Matrix transformations related to I-convergent sequences

Transcription

1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volue 22, Nuber 2, Deceber 2018 Available olie at Matrix trasforatios related to I-coverget sequeces Eo Kol Abstract. Characterized are atrix trasforatios related to certai subsets of the space of ideal coverget sequeces. Obtaied here results are coected with the previous ivestigatios of the author o soe trasforatios defied by ifiite atrices of bouded liear operators. 1. Itroductio ad preliiaries Let N = {1, 2,... } ad let X, Y be ored spaces over the field K of real ubers R or coplex ubers C. As usual, a liear subset of the vector space ω(x) of all X-valued sequeces is called a sequece space. A subset Φ of X is called fudaetal if the liear spa of Φ is dese i X. By B(X, Y ) we deote the space of all bouded liear operators fro X to Y. We write sup, li ad istead of sup N, li ad =1, respectively. Let λ(x) be a subset of ω(x), let µ(y ) be a subset of ω(y ), ad let A = (A ) be a ifiite atrix of operators A B(X, Y ) (, N). We say that A aps λ(x) ito µ(y ), ad write A : λ(x) µ(y ), if for all x = (x ) λ(x) the series A x = A x ( N) coverge ad the sequece Ax = (A x) belogs to µ(y ). For a sequece (A ) we defie so-called group ors (cf. [18], p. 5) (A ), := sup r sup x 1 r A x = (, N). It is ow that the sets c(x), c 0 (X) ad l (X) of all coverget, coverget to zero ad bouded sequeces x = (x ) ω(x) are Baach sequece spaces with the or x = sup x, ad the set l p (X) of Received April 24, Matheatics Subject Classificatio. 40A35; 40C05; 40J05; 46B15; 46B45. Key words ad phrases. Baach space; bouded liear operator; ideal; I-covergece; I-boudedess; atrix trasforatio; sequece space; statistical covergece

2 192 ENNO KOLK sequeces x such that x p < is a Baach space with the or x p = ( x p ) 1/p if 1 p <. For x X ad N, let e(x) = (x, x,... ) be a costat sequece ad let e (x) = (e j (x)) be the sequece with e j (x) = x if j = ad e j (x) = 0 otherwise. It is ot difficult to see that if Φ is a (coutable) fudaetal set i X, the E 0 (Φ):= {e (φ) : N, φ Φ} is a (coutable) fudaetal set i c 0 (X) ad l p (X), ad E 0 (Φ) E(Φ) with E(Φ):= {e(φ) : φ Φ} is a (coutable) fudaetal set i c(x). If a atrix ap A is defied o a Baach sequece space λ(x), the the operators A ( N) defied above are liear ad bouded, i.e., A B(λ(X), Y ). Therefore, by the ivestigatio of atrix trasforatios the followig two well-ow theores of fuctioal aalysis (see, for exaple, [8] or [17]) are useful. Theore 1.1 (Priciple of uifor boudedess). Let X, Y be Baach spaces ad A B(X, Y ) ( N). If sup A x < for every x X, the sup A <. (1.1) Theore 1.2 (Baach Steihaus). Let X, Y be Baach spaces, A B(X, Y ) ( N), ad let Φ be a fudaetal set of X. The liit li A x exists for ay x X if ad oly if sup A < (1.2) ad li A φ exists for every φ Φ. Moreover, the liit operator A, Ax = li A x (x X), is bouded ad liear, i.e., A B(X, Y ), ad A sup A. The equality li A x = Ax (x X) with A B(X, Y ) is true if ad oly if (1.2) holds ad li A φ = Aφ (φ Φ). Based o Theores 1.1 ad 1.2, Zeller [23] (see also [18]) ad Kagro [9] characterized the atrix trasforatios A fro c(x), c 0 (X) ad l 1 (X) to c(y ) as follows. Theore 1.3. Let X, Y be Baach spaces ad let A = (A ) be a ifiite atrix with A B(X, Y ). The the followig stateets hold. (i) A : c(x) c(y ) if ad oly if li (A ),1 < ( N), (1.3) =1 A x ( N, x X), (1.4)

