I I L an,klr ~ Mr, n~l

Transcription

1 MATHEMATICS ON MATRIX TRANSFORMATIONS OF CERTAIN SEQUENCE SPACES BY H. R. CHILLINGWORTH (Commuicated by Prof. J. F. KoKS!IIA at the meetig of Jue 29, 1957) I this paper we determie the characteristics of certai matrix trasformatio spaces ([1], 298), with special emphasis o the trasformatios of the space of bouded sequeces. A extesio of part of a theorem of H. S. ALLEN ([2], Theorem 3, (a)) to covergece free spaces is also give. We have, from Alle's theorem, the followig: If Z<,ex<,a ([3], 273), the Z--+- f3=ex--+- f3=a --+- {3, where f3 is ay perfect sequece space. ([3], 275). Thus i cosiderig the trasformatios of such a ex to various perfect sequece spaces, we may cosider merely the trasformatios of a ; ad this we shall do. The results will the be immediately applicable to ex --+- f3 with ex as defied. I particular, ex may be z or r. Theorem I. a -+- a, (r;;.l) is the set of matrices A= (a,) such that I I L a,lr ~ Mr, ee where E is a arbitrary subset of the positive itegers, ad M is idepedet of E. To prove ecessity we shall employ the method of proof adopted by LoRENTZ ([4], 244) for the particular case r= l. For sufficiecy, which we cosider first, we employ a theorem of KoTHE ad ToEPLITz. (See [1 ], 288.) Let E be a arbitrary subset of the positive itegers. Let X be the set of all X= {x} such that lxl < l, (= l, 2,... ). The X is p-bd. i a ([3], 298, (10.4, III).) Hece, if A= (a.) E a -+- a,, the AX is p-bd. i a, ([1], 288, (6.2, VIII)), i.e., if y=ax, x EX, the L IYl' <,111\ where M is a costat, for every y ([3], 299, (10.4, V); 298, (10.4, IV).) Thus for every x i X. I I L a,xlr ~ M'

2 579 Tae x" = I, E E, ad x = 0 otherwise. The x E X, ad hece co L I L a,l'' ~ Mr, EE ad the coditio has bee proved ecessary. I provig sufficiecy, we cosider first real (a. ) ad real bouded sequeces, ad the exted the result to complex sequeces ad matrices. Let {x} be a sequece of positive terms, such that, as varies, X~.; assumes oly a fiite umber of differet values, ad let bd lxl <I. We may the always write x=.l bpxrpl, where the xrpl = {xlfl} are sequeces p of O's ad I's, ad bp> 0,.L bp< I, ad p assumes oly a fiite umber p of values i the summatio. Cosider the fuctio I this case, co F(x) == { L I L a, X lr}iir. =l F(x) ={ L I La L bp xlfl 1"}11", p ad, the bp beig fiite i umber, this may be rearraged as F(x) = { L I L bp L a, xlfl llr. p Hece, by Miowsi's iequality, F(x) ~.L bp F(xrPl) ~ M p if the coditio o A= (a. ) is satisfied, sice F(xrPl) < M ad _LbP <I. If egative values of x are admitted, we obtai F(x) <2M (agai by a simple applicatio of Miowsi's iequality). But these {x}, with x assumig a fiite umber of differet values betwee -I ad +I as varies, are dese i the set of real bouded sequeces with uit boud, sice for a give e > 0, the rage - I to + I divided equally ito [2/e] +I segmets will provide a value of x, for each, withi a distace e of every poit i the rage. Hece F(x) <2M for ay sequece i G with llxll <I, ad F(x) <oo for all sequeces i G This proves the theorem for real sequeces ad matrices. Now cosider the sufficiecy of the coditio if the matrix, still regarded as real, is applied to complex bouded sequeces {z}, where z=x+iy, ad the X~.; ad Y are both real ad bouded. We have, if the coditio holds, L I L a,lr~ Mr. EE

