Minimal commutativity of compositions of operators of mixed type I
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- Minimal commutativity of compositions of operators of mixed type - I Miryana Hristova
- ABSTRACT
- МИНИМАЛНА КОМУТАТИВНОСТ НА КОМПОЗИЦИИ ОТ ОПЕРАТОРИ ОТ СМЕСЕН ТИП Миряна Христова
- I ntroduction
- The case of preserving the powers Description of the commutant
- The case of increasing the powers
| ГОДИШНИК на Минно-геоложкия университет “Св. Иван Рилски”, Том 56, Св.IІІ, Механизация, електрификация и автоматизация на мините, 2013
ANNUAL of the University of Mining and Geology “St. Ivan Rilski”, Vol. 56, Part ІІІ, Mechanization, electrification and automation in mines, 2013 Minimal commutativity of compositions of operators of mixed type - I Miryana Hristova University of National and World Economy, Department of Mathematics, Studentski grad, 1700 София, Bulgaria E-mail: miryana_hristova@abv.bg ABSTRACT. The operators  are considered with non-negative integer parameters  in the case when  in the space  of the functions analytic in neighbourhoods of the origin  of the complex plane  . Using the power series descriptions of the commutants of compositions of operators of the type  with different parameters  and  from previous author's papers, here the question about the minimal commutativity (in the sense of (Raichinov 1979)) of compositions is considered.
МИНИМАЛНА КОМУТАТИВНОСТ НА КОМПОЗИЦИИ ОТ ОПЕРАТОРИ ОТ СМЕСЕН ТИП Миряна Христова Университет за национално и световно стопанство, Катедра Математика, Студентски град, 1700 София, България Резюме. Операторите са разгледани с неотрицателни цели параметри в случая, когато , в пространството на функциите аналитични в околности на координатното начало на комплексната равнина . Използвайки описанието чрез степенни редове на комутантите на композиции на оператори от вида с различни параметри и от предишни свои статии, авторът разглежда тук въпроса за минималната комутативност (в смисъл на (Райчинов 1979)).
Introduction Let  be the space of functions analytic in (possibly different) neighbourhoods of the origin  in the complex plane  or its subspace  of the polynomials of the complex variable  . We want to consider a generalization of the usual operator of integration  multiplying it by a non-negative power  and then differentiating  times, i.e. we consider the operators of mixed type
 (1)
In (Hristova 2012) the commutational properties of a single operator  in the case  were investigated, and in (Hristova 2013a, 2013b) the case  is presented. Here we will combine the results from these papers to describe the commutants of compositions of operators of the type with different parameters and  when they increase or preserve the powers. We will discuss also the question about the minimal commutativity of compositions (in the sense of (Raichinov 1979)).
Let us represent first the action of only one operator on a single power  :
 (2)
Denoting  by  , we have  and can write shortly
 (3)
Now an arbitrary power  of acts on as
 (4)
In order to avoid writing the long products in (4) we will use again a short representation denoting them by one letter:  (5)
and then we can write simply  (6)
In fact, if  is an analytic function from with coefficients  , then we have the short representation
 (7)
with  from (5) and (6) and from (3).
Let us give some definitions: Definition 1. It is said that a continuous linear operator  commutes with a fixed operator  , if  . The set of all such operators is called the commutant of and will be denoted by  .
Definition 2. It is said that a continuous linear operator  is generated by an operator , if is a polynomial of with complex coefficients, i.e.  ,  . The set of all operators generated by will be denoted by  .
Obviously every operator , which is generated by , i.e.  , also commutes with , i.e.  , and hence  . The opposite inclusion  is, in general, not true. Therefore the following definition is natural:
Definition 3. (Raichinov 1979) An operator is called minimally commutative if , i.e. if the commutant consists only of operators generated by and hence if  .
In general we can consider compositions  (8)
where  are operators of the form (1),  are nonnegative integers, and  is considered as a multipower.
In the papers (Hristova 2013c -2013f) the author considers for the sake of simplicity only compositions of two operators  (9)
The description of the commutants of compositions is given there in different cases: when the powers are preserved or increased by both operators and also in the mixed cases when one of the operators increases, while the other one preserves the powers. It is convenient to define the numbers  (10)
which show how each of the operators in the composition changes the powers of the complex variable  .