3 MATRIX TRANSFORMATIONS RELATED TO I-CONVERGENT SEQUENCES 193 A = O(1), (1.5) li A x =: A x ( N, x X), (1.6) li A x (x X). (1.7) (ii) A : c 0 (X) c(y ) if ad oly if (1.3) (1.6) hold. (iii) A : l 1 (X) c(y ) if ad oly if (1.6) holds ad H := sup A < ( N), (1.8) H = O(1). (1.9) Necessary ad sufficiet coditios for the atrix trasforatio A : l (X) c(y ) are cotaied i the followig theore of Maddox (see [18], Theore 4.6; cf. also [9], Theore 2). Theore 1.4. Let X, Y be Baach spaces ad let A = (A ) be a ifiite atrix with A B(X, Y ). The A : l (X) c(y ) if ad oly if (1.3) (1.7) are satisfied ad li (A ), = 0 ( N), li sup (A A ), = 0. Rear 1.5. It is ot difficult to see, usig Theore 1.2, that i Theores 1.3 ad 1.4 it suffices to require the fulfillet of coditios (1.4), (1.6) ad (1.7) oly for all eleets φ fro a fudaetal set Φ of X. The classical suability theory deals ostly with the trasforatios defied by ifiite atrices of real or coplex ubers. Characterizatios of such atrix trasforatios fro (ad to) various spaces of uber sequeces ay be foud, for exaple, i [22]. As a geeralizatio of usual covergece, Fast [4] (see also [21] ad [20]) itroduced the statistical covergece of uber sequeces i ters of asyptotic desity of subsets of N. Later several applicatios ad geeralizatios of this otio have bee ivestigated (for refereces, see [2] ad [3]). For istace, Maddox [19] ad Kol [10, 11] cosidered the statistical covergece of sequeces taig values i a locally covex space ad a ored space, respectively. Aother extesio of statistical covergece is related to geeralized desities. Let T = (t ) be a o-egative regular atrix of scalars (i.e., t 0 (, N) ad li t u = li u for ay coverget scalar sequece (u )). A set K N is said to have T-desity δ T (K) if the liit δ T (K) := li t K

4 194 ENNO KOLK exists (cf. [6]). A sequece x = (x ) ω(x) is called T -statistically coverget to a poit l X, briefly st T -li x = l, if δ T ({ : x l ε}) = 0 for every ε > 0 (see [1], Defiitio 7; [11], p. 44). If T is the idetity atrix, the the T -statistical covergece is just the usual covergece i X, ad if T is the Cesàro atrix C 1, the the T -statistical covergece is just the statistical covergece as defied by Fast [4]. A further extesio of statistical covergece was give i [16] by eas of ideals. Recall that a subfaily I of the faily 2 N of all subsets of N is a ideal if for each K, L I we have K L I ad for each K I ad each L K we have L I. A ideal I is called o-trivial if I ad N / I. A o-trivial ideal I is called adissible if I cotais all fiite subsets of N. Ay o-trivial ideal I defies a filter F(I) := {K N : N \ K I}. A sequece x = (x ) ω(x) is said to be I-coverget to l X, briefly I-li x = l, if for each ε > 0 the set { N : x l ε} belogs to I (see [16], Defiitio 3.1). For exaple, I T := {K N : δ T (K) = 0} is a adissible ideal ad the I T -covergece coicides with the T -statistical covergece. The followig two otios are closely related with the I-covergece. A sequece x = (x ) ω(x) is said to be I -coverget to l X, briefly I -li x = l, if there exists a idex set K = ( i ) such that K F(I) ad li i x i = l i X (see [16], Defiitio 3.2). A sequece x = (x ) ω(x) is said to be I-bouded, briefly x = O I (1), if for soe K = ( i ) F(I) the subsequece (x i ) is bouded i X (cf. [7]). We rear that the I -covergece of uber sequeces was itroduced already by Freeda [5] as I-ear covergece. It is easy to see that I -covergece iplies I-covergece ad every I - coverget sequece is I-bouded. A adissible ideal I 2 N is said to have property (AP) if for every coutable faily of utually disjoit sets K 1, K 2,... fro I there exist sets L 1, L 2,... fro 2 N such that the syetric differeces K i L i (i N) are fiite ad L = i L i I. It is ow that the ideal I T defied above has the property (AP) (see [6], Propositio 3.2). The followig characterizatio of I-covergece is iportat for us (see [16], Theore 3.2).

