Nazariyai operatorho

Nazariyai operatorho, yak fasli tahlili, funksionali buda, ba omuzishi khosiyathoi operatorho va istifodai onho dar halli maslahoi gunogun oid ast. Mafhumi ope­rator yake az mafhumhoi umumitarini matematika ba hisob meravad.

Nazariya umumii operatorho dar natiqai inkishofi nazariyai muo- dilahoi integrali, halli masalaho oid ba yoftani funksiyahoi khususi va qimathoi khususii operatorhoi differensiali, inchunin inkishofi digar faslhoi tahlili matematikai klassiki ba vuqud omad. Nazariyai operatorho bayni in sohahoi matematika aloqai mus­tahkam ba vujud ovarda, ba inki- shofi minbadai onho hissa guzosht. Usulhoi operatori to paydoishi mafhumi tomi operator ham dar halli navhoi gunoguni muodilahoi diferensialii oddi va muodilahoi dtsfferensialii hosilahoyash khususi istifoda meshudand (nig. Hisobi operatsioni). Nazariyai operatorho dar mekha- yaikai kvanti apparati matemati­kii asosi ba hisob meravad. (nig. Operatorho).

Operatorhoe, ki dar fazoi khattii normironidashuda amal mekunand, beshtar mamuland. In sinfi operatorho funksiyahoi adadi, tabdilothoi khattii fazoi Evklid, operator­hoi differensiali va integrali barin mafhumhoro dar bar megirad. Operatori khatti az hama beshtar tadqiq shudaast va dar amaliya niz beshtar istifoda meshavad., Masalan, bigzo k (s, t) funksiyai befosilai dutagyiryobanda boshad, ki dar kvadrati a<s<b, a<.t<b doda shudaast. Formulai operatori integralii khattiro muayyan mekunad va operatori Fredgolm nom dorad.

A m a l h o bo operatorho. Farz kunam, ki operatori u = A(kh) doda shuda boshad va dar on du ele­menti gunoguni kh va kh’ ba khudi hamon yak elementi u ivaz nameshavand. On goh ba har yak elementi u (ob­raz) yak elementa kh (proobrazi u) muvofiq guzoshta meshavad. In muvofiqguzoriro operatori baraks menomand va chunin ishorat mekunand;

                                                          kh = A—¹ (u)

Muqarrar namudani operatori bar­aks ba halli muodilai u = A (kh) ek­vivalent meboshad.

Agar A₁ va A₂ operatorhoe boshand, ki fazoi R- ro ba fazoi R ‘ inikos mekunand, operatori A = A₁+ A₂–ro summai onho menomand, ki bo ba­robarii A(kh)= A₁ (kh) + A₂(kh) muayyan karda meshavad. Agar operatori A₁ fazoi R–po ba R ‘ va operatori A₂ R ‘-ro ba R“ tabdil dihad, ope­ratore, ki R -ro ba R ” tabdil medihad, zarbi (A₂A,) operatorhoi A₁ va A₂ nomida meshavad. Operatori E, ki nar yak elementro ba khudash tabdil medinad, operatori vo­hidi,operatore, ki har yak ele­mentro ba sifr tabdil medinad, operatori  s i fri  nom dorad. Baroi har guna operatori A barobarihoi

AE= EA= A₁A +0 = 0+A=A, A^-1A=AA^-1=E

(agar A–¹ mavjud boshad) joy dorand.

Ad,: Kolmogorov A. N.. Fomin S. V., Elementi teorii funktsiy i funk­tsionalnogo analiza, 3 izd., M.. 1972.