3 Now 580 L I L (a.,.x,+iay!jir~ LIA+iBlr, where A= L ax, B- LaY, ad A ad B are real. Sice the matrix applies to all real bouded sequeces ad trasforms them to sequeces i a., we have also Now LIAl 7 < oo, Ll Bl 7 < r> ( L I A+ i B IT) l/r ~ ( L I A IT) llr + ( L I B I'< ' by Miowsi's iequality. Thus L I L ai' (x+i Y~c)lr< ad the result follows for real matrices. Similarly, if a=b~c+ie' with b ad e real for all ad, the for a real sequece {x} we require that oo, L I L b e+i L e x,j ~ L I P+i QiT say, < oo. The proof ow proceeds as before, ad we see that the covergece of L IPlr ad L IQlr is ecessary ad sufficiet for the covergece of the left had expressio, which meas that the matrices (b, ) ad (e. ~c) separately satisfy the give coditio. The argumet may ow be exteded to complex sequeces ad complex matrices, ad we see that it is ecessary ad sufficiet for the real ad imagiary parts of the matrix elemets separately to satisfy the give coditio. It remais oly to show that, if the L I L a. IT~ Mr, ee L I L b. lr ad L I L e. lr ee ee are each bouded; ad this is obvious, sice The theorem is ow proved. Er is the space of all sequeces such that lx~cl < N (r > 0) for every ([3], 274, xiv). Fr is the space of all sequeces such that L Tix~cl coverges (r> 0) ([3], 274, xv).

4 581 These spaces are perfect, ad E; = F, ([3], 277, (10.1, IX)). The followig result will be required i the ext matrix trasformatio theorem. Lemma. A set Y is Er-bd. if, ad oly if, IYfrl is bouded for all yi Y ad for every ; i.e., if IYl <.M' for every yi Y. Suppose first that this coditio is satisfied. The if u E F, ILYul~~llf L'lul, which coverges by defiitio of F,. Thus the coditio is sufficiet. To prove ecessity, let v ='u, the v is i av if, ad oly if, u is i F,. Thus if y E Y, I LY ul =I L~ rul =I L ~ vl ~N(u), by p-boudedess. But {v} is i a 1, ad otherwise arbitrary. Hece {Yf'} is 0" - coditio for which is IYfrl <.M ([3] 10.4, III). This proves the lemma. bd., the Theorem II. 0" --+ E, is the set of matrices A= (a. ) such that L Iaiei <.Nr, for some positive N ad all. The coditio is sufficiet, sice where M' =bd. lxl Thus if A satisfies the coditio of the theorem, I Y I""' I La x,, I~ M' N '. Replacig J.lf'N by M gives IYl <.M', whece y E E,. To prove ecessity, cosider the trasformatio of the set of sequeces i 0" give by x = ei 0. This is collectively bouded as the () vary, so will trasform, by [I], 288, (6.2, VIII(b)), ito a Er-bouded set; i.e., from the lemma, I Y I """ I La eio I ~ M ' for some positive M idepedet of the choice of the (), ad for all. Choose the () such that, for some fixed, arg a = -fj (= l, 2,... ). The IYl = L ia. l <.M' for this. But by varyig the () we may obtai this result for all. Hece the coditio is ecessary. Theorem III. a,--+f, is the set of all matrices A=(a,) such that L 'l L a,,l ee coverges for ay arbitrary set E of the positive itegers. By a method similar to that employed i the lemma precedig Theorem