5 MATRIX TRANSFORMATIONS RELATED TO I-CONVERGENT SEQUENCES 195 Propositio 1.6. Let I be a adissible ideal with property (AP). If I-li x = l i a Baach space X, the I -li x = l i X. By the ivestigatio of atrix trasforatios related to the I-covergece, the sets c I (X), c I 0 (X) ad li (X) of all I-coverget, I-coverget to zero ad I-bouded sequeces x ω(x) appear istead of c(x), c 0 (X) ad l (X), respectively. For X = K we oit the sybol X i otatios. I the followig two sectios we characterize atrix trasforatios A related to soe subsets of c I (X). 2. Matrix trasforatios of c I (X) Let X, Y be Baach spaces, ad let A = (A ) be a ifiite atrix of operators A B(X, Y ) (, N). For a set K = ( i ) N we defie the K-colu-sectio of A as A [K] = (A [K] ), where, for ay N, A[K] = A if K ad A [K] = 0 otherwise. Aalogously, the K-sectio of a sequece x = (x ) is defied by x [K] = (z ), where z = x if K ad z = 0 otherwise. Let I 2 N be a ideal. We say that a sequece space λ(x) is I-sectioclosed if for every x λ(x) ad for ay K I F(I) we have x [K] λ(x). Theore 2.1. Let I be a adissible ideal. Let λ(x) be a I-sectioclosed sequece space cotaiig the set E(X) = {e(x) : x X}, ad let µ(y ) be a arbitrary sequece space. If A : c I (X) λ(x) µ(y ), the A : c(x) λ(x) µ(y ), (2.1) A [K] : λ(x) µ(y ) (K I). (2.2) If I has property (AP), the (2.1) ad (2.2) iply that A : c I (X) λ(x) µ(y ). Proof. Let A : c I (X) λ(x) µ(y ). The (2.1) holds i view of c(x) c I (X) because the ideal I is adissible. Now, let K I ad x λ(x). The sequece y = x [K] is obviously I - coverget to 0 ad so, y c I (X). Moreover, sice λ(x) is I-sectio-closed, we have that y λ(x). Thus y c I (X) λ(x) ad so, Ay µ(y ). By A [K] x = A y ( N) we get A [K] x µ(y ), i.e., (2.2) holds. Coversely, suppose that (2.1) ad (2.2) hold ad I has property (AP). If x c I (X) λ(x), the for soe l X the sequece y = (y ) with y = x l is I-coverget to 0 ad, by Propositio 1.6, I - li y = 0. Thus, for soe K I, the sequece z = y [N/K] belogs to c 0 (X) which gives Az µ(y ) by (2.1). Further, sice y λ(x) because of E(X) λ(x), by (2.2) we get A [K] y µ(y ). Now, usig the equality Ay = Az + A [K] y, we get Ay µ(y ). But this shows that Ax µ(y ) with I- li x = l.

