Transpose of the Weighted Mean Matrix on Weighted Sequence Spaces

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1 Transose of the Weighted Mean Matri on Weighted Sequence Saces Rahmatollah Lashkariour Deartment of Mathematics, Faculty of Sciences, Sistan and Baluchestan University, Zahedan, Iran Fa: Abstract: In this aer, we concern with transose of the weighted mean matri This is uer triangular matri. on weighted sequence saces l w and L w which is considered by the author in [8] and [9] for secial case of these oerator, such as Coson on l w and dw,. Also, in a recent aer[7], the author has discovered the uer bound for the Coson oerator on the weighted sequence saces dw,. Also, we establish analogous uer bound for the continuous case. The weighted mean matrices are considered by the author in []. Key Words:Transose of Weighted Mean Matri,Weighted Sequence Sace. 2 Mathematics subject classification. 26D5, 26A48, 47B37.. Introduction and Notations: In this note, we consider the roblem of finding the norm of transose of the weighted mean matri A d a n,k, denoted by A t d, where a n,k { dk D n for k n for k > n. where the d n s are non-negative numbers with artial sum D n d d n We insist that d >, so that each D n is ositive.. These results are etension of some results which is considered by the author in [8] and [9] and Bennett[2] and [4]. If r n n w k d k D n, and also R n and are defined as usual, then the norm of A t d on l w is the suremum of Rn. Let w be a decreasing, non-negative sequence with lim n and divergent. Write w Then l w and the Lorentz sequence sace dw,, where, is the sace of sequences n with lw n /, w, n / convergent, where n is the decreasing rearrangement of n. We now consider the oerator A defined by A y, where y n a n,k k. We shall write A lw for the norm of A when regarded as an oerator from l w to l w, where A lw su{ A lw : lw }, A w, su{ A w, : w, }. Also, we define M w, A su{ A lw : lw }, where n is regarded as a decreasing, nonnegative sequences in l w. We assume that a n,k for all n, k. This imlies that A A for all, and hence the nonnegative sequences are sufficient to determine A lw. We assume further that each Ae k is in l w, that is:

2 2 a n,k is convergent for each k, that garantte each Ae k is in l w. For two finite sequence n and y y n, write y if Y k X k k, where X k k i i and Y k k i y i. Lemma : Suose, y IR n with y and a i is decreasing. Suose also either a n, or X n Y n. Then a k k a k y k. Proof: By Abel summation, it follows that a k y k a k Y k Y k Y n Y k a k a k+ + a n Y n. Now, alying the hyothesis in both cases, we deduce that n n Therefore Y k a k a k+ + a n Y n X k a k a k+ + a n X n a k k a k y k. a k k. Corollary: Let, y be decreasing, nonnegative elements of IR n or l w with y. Then y l w. l w Proosition [8], Proosition.4.: Suose that holds, and that 3 for all subsets M, N of IN having m, n elements resectively, we have a i,j i M j N m a i,j. i j Then A l w A l w A w, A w, for all non-negative elements of l wdw,, where is the decreasing rearrangement of n. Hence decreasing, non-negative sequences are sufficient to determine A l w A w,. Proosition 2[3], Lemma 9: Let A a i,j i,j be a matri oerator with non-negative entrie s, and consider the associated transformation, y, given by y i j a i,j j. Then the following conditions are equivalent: i y y 2... whenever ii r i,n r i+,n i, n, 2,..., where r i,n n j a i,j. Proof: i ii follows by taking to be the sequence,...,,,... of n ones followed by zeros. ii i: By Abel summation, it follows that y i a i,j j r i,n n n+. j Since r i,n r i+,n i, n, and also n is decreasing, non-negative sequence, then r i,n n n+ r i+,n n n+ a i+,j j y i+. j This comletes the roof of the statement. Lemma 2: Suose u u n and w are sequences of ositive numbers. i If m un M for all n, then m Un M for all n. un ii If so is Un. u n is increasing or decreasing, then iii If U as n, then Un U as n also with U. Proof. Elementary. 3. Transose of the Weighted Mean oerator Let now A d be the weighted mean matri with roerties, 2 and 3, and A t d be its transose which is defined as follows: A t dn kn d n k D k.

3 This is an uer triangular matri. Recall that is said to be -regular if r w su n n is finite[]. A leasently simle statement can also be made about the norm of the weighted mean matri oerator for general w. With the revious notation, r n n w n. Theorem : Suose A t d is a weighted mean oerator defined as before and also d d n is such that nd n D k k n. If w is -regular, then for >, we have: A t d w, r w & A t d lw r w. Proof: As mentioned before, it is sufficient to consider decreasing, non-negative sequences. Let be in l w or dw, such that 2..., and so w, lw and also the same is true for the norm of A t d. Then alying [], Theorem 4..6, we deduce that: A t d d n k w, D kn k k k kn r w n. r w w,. Hence A t d w, r w. This comletes the roof. Theorem 2: Suose A t d is an oerator on l w. If R su R n <, where r n n w k d k D n, and R n r r n and as usual. Then A t d is a bounded oerator from l w into itself, and we have M w, A t d R A t d w,. Proof: Since A t d n At d n for all n, it is sufficient to consider decreasing, nonnegative sequences. Let be in l w such that Then A t d w, A t d l w R R kn r n n d n k D k R n n n+ n n+ n. Hence A t d w, M w, A t d R. We have to show that this constant is the best ossible. We take... n and k for all k n +. Then l w & A t d l w R n. Therefore M w, A t d R At d w,. 3. Coson Oerator on Weighted Sequence Saces: We now consider the Coson oerator C on l w and dw,, which is defined by y C, where y i ji j j. It is given by the transose of the matri of the Averaging oerator A: a i,j { j for i j for i > j This is an uer triangular matri. The classical inequality of Coson [5] and [6] states that C A t > as an oerator on l saces. Proosition 3: If w is -regular, then C mas l w into l w. Also, we have C w, M w, C C l w r w. Proof: Since r n n r w n,