5 582 II, we easily establish that the coditio that a set Y E F, is p-bd. is that I 'IYl < M for some positive M ad all y E Y. Let X be the set of all x={x.,} such that lxl<l (=l, 2,... );the, exactly as i theorem I, we have that, if y=ax, x EX, the I 'IYl.;;;M. i.e. I,IIa.xi~M. ll Let x"= l, E E, x=o otherwise. The x EX, ad hece I,l I a.l ~M. ~! EE This shows that the coditio is ecessary. The proof of sufficiecy follows exactly o the lies of the correspodig result for a -+ a 10 so will ot be give. We metio also without proof two further trasformatios of a, both of which are sufficietly obvious. (a) <r -+ a is the set of all matrices with rows i a 1 (b) a -+ cp is the set of all colum-bouded matrices with rows i a 1 As i all these cases {J, the space to which a has bee trasformed, is perfect, we may apply Alle's theorem [2], 3(b ), that (<X -+ {J)' = fj* -+ <X*, ad obtai the followig corollaries to the above theorems. The coditios o A i each case are as follows. I(a) I a.-+ a 1, I I I a,. t.lr < M" for all arbitrary subsets E of the positive itegers. EE II( a) I Fr ->- a 1, I I a. l < Nr for all ad some positive N. ~! III( a) I E,-+ a 10 I 'l I a,. l coverges for all arbitrary subsets E EE of the positive itegers. (a)' I 4>-+ a 10 colums of A are i a 1 (b)' I a-+ a 10 (i) A is row-bouded, (ii) Colums of A are i a 1 Theorem IV. If (i) <X> cp, (ii) fj is covergece-free, the <X -+ fj is the set of all matrices A= (a ) such that (a) rows of A are i <X*. (b) row suffixes of o-zero Tows of A form a W-set for {J. (See [3], 281 for defiitio of W-set.) Coditio (a) is ecessary i order that A shall apply absolutely to <X. (a) ad (b) are obviously together sufficiet. To show that (b) is ecessary, assume that the pi" row is ot a zero row, ad that pis ot i a W-set for {J. Let the first o-zero term of the row be ap., O applyig the matrix to e',> we obtai yp (A e''>)p = ap., =1= 0. But for all sequeces i {J, yp = 0. This cotradictio proves the ecessity of the coditio, ad thus completes the proof of the theorem.

6 583 Theorem V. If (i) IX is ormal ad cotais cp, (ii) {3 is covergecefree, (iii) ~X**?A?IX, the IX--+ f3=a.--+ f3=l(a., {3)=1X**--+ {3. (cf. [2], 375, theorem 3(a)). For, sice X*** =lx*, it follows from theorem IV that IX--+ {J=~X**--+ {3. Ad sice, from (iii), ~X***;;;.A.*;;;.~X*, it follows that A.*=~X*, theorem IV the gives lx --+ {3 = }, --+ {3 =IX** --+ (3. But sice IX IS ormal, IX --+ {3 = L(IX, {3); ad as IX --+ {3 =A. --+ {3 < L(A., {3) < L( IX, {3) =IX --+ {3' the theorem follows. If IX is ot ormal, but IX;;;. cp, we still have IX->- {J=A.--+ {3=1X** ->- {3, where {3 is covergece-free. We coclude with a result derived from Alle's theorem ([2], 375) quoted above. Theorem VI. If (i) lx is ormal ad cotais cp, (ii) X<; A.<:, ~X**, the _L(A.) is a rig ([I], 3ll, 6.5). For _L(A.) <A. --+ lx **<IX --+ lx **, from the defiitios; ad from (i) ad (ii), IX --+IX**= _L(IX **), by [2], 375, 3(a) (sice IX** is :perfect). Thus _L(A.)< _L(~X**)- _L(A.**), from (ii). Hece, by [I], 3I2, (6.5, III), _L(A.) is a rig. As a example, cp < Z < r < a, ad Z is ormal. Thus _L( Z) ad _L( T) are rigs (cf. [I], 3I3, (6.5, IV) ad ote). I wish to tha Dr. R. G. CooKE for a umber of useful commets o the mauscript. Postscript. Sice this paper was submitted for publicatio, a ote by H. S. ALLEN appeared i the Oxford Quarterly Joural of Mathematics, Volume 8, No. 30, Jue 1957, givig the result which appears as my Theorem VI above. Birbec College, Lodo, W.O. I REFERENCES l. CooKE, R. G., Liear operators, Macmilla (1953). 2. ALLEN, H. S., Trasformatios of Sequece Spaces, Joural Lodo Math. Soc. 31, (1956). 3. CooKE, R. G., Ifiite Matrices, ad Sequece Spaces, Macmilla (1950). 4. LORENTZ, G. G., Direct Theorems o Methods of Summability, II, Caadia Joural of Mathematics, III, (1951).