6 196 ENNO KOLK It is ot difficult to see that l (X) ad l I (X) are exaples of I-sectioclosed sequece spaces which cotai E(X). Let st T (X) deote the set of all T -statistically coverget sequeces x ω(x). Sice the ideal I T is adissible ad has property (AP), fro Theore 2.1 we iediately get a geeralizatio of Theore 4.1 fro [12]. Propositio 2.2. Let T = (t ) be a o-egative regular atrix of scalars. Assue that λ(x) is a I T -sectio-closed sequece space cotaiig E(X). The, for a arbitrary sequece space µ(y ), A : st T (X) λ(x) µ(y ) if ad oly if (2.1) holds ad A [K] : λ(x) µ(y ) (δ T (K) = 0). Theore 2.1 reduces, for λ(x) = l (X), to the followig result. Propositio 2.3. Let µ(y ) ω(y ) be a sequece space ad let I be a adissible ideal. If A : c I (X) l (X) µ(y ), the (2.1) ad A [K] : l (X) µ(y ) (K I) (2.3) are satisfied. If I has property (AP), the (2.1) ad (2.3) iply A : c I (X) l (X) µ(y ). Propositio 2.3 together with Theores 1.3(i) ad 1.4 gives the followig characterizatio of the atrix trasforatio A : c I (X) l (X) c(y ). Corollary 2.4. Let X, Y be Baach spaces, A = (A ) be a ifiite atrix with A B(X, Y ), ad let I be a adissible ideal. If A aps c I (X) l (X) to c(y ), the (1.3) (1.7) hold ad, for ay K = ( i ) I, the atrix A [K] = (A [K] ) satisfies the coditios li (A [K] ), = 0 ( N), (2.4) li sup (A [K] A[K] ), = 0, (2.5) If I has property (AP), the (1.3) (1.7), (2.4) ad (2.5) are also sufficiet for A : c I (X) l (X) c(y ). Let B = (b ) be a ifiite atrix of scalars. Usig the ow characterizatios of atrix trasforatios B : c c ad B : l c (see, for exaple, [22]), Propositio 2.3 perits to forulate also a extesio of Corollary 5.1 fro [12]. Corollary 2.5. Let I be a adissible ideal. If B : c I l c, the sup b <, (2.6) li b =: b ( N), (2.7)

7 MATRIX TRANSFORMATIONS RELATED TO I-CONVERGENT SEQUENCES 197 li li b, (2.8) b b = 0 (K I). (2.9) K If I has property (AP), the (2.6) (2.9) are also sufficiet for B : c I l c. 3. Matrix trasforatios to c I (Y ) Let I be a adissible ideal ad let X, Y ad A be the sae as i Sectio 2. The followig characterizatios of atrix trasforatios A to the space of I-coverget sequeces are ow. Theore 3.1 (see [14, 15]). Let Φ X be a fudaetal set of X. The followig stateets are true. (i) If A : c(x) c I (Y ) l (Y ), the (1.3), (1.5) hold ad li =1 A φ ( N, φ Φ), (3.1) I- li A φ =: A φ ( N, φ Φ), (3.2) I- li A φ (φ Φ). (3.3) Coversely, if Φ is coutable ad I has property (AP), the (1.3), (1.5) ad (3.1) (3.3) are also sufficiet for A : c(x) c I (Y ) l (Y ). (ii) If A : c 0 (X) c I (Y ) l (Y ), the (1.3), (1.5) ad (3.2) hold. If Φ is coutable ad I has property (AP), the (1.3), (1.5) ad (3.2) iply A : c 0 (X) c I (Y ) l (Y ). (iii) If A : l 1 (X) c I (Y ) l (Y ), the (1.8), (1.9) ad (3.2) are satisfied. Coversely, if Φ is coutable ad I has property (AP), the (1.8), (1.9) ad (3.2) are also sufficiet for A : l 1 (X) c I (Y ) l (Y ). If O(1) is replaced by O I (1) i (1.5) ad (1.9), the (i) (iii) give the characterizatios of atrix aps A : c(x) c I (Y ) l I (Y ), A : c 0 (X) c I (Y ) l I (Y ) ad A : l 1 (X) c I (Y ) l I (Y ), respectively. Our purpose is to cosider the sae type trasforatios A without the separability assuptio of the space X. As oe ay expect, results obtaied i this case are i soe respects weaer i copariso with the results i Theore 3.1. Let N = ( i ) be a set fro the filter F(I). We say that a sequece x ω(x) is I,N-bouded if (x i ) l (X). If l I,N (X) deotes the set of all I,N-bouded sequeces x ω(x), the it is clear that l (X) l I,N (X) ad l I (X) = l I,N (X). N F(I)