4 then by Lemma 2i and Theorem 2, it follows that C w, M w, C C l w r w. Corollary [8], Theorem 2.3.: If su W k k <, then the Coson oerator is a bounded oerator from dw, into itself, and also we have Write M w, C C w, su u n n r, r > v n n n W k k. t r dt andas usual U n u u n, etc. For r <, the usual integral comarison gives or v v n U n V n, r n r U n n r r, we need to know that Un V n is increasing. The following is the key lemma. Lemma 3: With v n as above for any r >, n r v n decreases with n and n r v n+ increases with n. Proof: Write t n n r v n. Then t n+ n+ r n+ n ds s n+r For n s n, we have n+ n+ r s+ r n. Simi- nr s. Therefore t r n+ t n larly for the second statement. n n n s+ s ds s + r., hence Proosition 4: Let < r < and let U n n j j. Then Un increases and tends n r to r. u n vn Proof: With v n as above, by Lemma 3, increases with n, and so alying Lemma, we U deduce that n V n is increasing. The limit follows from the inequalities above. We now consider the tail of the series for ζ+. For the tail of a series, the analogous result to Lemma 2ii is the following. Lemma 4: Suose that v n >, u n > for all n and that v n and u n are convergent. Let U n jn u j, similarly V n. un vn If is increasing or decreasing, then so is Un V. n Proof: Elementary. Proosition 5: Let r > and let U n jn. Then n j r U +r n decreasing, n r U n increasing. Both tend to r as n. Proof: Let u n and v n +r n n n dt. t +r Then V n+ rn. By the usual integral comarison, r rn r U n rn r, which imlies the stated limits. By Lemma u 3, n v n+ is decreasing, so by Lemma 2ii, U n V n+ rn r U n is decreasing. Similarly, U n V n increasing. Remark: This is stated without roof in [], Remark 4.. Theorem 3: If w n, <, then the Coson oerator C is a bounded oerator from l wdw, into itself. Also, we have M w, C C w, C l w. Proof: Since r n wn n, then r n n n. Also, since is the U n of the Proosition 4, W then n increases with n and tends to n. Hence alying Proosition 2, we deduce that M w, C C w, C l w. Remark: When, so that n, we have r n as n, so the Coson oerator C is not a bounded oerator on dw,, although of course is satisfies condition 2. is

5 Theorem 4: Let be defined by n, where < <. Then the Coson oerator is a bounded oerator from dw, into itself. Also, we ahve C w, M w, C Proof: We now have R n W k k n. k, so the new Rn is eactly the rn of Theorem 3 and Proosition 4 again gives the statement. 4. Continous Version of the Coson Oreator: In this section, we consider the analogous roblem for the continuous case concern the sace L w. In the continuous case, the Coson oerator C is given by: Cf ft dt. t Let w be a decreasing, non-negative function on,. We assume that W wtdt is finite for each Hence is ermited for < α <, but not for α.. α Then L w is the sace of functions f having w f d convergent, with norm / f LP w w f d. Proosition 6: Let f be in L w, a w W f, and also A atdt. Then A is finite and also we have: A Lw f L w. Proof: Fi. For any <, let atdt A. Then d d A A a, and so A A A A t atdt. Hence, alying Holder s inequality, we deduce that: wa d w A t atdtd A t at t wddt A t atw tdt wta t atftdt / / wtft dt wta t dt. Therefore wta t dt / f Lw. The above inequality is true for all >, and so true with relacing by infinity. This comletes the roof. Proosition 7: If W w r w >, then C Lw r w. Proof: We have t r w wt W t, and so wt Cf r w W t ftdt r wa. This establishes the statement. Theorem 5: If w, where α <, then α C Lw α. Attained by action of C on decreasing ositive functions. Proof: i We have W α α, and so Hence W w α C Lw >. α.

6 ii Now, by taking ε >, and define r by: α + r + ε, we deduce that: f { r for for <. Then f is decreasing and in L w, since d α and d are convergent. Also, we have: α+r Cf and also we have: dt tr+ r r for, [9] R. Lashkariour, Oerators on Lorentz sequence saces I, Indian Journal of Pure and Alied Mathematics, to aear. [] R. Lashkariour, Weighted Mean Matri on Weighted Sequence Saces, WSEAS TRANSACTION on MATHEMATICS, Issue 4, Volume 3, 24, [] S. Reisner, A factorization theorem in Banach lattices and its alications to Lorentz saces, Ann. Inst. Fourier 398, Cf Cf r for < <. Hence Cf r f >, and so Cf Lw r f L P w, where r α+ε. Now, alying i and ii imlies the statement. References [] G. Bennett, Factorizing the Classical Inequalities, Mem. Amer. Math. Soc [2] G. Bennett, Inequalities comlementary to Hardy, Quart. J. Math. Oford , [3] G. Bennett, Lower bounds for matrices II, Canadian J. Math , [4] G. Bennett, Some elementary inequalities, Quart. J. Math. Oford , [5] E. T. Coson, Notes on series of ositive terms, J. London Math. Soc. 2927, 9-2. [6] G. H. Hardy, J. Littlewood and G. Polya, Inequalities, Cambridg Univ. Press, 2. [7] G.J.O. Jameson and R. Lashkariour, Norms of certain oerators on weighted l saces and Lorentz sequence saces, Journal of inequalities in Pure and Alied Maths. Volum3, Issue, 22 Article 6-7. [8] R. Lashkariour, Lower bounds and norms of oerators on Lorentz sequence saces, Doctoral dissertation Lancaster, 997.