In the general case of composition of more than two operators the reasonings are the same but the written form of the results becomes more complicated. Let us note that descriptions of commutants are made by many mathematicians. In the references of this paper we have included only a very small part of the publications related to the commutants of operators similar to the one considered here, see all refferences. Additional huge number of publications related to commutants can be found in the bibliographies of the cited monographs. The case of preserving the powers Description of the commutant The description of the commutant in the case of composition of operators preserving the powers is proved in another paper (Hristova 2013c). The interesting fact is that it remains the same as the one for a single operator given in paper (Hristova 2013a). Theorem 1. Let the operators  and  be of the type (1) with  ,  . We can fix  and then to express  , , writing
 (11)
Then a linear operator  commutes with the composition operator  ,  , , if and only if it has the form
 (12)
where  is an arbitrary sequence of complex numbers, but such that the series in (12) converges.
For the sake of completeness we will give here only a Sketch of the proof A short expression of the action of either of the two operators  , , on an arbitrarily fixed single power of the complex variable is
 (13)
Then the action of the composition on can be written as
 (14)
If is an operator from the commutant  , we suppose that its action on an arbitrary power has the form
 (15)
with unknown coefficients  . Then the expressions of  and  are
 (16)
 (17)
If we equate the coefficients of the equal powers in (16) and (17), then  (18)
Taking into account that and the form of the coefficients in (13) and (14), we have that  and then it follows that
 (19)
This reduces the series in (15) to only one term:  (20)
Finally, if an arbitrary analytic function  has a power series expansion with coefficients  , then
 (21)
which is the desired representation (12). Minimal commutativity First we have to describe the operators generated by  and then this description will be compared with the one of the commutant  given in Theorem 1.
Theorem 2: Let us denote for simplicity of the writing the composition (14) and the coefficient in it by one letter  (22)
Then the operators  (23)
Proof: This follows immediately from the representation the action of the powers  of  on functions  :
 (24)
Let us make the definition of the minimal commutativity more precise. One can use two different variants of the definition, namely finite and infinite minimal commutativity If in an algebra  (25)
If the commutant  ,  , (26)
with finite sum, then Theorem 3. If the operator Proof: Let  be an arbitrary polynomial. Then by (6) any operator from the (finite) commutant  in must have a polynomial form
 (27)
with zero coefficients From (26) the action of an arbitrary operator i.e. Now, equating the coefficients of the equal powers in (27) and (28), we have to solve the linear system with unknowns We can suppose that  (30)
The determinant of the system is the non-vanishing Wandermonde's one  (31)
since Remark: We proved Theorem 2 for the subspace  (32)
The first  (17)
i.e. all The case of increasing the powers Due to the lack of space we are not able to consider here the case of increasing the powers by the operators in the composition (9) and also in the mixed cases. Let us only mention that the result which will be proved in Part II (Hristova 2013d) of this paper states that The composition  is minimally commutative if and only if the total change of the powers is exactly one.