8 198 ENNO KOLK Theore 3.2. Let λ(x) ω(x) be a Baach sequece space with the or λ ad a fudaetal set E. Let I, J be adissible ideals ad N = ( i ) F(J ). If A : λ(x) c I (Y ) l J,N (Y ), the sup sup A x < ( N), (3.4) x λ 1 =1 li =1 A x ( N, x E), (3.5) A i = O(1), (3.6) I- li A x (x E). (3.7) Coversely, if J I, coditios (3.4) (3.6) hold, ad there exists a set K = ( i ) F(I) such that the A : λ(x) c I (Y ) l J,N (Y ). li i A i x (x E), (3.8) Proof. Assue that A : λ(x) c I (Y ) l J,N (Y ). The the series A x = A x coverge for all x λ(x). Thus, because of Theore 1.2, coditios (3.4), (3.5) hold ad A B(λ(X), Y ) ( N). Further, sice Ax l I,N (Y ), (A i x) is bouded for ay x λ(x). So, i view of Theore 1.1, coditio (3.6) ust be satisfied. Coditio (3.7) holds by Ax c I (Y ) (x E). Now. assue that J I ad coditios (3.4) (3.6) ad (3.8) are satisfied. The the operator A is deteried o λ(x) ad A B(λ(X), Y ) ( N) i view of (3.4) ad (3.5). Coditio (3.6) shows, by Theore 1.1, that the sequeces Ax (x λ(x)) are i l J,N (Y ). Moreover, sice J I iplies N F(I), we have M := N K F(I). Cosequetly, the sequece of operators (A i ), where M = ( i ), satisfies the coditios of Theore 1.2 with respect to the fudaetal set E of λ(x). Thus the sequeces Ax (x λ(x)) ust be I -coverget, hece also I-coverget i Y. I the case λ {c, c 0, l 1 }, fro Theore 3.2 we get the followig result. Propositio 3.3. Let I, J ad N be the sae as i Theore 3.2. (i) If A : c(x) c I (Y ) l J,N (Y ), the coditios (1.3), (1.4), (3.6) hold ad I- li A x ( N, x X), (3.9) I- li A x (x X). (3.10)

9 MATRIX TRANSFORMATIONS RELATED TO I-CONVERGENT SEQUENCES 199 Coversely, if J I, coditios (1.3), (1.4), (3.6) are satisfied, ad there exists a set K = ( i ) F(I) such that the A : c(x) c I (Y ) l J,N (Y ). li A i i,x ( N, x X), (3.11) li A i i,x (x X), (3.12) (ii) If A : c 0 (X) c I (Y ) l J,N (Y ), the coditios (1.3), (3.6) ad (3.9) hold. Coversely, if J I ad coditios (1.3), (3.6), (3.11) are satisfied, the A : c 0 (X) c I (Y ) l J,N (Y ). (iii) If A : l 1 (X) c I (Y ) l J,N (Y ), the (1.8), (3.9) hold ad H = O J (1). (3.13) If J I ad coditios (1.8), (3.11) ad (3.13) are satisfied, the A : l 1 (X) c I (Y ) l J,N (Y ). Proof. Our stateets follow fro Theore 3.2 by reaso of the followig rears. (i). Sice c(x) has the fudaetal set E 0 (X) E(X), coditios (3.4), (3.5) reduce, respectively, to (1.3), (1.4). Moreover, (3.7) taes the for (3.9) if x E 0 (X), ad the for (3.10) if x E(X). Siilarly, (3.8) reduces to (3.11) ad (3.12), respectively (ii). We argue as above usig oly the fact that c 0 (X) has the fudaetal set E 0 (X). (iii). The proof is quite siilar if we observe that l 1 (X) has the fudaetal set E 0 (X) ad A = sup A ( N) (see [9], p. 113). If J = I f, the ideal of all fiite subsets of N, the l J,N (Y ) = l (Y ). Cosequetly, Propositio 3.3 gives the followig extesio of Theore 3.1. Propositio 3.4. The followig stateets hold. (i) If A : c(x) c I (Y ) l (Y ), the coditios (1.3) (1.5), (3.9) ad (3.10) hold. Coversely, if coditios (1.3) (1.5), (3.11) ad (3.12) are satisfied, the A : c(x) c I (Y ) l (Y ). (ii) If A : c 0 (X) c I (Y ) l (Y ), the coditios (1.3), (1.5) ad (3.9) hold. Coversely, if coditios (1.3), (1.5) ad (3.11) are satisfied, the A : c 0 (X) c I (Y ) l (Y ). (iii) If A : l 1 (X) c I (Y ) l (Y ), the coditios (1.8), (1.9) ad (3.9) hold. If coditios (1.8), (1.9) ad (3.11) are satisfied, the A : l 1 (X) c I (Y ) l (Y ). This propositio gives, i special case I = I T, the characterizatios of atrix trasforatios A : c(x) st T (Y ) l (Y ), A : c 0 (X) st T (Y )

