamma \le 1.\)
- 2.
\(\left\| \mathsf {C}^\mathsf {t}(\eta , \gamma )\right\| _{\mathcal {L}_p}=\frac{{\Gamma \left( {\eta + 1} \right) {\Gamma ^2}\left( {\frac{1}{p}} \right) \Gamma \left( {\gamma + 1} \right) }}{{\Gamma \left( {\eta + \frac{1}{p}} \right) \Gamma \left( {\gamma + \frac{1}{p}} \right) }}.\)
- 3.
\(\left\| \mathsf {H}^\mathsf {t}(\eta , \gamma ) \right\| _{\mathcal {L}_p}=\frac{1}{{\Gamma \left( \eta \right) \Gamma \left( \gamma \right) }}\int _0^1 {\int _0^1 {{{\left( {\alpha \beta } \right) }^{\frac{p-1}{p}}}|\log \alpha {|^{\eta - 1}}|\log \beta {|^{\gamma - 1}}\mathrm{d}\alpha \times \mathrm{d}\beta } } .\)
- 4.
\(\left\| {\Gamma }^\mathsf {t}\left( {\eta ,\gamma } \right) \right\| _{\mathcal {L}_p} =\frac{{{p^2}\eta \gamma }}{{\left( {p\eta - p + 1} \right) \left( {p\gamma - p + 1} \right) }},\,\,\, {\eta> \frac{p-1}{p},\gamma > \frac{p-1}{p}} .\)
Putting \(\eta =\gamma =1,\) the second part of the above corollary implies
This says us that the second part of Corollary 4.2 is a generalization of Copson’s inequality (see Corollary 3.2).
To the best of our knowledge the exact values of \(L_p\left( {\mathsf {H}}_{\mu \times \lambda } \right) \) and \(L_p\left( {\mathsf {H}}_{\mu \times \lambda }^t \right) \) for \(1<p<\infty \) have not been found yet. Keeping in mind factorization (4.7) and the results obtained by Chen and Wang in ([10], Theorem 2.2) for \(L_p(\mathsf {H}_{\mu })\) and \(L_p(\mathsf {H}^\mathsf {t}_{\mu })\) where \(p>1\) and \(\mathsf {H}_{\mu }\) is the classical (two-dimensional) Hausdorff matrix, we now enable to fill up this gap by the use of Theorem 2.1.
Theorem 4.3
Let p be fixed, \(1<p<\infty \) and \(\mathsf {H}_{\mu \times \lambda }\) be the four-dimensional Hausdorff matrix associated with the measures \(\hbox {d}\mu \) and \(\hbox {d}\lambda \). Then,
and
We refer the readers to the paper [18] in which the operator norm and lower bound of some non-factorizable matrices are founded.
References
Başar, F.: Summability Theory and Its Applications, Bentham Science Publishers, e-books. Monographs, İstanbul (2012)
Başar, F., Çapan, H.: On the paranormed spaces of regularly convergent double sequences. Result. Math. 72(1–2), 893–906 (2017)
Başar, F., Sever, Y.: The space \(\cal{L}_p\) of double sequences. Math. J. Okayama Univ. 51, 149–157 (2009)
Başar, F., Çapan, H.: On the paranormed space \(M_u(t)\) of double sequences. Bol. Soc. Parana. Math. 37(3), 97–109 (2018)
Bennett, G.: Lower bounds for matrices II. Can. J. Math. 44, 54–74 (1991)
Bennett, G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120(576), 1–130 (1996)
Çapan, H., Başar, F.: Some paranormed difference spaces of double sequences. Indian J. Math. 58(3), 405–427 (2016)
Çapan, H., Başar, F.: On some spaces isomorphic to the space of absolutely q-summable double sequences. Kyungpook Math. J. 58(2), 271–289 (2018)
Çapan, H., Başar, F.: On the paranormed space \(L(t)\) of double sequences. Filomat 32(3), 1043–1053 (2018)
Chen, C.-P., Wang, K.-Z.: Lower bounds of Copson type for Hausdorff matrices II. Linear Algebra Appl. 422, 563–573 (2007)
Copson, E.T.: Note on series of positive terms. J. Lond. Math. Soc. 3, 49–51 (1928)
Hardy, G.H.: An inequality for Hausdorff means. J. Lond. Math. Soc. 18, 46–50 (1943)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)
Mursaleen, M., Başar, F.: Domain of Cesáro mean of order one in some spaces of double sequences. Stud. Sci. Math. Hungar. 51(3), 335–356 (2014)
Rhoades, B.E.: Some classes of doubly infinite matrices. Indian J. Pure Appl. Math. 23(6), 419–427 (2003)
Salem, Sh, Zareen, A.A., Khan, R.Redheffer: Hardy’s inequality in several variables. J. Math. Anal. Appl. 252(2), 989–993 (2000)
Talebi, G.: On boundedness of four-dimensional Hausdorff matrices on the double Taylor sequence spaces. Indag. Math. 28(5), 953–961 (2017)
Talebi, G.: Operator norm and lower bound of four-dimensional matrices. Indag. Math. 28(6), 1134–1143 (2017)
Talebi, G.: Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces. Linear Multilinear Algebra 65(11), 2257–2267 (2017)
Talebi, G.: Lower bound of four dimensional Hausdorff matrices (2018) (preprint)
Yeşilkayagil, M., Başar, F.: Domain of Riesz mean in some spaces of double sequences. Indag. Math. 29(3), 1009–1029 (2018)
Yeşilkayagil, M., Başar, F.: A note on Abel summability of double series. Numer. Funct. Anal. Optim. 38(8), 1069–1076 (2017)
Yeşilkayagil, M., Başar, F.: Domain of Riesz mean in the space \(L_s\). Filomat 31(4), 925–940 (2017)
Yeşilkayagil, M., Başar, F.: A note on Riesz summability of double series. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 86(3), 333–337 (2016)
Yeşilkayagil, M., Başar, F.: Some topological properties of the spaces of almost null and almost convergent double sequences. Turk. J. Math. 40(3), 624–630 (2016)
Yeşilkayagil, M., Başar, F.: Mercerian theorem for four-dimensional matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65(1), 147–155 (2016)
Yeşilkayagil, M., Başar, F.: On the characterization of some classes of four-dimensional matrices and Steinhaus type theorems. Kragujev. J. Math. 40(1), 35–45 (2016)
Yeşilkayagil, M., Başar, F.: Comparison of four-dimensional summability methods. Aligarh Bull. Math. 34(1–2), 1–11 (2015)
Yeşilkayagil, M., Başar, F.: Four-dimensional dual and dual of the new sort summability methods. Contemp. Anal. Appl. Math. 3(1), 13–29 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fuad Kittaneh.
The paper is dedicated to the doyen of the martyrs, the chief of the youth of paradise, Imam Hossein ibn Ali (peace be upon him) in the memory of the 1379th occasion of his Arbaeen.
Rights and permissions
About this article
Cite this article
Talebi, G. Complementary Results on the Boundedness Problem of Factorizable Four-Dimensional Matrices. Bull. Malays. Math. Sci. Soc. 43, 609–618 (2020). https://doi.org/10.1007/s40840-018-00703-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-018-00703-7