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Inequalities involving upper bounds for certain matrix operators. (English) Zbl 1118.47024

The authors work with Hausdorff operators and some other classical matrix operators on weighted sequence spaces and obtain upper bounds of their norms (or the norms themselves) in some special cases. The results extend those of G.Bennett [Q. J.Math., Oxf.II.Ser.49, No.196, 395–432 (1998; Zbl 0929.26013)] and G.J.O.Jameson and the first author [J. Inequal.Pure Appl.Math.3, No.1, Paper No.6, 1–17 (2002; Zbl 1021.47019)].

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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References:

[1] Bennett G, Factorizing the classical inequalities,Mem. Am. Math. Soc. 576 (1996) 1–130 · Zbl 0857.26009
[2] Bennett G, Inequalities complimentary to Hardy,Quart. J. Math. Oxford (2) 49 (1998) 395–432
[3] Hardy G H, Divergent Series, 2nd edition (American Mathematical Society) (2000)
[4] Hardy G H, Littlewood J E and Polya G, Inequalities, 2nd edition (Cambridge: Cambridge University Press) (2001)
[5] Jameson G J O and Lashkaripour R, Lower bounds of operators on weightedl p spaces and Lorentz sequence spaces,Glasgow Math. J. 42 (2000) 211–223 · Zbl 0965.47016 · doi:10.1017/S0017089500020061
[6] Jameson G J O and Lashkaripour R, Norms of certain operators on weightedl p spaces and Lorentz sequence spaces,J. Inequal. Pure Appl. Math. 3(1) (2002) 1–17 · Zbl 1021.47019
[7] Lashkaripour R, Operators on Lorentz sequence spaceII,WSEAS Trans. on Systems 1(1) (2002) 16–22 · Zbl 1290.47029
[8] Lashkaripour R, Weighted mean matrix on weighted sequence spaces,WSEAS Trans. on Math. 3(4) (2004) 789–793 · Zbl 1205.40009
[9] Lashkaripour R, Transpose of the weighted mean operators on weighted sequence spaces,WSEAS Trans. on Math. 4(4) (2005) 380–385
[10] Pecari J, Peric, I and Roki R, On bounds for weighted norms for matrices and integral operators,Linear Algebra and Appl. 326 (2001) 121–135 · Zbl 0985.15021 · doi:10.1016/S0024-3795(00)00310-4
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