SUBLINEAR FUNCTIONALS AND KNOPP ’ S CORE THEOREM

In this paper we are concerned with inequalities involving certain sublinear functionals on m, the space of real bounded sequences. Such inequalities being analogues of Knopp's Core theorem.

l.cl m be tim linear space of real bounded sequences with the usu:i stq)rcmum nom.We write n m 0={xe m'suplE Xkl<Oo} n k=0 Let 1. be the sequence of infinite matrices (Ai) (ank(i)).Given a sequence x (x k) we write A(x)= Z ank(i) x k k=O ifitcxistsforeachnandi>0.WealsowriteAxfor(Ai )oo Thesequcnccx=(xk) i' said to be summable m the value s by the method (1.) if n(X) i,n=O" Atn(X) s (n ,,*, uniformly in i) If (1.2) holds, then we write xs(,.).
It is well known, (Sticglitz 131), that () is regular if an(!only if the tllxing hold: X ank(i)1< (fi,r all n, for all i).k and there exist an integer m such that sup lira ank(i) 0, uniformly in i, I.'1 n lira T. ank(i) unifinnly in i.
(1 n k "i'hrough(,t tile paper we write I11 sup k ank(i) < n,! to mean that, there exists a constant M such that Z] a,lk(i) <M If, for every bounded sequence x, x---s(l.)then (,,,'1.) is said to be a Schur Throughout the paper we consider only real matrices and real tmndcd scqucncc,.
In this paper we arc concerned with inequalities involving certain sublinear fnctimal,; m .the space of real bounded sequences.Such inequalities being analogues of Knopp's ('ore hcretn Th:t theorem determines a class of regular matrices for which limsup Ax < limsup x for all x e m, see e.g Cooke I41, Maddox 151, Simons 161.This result has also been cxtcntlcd corcgular matrices by Rhoades 171, Schaefer 181, and, Das 191. Before stating tile theorems to be proved, we introduce some flrther notalion.If f, g arc any two of the atx)ve functionals, we shall write fA <_ gB to denote that, for every I))tll(lctl .,,tlUClt'x, tle tran.,l'ormsAx and Bx are defined an(I boltndcl aml f(Ax) < g(llx).
PR()()F. in Iheorem 1. il is enough to lake l/n+ ank(i) 0 _< k _<.i-n otherwise We de(lute at once from Corollary 2 that if a sequence x is convergenl to s. tllelv it is altvu)',t convergent to s which is a well-known result.
We note in passing thal a matrix A is strongly regular, Lorcntz [l ]. if an(l (rely if i i, vcgtlar and that Y: hnkan,k+ll --> 0 (n -+ oo) (2.1) k Sufficiency.Given : > 0, we can find a positive integer p such that for x m anl for all k>O, k+p rE=k xr < L*(x) + P + (2.5) (We fix p throughout the analysis).
As in l.x)rentz's proof (see 111; Th.Siucc A is strongly regular, (2.4) holds.Thus the expression in (2.7) tends to zero as n---,,,,.11cncc we t'iml that lly (2.5), wc have I.(Ax) _< (l.*(x) c)limsup , ttkl + Ilxli iimsul,t (k,tkl-auk) tim regularity of A and (2.3) we get that L(Ax) < L*(x) + e. Stut.'c t" in arbitrary, sufficiency follows.I'ROOF.Recall that A is called F-regular if it maps F, the class of all almost converge,t .SCtlUenCes, imo itself and f-lira Ax f-lim x.Corollary to Theorem 4 in IOI gives the necessary a,d .sufficient conditions for A to be F-regular.
We now come to the proof of necessity.
"!' gc the necessity of (2.8), we define (bnk(i)) by bnk(i) ig[ Z. ark E.ark r=l then (2.6) with ank relaced by bnk(i) holds.Since xe rn and A is F-regular, Corollary to "i'hcorem in lOi yields that the second and third sigmas with ank replaced by bnk(i), tend to zero as n--oo, uniformly in i.On the other hand IFnpl, with ank replaced by bnk(i) is not greater than i+n P -2 Ilxll k Ibnk0) -" bn,k+ l(i)l -P Ilxll k I--l--n + i (ark-ar'k+ )1 Since A is F-regular, the last sigma tends to zero as n.-oo, uniformly in i. lcncc we have, hy 2. + Ilxll limsUPn st.pt k (I-!-n + I i arkl" --]-r = E " ark) Using (2.8) and the fact that A i,; F-regular, we get L*(Ax) < L*(x) + r.
Since e is arbitrary, the required conclusion follows.
We now give another ingcquality shaqer than that of'i'heorcm 4. (See "i'hc,,en 6 I,clx ).It i, also an analogue of Theorem 3 given by Devi !1.We first need to prove a l.cnma.
Before proving the theorem we note that W* is well-defined (see Devi il !1).
We now come to the proof.Suppose that i.*A < W*.Since W* < L*, it follows from Theorem 4 that A is F-rcgla that (2.8) holds.
We conclude the paper with the following remark: Since no Schur mcthd is rcglr.l'hc,ct 7 includes the result that QA(X) < qB(x) is impossible when (I],) is a regular method.For example, QA(x) _< [(x) (for every x m), is impossible.