References Райчинов, И. 1979. Върху някои комутационни свойства на алгебри от линейни оператори, действуващи в пространства от аналитични функции, I, Год. ВУЗ, Прил. мат., 15, 3, 27-40. Райчинов, И. 1981. Върху някои комутационни свойства на алгебри от линейни оператори, действуващи в пространства от аналитични функции, II, Год. ВУЗ, Прил. мат., 17, 1, 61-71. Фаге, М. К., Н. И. Нагнибида. 1987. Проблема эквивалентности обыкновенных линейных дифференциальных oператоров, Nовосибирск, Издательство Наука, Сибирское отделение. Христова, М. С. 1979. Върху един клас минимално комутиращи оператори в пространстото на полиномите над полето Христова, М. С. 1982. Върху някои изоморфизми, осъществявани от оператори, действуващи в пространства от аналитични функции, Год. ВУЗ, Прил. мат., 18, 2, 63-72. Христова, М. С. 1984. Общ вид на линейните оператори, дефинирани в пространства от аналитични функции и комутиращи с обобщения оператор на Харди-Литълвуд, Год. ВУЗ, Прил. мат., 20, 2, 67-74. Христова, М. С. 1986a. Някои аналитични характеристики на оператори, комутиращи с фиксирана степен на обобщения оператор на Харди-Литълвуд, Научни трудове ВИИ Карл Маркс, 1, 149-159. Bozhinov, N. 1988. Convolutional representations of commutants and multipliers, Sofia, Publishig House of the Bulgarian Academy of Sciences. Dimovski, I. H. 1990. Convolutional Calculus, Dordrecht, Kluwer, Bulgarian 1st Ed.: In Ser. "Az Buki, Bulg. Math. Monographs", 2, Sofia, Publ. House of Bulg. Acad. Sci., 1982. Dimovski, I. H., V. Z. Hristov, M. Sifi. 2006. Commutants of the Dunkl operators in Hristov, V. H. 2009. Description of the commutant of compositions of Dunkl operators, Mathematical Sciences Research Journal, 13, 9, 213-219. Hristova, M. S. 1986b. On the generalized Hardy-Littlewood operator, In: Юбилеен сборник ,,100 години от рождението на академик Л. Чакалов“, София, 143-149. Hristova, M. S. 2012. Commutational properties of an operator of mixed type increasing the powers, AIP Conf. Proc., 1497, 334-341. In: Applications of Mathematics in Engineering and Economics - AMEE '12, Sozopol, Bulgaria. http://dx.doi.org/10.1063/1.4766802 Hristova, M. S. 2013a. Commutational properties of operators of mixed type preserving the powers - I, to appear Hristova M. S. 2013b. Commutational properties of operators of mixed type preserving the powers - II, to appear Hristova M. S. 2013c. Commutational properties of compositions of operators of mixed type preserving the powers – I, to appear. Hristova M. S. 2013d. Minimal commutativity of compositions of operators of mixed type not decreasing the powers – II, to appear. Linchuk, Y. S. 2012. Commutants of composition operators induced by a parabolic linear fractional automorphisms of the unit disk, Fract. Calc. Appl. Anal., 15, 1, 25-33, DOI: 10.2478/s13540-012-0003-6. Raichinov, I. 1978. Linear operators defined in spaces of complex functions of many variables and commuting with the operators of integration, Serdica, Bulgaricae mathematicae publications, 4, 4, 316-323. Samko, S. G., A. A. Kilbas, O. I. Marichev 1993. Fractional integrals and derivatives: Theory and applications, Switzerland and Philadelphia, Pa., USA, Gordon and Breach Science Publishers. Каталог: sessions sessions -> Изследване чистотата на слънчогледово масло за производство на експлозиви anfo sessions -> Laser “Raman” spectroscopy of anglesite and cubanite from deposit “Chelopech” Dimitar Petrov sessions -> Св иван рилски sessions -> Modeling of sessions -> Управление на риска от природни бедствия sessions -> Oценка на риска от наводнениe в елховското структурно понижение в района на гр. Елхово красимира Кършева sessions -> Гравиметрични системи използвани в република българия и оценка точността на системи igsn-71 и unigrace при точки от гравиметричните и мрежи sessions -> Toxicological assessment of photocatalytically destroyed mixed azo dyes by chlorella vulgaris sessions -> Field spectroscopy measurements of rocks in Earth observations Изтегляне 91.94 Kb. Сподели с приятели:
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generated by the operator 
have the form
of operators the notion convergence of sequences 
is defined and it is compatible with the algebraic operations at least so that 
imply 
, 
, and 
, 
consists only of elements of the form
of the polynomials, then it is finitely minimally commutative.
of the highest degrees if 
.
generated by 
is
(28)
is also a polynomial of degree at most 
:
(29)
since if 
, we can take 
and the system (29) becomes with equal number of equations and unknowns:
for 
and the operator 
of the polynomials. If we consider the whole space 
are chosen different from zero. Indeed, if we suppose that the operator 
equations have an unique solution 
as in Theorem 2. But from the next equations for 
we see that
, and cannot be arbitrarily chosen. This contradicts the description of the commutant 
на комплексните числа, Год. ВУЗ, Прил. мат., 15, 3, 55-68.
, Fractional Calculus & Applied Analysis, 9, 3, 195-213.