Science & Technologies COMMUTATIONAL PROPERTIES OF OPERATORS OF MIXED TYPE PRESERVING THE POWERS - I

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1 COMMUTATIONAL PROPERTIES OF OPERATORS OF MIXED TYPE PRESERVING TE POWERS - I Miryaa S. ristova Uiversity of Natioal ad World Ecoomy, Deartmet of Mathematics Studetsi Grad "risto Botev", 17 Sofia, BULGARIA miryaa_hristova@abv.bg ABSTRACT The followig oerator of mixed tye deedig o two arameters ad d y( ), y( ) y( ) d,,. d is cosidered i the sace A of the fuctios aalytic aroud the origi i the comlex lae. We study here the oerator, : A A i the case whe it reserves the owers, i.e. if 1. The author has cosidered the cases of icreasig ad decreasig the owers i the aers [2] ad [3]. I this art I we rove that the oerators L of the commutat L A A L L of have the form : ( ) y () Ly( ) d where { d } is a arbitrary seuece of comlex umbers. Key words: commutat of liear oerator, miimal commutativity Itroductio Let A be the sace of the fuctios aalytic aroud the origi i the comlex lae. We wat to cosider a oerator of mixed tye by first itegratio y( ) d, the multilicatio by a o-egative ower ad fially times differetiatio, i.e. we cosider the followig oerator of mixed tye d y( ), y( ) y( ) d,,. d (1) : A It is suitable to rereset the actio of the oerator A o a sigle ower whe 1 : 1 ( 1)(( 1) 1) (( 1) 1) ( 1). 1 The two arameters ad, ad esecially the umber 1, determie if the oerator will icrease, reserve, or decrease the owers. If 1, the the oerator icreases the owers by : 1 ( 1)! b ; b, 1. 1 ( )! The author has cosidered the commutatioal roerties of (2) i [2]. (2) Volume III, Number 3, 213 Natural & Mathematical sciece 25

2 If, we ca deote 1 ad the the oerator decreases the owers by 1 ( 1)! 1 b ; b 1 ( )! (3) 1. The commutatioal roerties of (3) were cosidered by the author i [3]. I this aer the case 1 will be cosidered, i.e. 1. The the oerator, defied by (1) reserves the owers: 1 ( 1)!, b ; b. (4) 1 ( d I fact, if y() ) () a is a aalytic fuctio from A with coefficiets a, the we ca use the short reresetatio with b from (4). y() a b (5) There are differet articular cases of the oerator, give by (1) or (4) cosidered i the mathematical literature. ere we will metio as a examle the Libera oerator [4] 2 y() t dt, whe the arameters are, 1. The oerator, ca be cosidered also as a adamard roduct y( ) y( ) b( ) a b ab ad to cosider such oerator as a geeralied itegratio, oe has to assume lim b (see Samo S. G., A. A. Kilbas, O. I. Marichev [5, Sect. 22]. Let us reset ow the defiitio of commutat: Defiitio 1: It is said that a cotiuous liear oerator L commutes with a fixed oerator, if L L. The set of all such oerators is called the commutat of ad will be deoted by C. Descritio of the commutat The followig theorem describes the commutat C of the oerator. Theorem 1: Let : A A be the geeral oerator defied by (1) with 1, i.e. by (4). The a liear oerator L : A A belogs to the commutat C of if ad oly if it has the form Volume III, Number 3, 213 Natural & Mathematical sciece 26

3 where { d } coverget. ( ) y () Ly( ) d (6) is a arbitrary seuece of comlex umbers, but such that the series i (6) is ( y Proof: Let y() ) () c A with c, ad let the ower series exasio of Ly() for a arbitrary oerator L of the commutat C be, (7) Ly() c with uow coefficiets,. I the articular case y() L, we ca use. (8) Now istead of Ly( ) Ly( ) (9) we ca cosider the commutatioal roerty L L,1, 2, (1) for arbitrarily fixed i order to fid the umbers,, sice the owers basis of A. Let us exress L ad L usig (4) ad (8):,, L L b b L b b,1, 2,, form a (11) L,,, b. (12) Euatig the coefficiets of the eual owers i (11) ad (12), we have, b, b, ad usig b b for, we ca exress, d, where d,,1, 2,, ca be arbitrarily chose comlex umbers. Thus L ca be writte as L d d - arbitrary, (14) ad the (13) Ly( ) c d (15) shows that the desired descritio (6) is a ecessary coditio for L to commute with. The sufficiecy of (6) or (15) follows immediately: () Ly L c L c b c b L Volume III, Number 3, 213 Natural & Mathematical sciece 27

4 ad the theorem is roved. cb d cd cd Ly() (16) Remar: Let us ote that a sufficiet coditio for covergece of the series i the descritio (6) or (15) i the sace A is lim su d. To rove this, we have to use the Cauchy-adamard formula for the radius of covergece ad have to show that 1 RL or li msu cd. limsu c d But Fially, y() c has a o-ero radius of covergece ad R y 1 or lim su c. lim su c esures the covergece i (6) or (15). limsu c d li msu c. limsu d Let us aouce here that i art II of this aer we will rove the miimal commutativity of the oerator,. The followig two defiitios are give by Raichiov i [1]: Defiitio 2: It is said that a cotiuous liear oerator T is geerated by a oerator, if T is a fiite or ifiite sum T d, d. Every oerator T geerated by commutes with, but the oosite is ot true i geeral. Therefore the followig defiitio is atural: Defiitio 3: A oerator is called miimally commutative if its commutat cosists oly of oerators T geerated by. The followig theorem will be roved i art II of this aer: Theorem 2. If the oerator defied by (1) or (4) is cosidered i the subsace S A of the olyomials, the it is fiitely miimally commutative. Volume III, Number 3, 213 Natural & Mathematical sciece 28

5 To mae this statemet comletely clear, let us metio that the miimal commutativity ca be treated i two differet ways, amely fiite ad ifiite miimal commutativity. If the commutat C cotais oly elemets of the form A a, a, with fiite sums, the is called fiitely miimally commutative. BIBLIOGRAPY 1. Райчинов, И., Върху някои комутационни свойства на алгебри от линейни оператори, действуващи в пространства от аналитични функции, I, 1979, Год. ВУЗ, Прил. мат., 15, No. 3, ristova M. S., 212, Commutatioal roerties of a oerator of mixed tye icreasig the owers, AIP Cof. Proc. 1497, (Alicatios of Mathematics i Egieerig ad Ecoomics - AMEE '12, Soool, Bulgaria). 3. ristova M. S., 212, Commutatioal roerties of a oerator of mixed tye, Mathematical Scieces Research Joural, 16, 5, , Global Publishig Comay. 4. Libera, R. I., Some classes or regular uivalet fuctioas, 1965, Proc. AMS, 16, Samo S. G., A. A. Kilbas, O. I. Marichev, 1993, Fractioal itegrals ad derivatives: Theory ad alicatios, Gordo ad Breach Sciece Publishers, Switerlad ad Philadelhia, Pa., USA. Volume III, Number 3, 213 Natural & Mathematical sciece 29