On Operators of Multiplications Acting on Sequence Spaces Defined by Modulus Functions

Transcription

1 Iteratioal Mathematical Forum, 5, 2010, o. 62, O Operators of Multiplicatios Actig o Sequece Spaces Defied by Modulus Fuctios Kuldip Raj ad Viay Khosla School of Mathematics Shri Mata Vaisho Devi Uiversity Katra J & K, Idia kuldeepraj68@hotmail.com Abstract I this paper we characterize multiplicatio operators ad obtai some criteria for closed rage,ivertibility ad Fredholmess for these operators. Mathematics Subject Classificatio: Primary 47B20, Secodary 47B38 Keywords: Multiplicatio operator, Fredholm operator, Closed rage ad Ivertible operator 1. Itroductio ad Prelimiaries: A modulus fuctio is a fuctio f :[0, ) [0, ) such that (i) f(x) = 0 if ad oly if x =0. (ii) f(x + y) f(x)+f(y) for all x 0,y 0. (iii)f is icreasig. (iv) f is cotiuous from right at o. It follows that f must be cotiuous everywhere o [0, ).The modulus fuctio may be bouded or ubouded.for example take f(x) = x,the f(x) x+1 is bouded. If f(x) =x p, 0 <p<1, the the modulus fuctio f(x) is ubouded.let f be a modulus fuctio ad A =(a k ) be a o-egative matrix such that sup a k is fiite.if we deote by C,the space of all complex se- queces x = {x k }, the by W o (A, f), we mea the class of all sequece x C such that lim a k f( x k ) = 0.The class W o (A, f) is a liear space over the

2 3074 Kuldip Raj ad Viay Khosla complex field C.For every x W o (A, f),we defie x A,f = sup a k f( x k ). Bhardwaj ad Sigh [1] proved that. A,f is a paraorm o W o (A, f) ad (W o (A, f),. A,f ) is a complete liear topological space. Let u : N C be a mappig. The a bouded liear trasformatio m u : W o (A, f) W o (A, f) defied by (m u f)(x) =(u.f)(x) =u(x)f(x) is called a multiplicatio operator iduced by u. Recetly several authors have studied multiplicatio ad weighted compositio operators o differet fuctio spaces. For example oe ca refer to ([2],[3],[5],[6]). For more details see [4]. I this paper we characterize multiplicatio operator ad also obtaied some criteria for compactess,closed rage,ivertibility ad Fredholmess of these operators. 2.Multiplicatio operators actig o sequece spaces defied by modulus fuctios Theorem 2.1: Let m u : W o (A, f) W o (A, f) be a liear trasformatio. The m u is a bouded operator if ad oly if there exists M>0such that f( u(p)y ) Mf(y), for all p N ad y R +....(i) Proof: Suppose that the coditio of the theorem is true. We ca assume that M is a iteger.for x W o (A, f),we have lim ak f( u(k)y ) M lim a k f( y ). Thus m u x W o (A, f).further m u x A,f = sup{ a k ( u(k)x k )} sup{ a k Mf( x k )} M x A,f. This proves the cotiuity of m u at the origi ad hece everywhere i view of liearity of m u. Coversely, if the coditio of the theorem were false, the for every iteger M > 0 there exists p m N ad y m R + such that f( u(p m )y m ) >Mf(y m ). Let g m = y m χ {pm}. The m g m A,f = y m χ {pm} A,f = m sup < sup a pm f(y m ) a pm mf( u(p m )y m ) = m u y m χ {pm} A,f = m u g m A,f.