10 200 ENNO KOLK l (Y ) ad A : l 1 (X) st T (Y ) l (Y ). Siilar characterizatios are proved, for separable X, i [14] ad, for X = Y = K, i [13]. Acowledgeets The author is grateful to the referee for useful suggestios. This research was supported by istitutioal research fudig IUT20-57 of the Estoia Miistry of Educatio. Refereces [1] J. Coor, O strog atrix suability with respect to a odulus ad statistical covergece. Caad. Math. Bull. 32 (1989), [2] J. Coor, A topological ad fuctioal aalytic approach to statistical covergece, i: Aalysis of divergece (Oroo, ME, 1997), Applied ad Nuerical Haroic Aalysis, Birhäuser Bosto, Bosto, MA, 1999, pp [3] G. Di Maio ad Lj. D. R. Ko ciac, Statistical covergece i topology, Topology Appl. 156 (2008), [4] J. Fast, Sur la covergece statistique, Colloq. Math. 2 (1951), [5] A. R. Freeda, Geeralized liits ad sequece spaces, Bull. Lodo Math. Soc. 13 (1981), [6] A. R. Freeda ad J. J. Seber, Desities ad suability, Pacific J. Math. 95 (1981), [7] J. A. Fridy ad C. Orha, Statistical liit superior ad liit iferior, Proc. Aer. Math. Soc. 125 (1997), [8] H. Heuser, Futioalaalysis, B. G. Teuber, Stuttgart, [9] G. Kagro, O atrix trasforatios of sequeces i Baach spaces, Izv. Aad. Nau Ésto. SSR. Ser. Teh. Fiz.-Mat. Nau 5 (1956), (Russia) [10] E. Kol, Statistically coverget sequeces i ored spaces, i: Reports of Coferece Methods of Algebra ad Aalysis (Septeber 21 23, 1988, Tartu, Estoia), pp (Russia) [11] E. Kol, The statistical covergece i Baach spaces, Tartu Ül. Toietised 928 (1991), [12] E. Kol. Matrix suability of statistically coverget sequeces, Aalysis 13 (1993), [13] E. Kol. Matrix aps ito the space of statistically coverget bouded sequeces, Proc. Estoia Acad. Sci. Phys. Math. 45(2/3) (1996), ; 46(1/2) (1997), 150. [14] E. Kol, Baach-Steihaus type theores for statistical ad I-covergece with applicatios to atrix aps, Rocy Moutai J. Math. 40 (2010), [15] E, Kol, O statistical ad ideal covergece of sequeces of bouded liear operators, Eur. J. Pure Appl. Math. 8 (2015), [16] P. Kostyro, T. Salát, ad W. Wilczyńsi, I-covergece. Real Aal. Exchage 26 (2000/2001), [17] G. Köthe, Topologische Lieare Räue. I, Die Grudlehre der Matheatische Wisseschafte, Bad 107, Spriger-Verlag, Berli Heidelberg New Yor, [18] I. J. Maddox, Ifiite Matrices of Operators, Lecture Notes i Matheatics 786, Spriger-Verlag, Berli Heidelberg New Yor, [19] I. J. Maddox, Statistical covergece i a locally covex space, Math. Proc. Cabridge Philos. Soc. 104 (1988),

11 MATRIX TRANSFORMATIONS RELATED TO I-CONVERGENT SEQUENCES 201 [20] I. J. Schoeberg, The itegrability of certai fuctios ad related suability ethods, Aer. Math. Mothly 66 (1959), [21] H. Steihaus, Sur la covergece ordiaire et la covergece asyptotique, Colloq. Math. 2 (1951), [22] M. Stieglitz ad H. Tietz, Matrixtrasforatioe vo Folgeräue. Eie Ergebisübersicht, Math. Z. 154 (1977), [23] K. Zeller, Verallgeeierte Matrixtrasforatioe, Math. Z. 56 (1952), Istitute of Matheatics ad Statistics, Uiversity of Tartu, Tartu, Estoia E-ail address: eo.ol@ut.ee