3 Operators of multiplicatios actig o sequece spaces 3075 This proves that m u is ot bouded. Hece the coditio(i) must be true. Theorem 2.2: Let Am u = m u A. The A is a multiplicatio operator. Proof: Let v = Ae.The Ae = Ame e = me Ae = me v = e v = ve = m v e. We ow proves that v iduces a bouded operator, the for every m N, there exists p m N such that f( v(p m )y m ) >mf(y m ) Take g m = y m e pm. The m g m A,f = M sup sup a pm f(y m ) a pm f( v(p m )y m ) = m v y m e pm A,f = Ay m e pm A,f = Ay m A,f which cotradicts the cotiuity of A. Hece A must be a bouded operator ad A = m v. Theorem 2.3:Let m u B(W o (A, f)). The m u has closed rage if ad oly if there exists δ>0such that f( u(p)y ) δf(y) for all p [Z(u)] ad y R +. Proof:Assume that the coditio of the theorem is true. Let h ram u. The there exists a sequece {h } such that m u h h i.e m u h h A,f 0 as.now{m u h } is a cauchy sequece. Therefore for every ɛ>0 there exists o N such that m u h m u h m A,f <ɛfor all, m o.now δ sup a k ( h (k) h m (k) ) sup ak f( u(k) h (k) h m (k) ) [Z(u)] [Z(u)] <ɛfor all m, o....(i) Defie { h h (k), if k [Z(u)] (k) = 0, elsewhere. The from (i) it follows that {h (k)} is a cauchy sequece i W o(a, f). But W o (A, f) is complete. Therefore there exists h W o (A, f) such that h h. Hece by cotiuity of m u, we get m u h = m u h m uh. Hece h = m u h so that h ra m u.thusm u has a closed rage. Coversely,if the coditio of the theorem were false,the for every positive iteger M there exists p m N ad y m R + such that f( u(p m )y m ) < 1 f(y M m).

4 3076 Kuldip Raj ad Viay Khosla Let g m = y m χ {pm}.the m u g m = sup a k f( (ug m )(k) ) < 1 sup a M pm f(y m ) = 1 g M m A,f. This proves that m u is ot bouded away from zero so that m u does ot have closed rage. Theorem 2.4:Let m u B(W o (A, f)).the m u is ivertible if ad oly if there exists ɛ>0 such that f( u(p)y ) ɛf(y) for all p N ad y R +. Proof: we first assume that there exists ɛ>0such that f( u(p)y ) ɛf(y) for all p N ad y R +.Now y ɛf( ) f( u(p) y )=f(y) orf( 1 y) 1 f(y) for all p N ad u(p) u(p) y u(p) ɛ y R +.This proves that m v is a bouded operator, where v = 1. Clearly m u v is the iverse of m u. Coversely,suppose that m u is ivertible with m v as its iverse.clearly v = 1. Hece by cotiuity of m u v, there exists M > 0 such that f( v(p)y ) Mf(y) for all p N ad y R + 1 or equivaletly f( y) Mf(y). Takig u(p) y = u(p)z, we get f(z) Mf( u(p)z )orf( u(p)z ) 1 f(z) for all p N M ad z R +. Takig ɛ = 1, we get f( u(p)z ) ɛf(z). Hece the coditio M must be true. Theorem 2.5:Let m u B(W o (A, f)).the m u is Fredholm if ad oly if (i) [Z(u)] is a fiite set. (ii) There exists ɛ>0such that f( u(p)y ) ɛf(y) for all p [Z(u)] ad y R +. Proof :If[Z(u)] is a fiite set, the ker m u is fiite dimesioal. From the coditio (ii), m u has closed rage. Moreover dim(w o (A, f)/ram u ) is fiite. This proves that m u is Fredholm. The coverse of the theorem is obvious. Refereces 1. Bhardwaj Viod,K ad Niraja Sigh: O some sequece spaces defied by a modulus, Idia J.Pure Appl.Math.30(8)(1999), Komal,B.S. ad Raj Kuldip : Multiplicatio operators iduced by operator

5 Operators of multiplicatios actig o sequece spaces 3077 valued maps,it.j.cotem.math.scieces,vol.3,2008,o.14, Komal,B.S. ad Gupta,S.: Multiplicatio operators betwee Orlcz spaces, Iteg. Eqs. Oper. Theory, 41(2001), Sigh,R.K. ad Mahas,J.S.: Compositio operators o fuctio spaces, North-Hollad, Takagi,H.; Fredholm Weighted compositio operators, Iteg. Eqs. Oper. Theory, 16(1993), Takagi,H. ad Yokouchi,K.; Multiplicatio ad compositio operators betwee L p -spaces, Cotemporary Math. 232(1999), Received: May, 2010