Ohranjevalci relacij ekvivalentnosti

doktorska disertacija

Gordana Radić (Avtor), Tatjana Petek (Mentor)

Povzetek

V teoriji linearnih ohranjevalcev se srečujemo s problemi karakterizacije linearnih preslikav na vektorskem prostoru/algebri matrik ali operatorjev, ki ohranjajo določene lastnosti elementov. V doktorski disertaciji se bomo omejili na tiste preslikave, ki ohranjajo relacijo ekvivalentnosti, unitarne ekvivalentnosti ali kongruentnosti na ▫$\beta(\chi)$▫ oziroma ▫$\beta(\mathcal{H})$▫. V vseh obravnavanih primerih se izkaže, da lahko zastavljen problem zreduciramo na problem ohranjanja množice operatorjev ranga ena. Najprej podrobneje preučimo bijektivne linearne preslikave ▫$\phi)$▫ na ▫$\beta(\chi)$▫, algebri omejenih linearnih operatorjev na refleksivnem kompleksnem Banachovem prostoru ▫$(\chi)$▫, ki ohranjajo relacijo ekvivalentnosti. To pomeni, da sta ▫$\phi(A)$▫ in ▫$\phi(B)$▫ ekvivalentna, kakor hitro sta ▫$A, B$▫ ▫$\in$▫ ▫$\beta(\chi)$▫ ekvivalentna, tj. obstajata taka obrnljiva operatorja ▫$S,T$▫ ▫$\in$▫ ▫$\beta(\chi)$▫, da je ▫$A = SBT$▫. Če pri tem ▫$S$▫ in ▫$T$▫ zapišemo kot končen produkt involucij na ▫$\chi$▫, rečemo, da sta ▫$A$▫ in ▫$B$▫ involutivno ekvivalentna. V duhu te na novo definirane relacije preoblikujemo zastavljen problem in opišemo surjektivne linearne preslikave, ki involutivno ekvivalentna operatorja preslikajo v ekvivalentna. Še več, celo brez predpostavke linearnosti klasificiramo surjektivne preslikave, a tokrat z močnejšim privzetkom, da je operator ▫$A-B$▫ ekvivalenten operatorju ▫$C$▫ natanko takrat, ko je operator ▫$\phi(A) - \phi(B)$▫ ekvivalenten operatorju ▫$\phi(C)$▫, za vse ▫$A,B,C$▫ ▫$\in$▫ ▫$\beta(\chi)$▫. V posebnem primeru, kadar sta ▫$S$▫ in ▫$T$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫, kjer je ▫$\mathcal{H}$▫ kompleksen Hilbertov prostor, unitarna, pravimo, da sta ▫$A,B$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫, unitarno ekvivalentna. Poiskali bomo natančno strukturno obliko bijektivnih linearnih preslikav na ▫$\beta(\mathcal{H})$▫, ki unitarno ekvivalentna operatorja preslika v unitarno ekvivalentna. Pokazali bomo, da takšni linearni ohranjevalci pravzaprav ohranjajo množico unitarnih operatorjev, nato pa z uporabo znanega rezultata, ki te preslikave opiše, podali rešitev problema. Če se zgodi, da je ▫$A = SBS*$▫, za nek obrnljiv operator ▫$S$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫, rečemo, da sta ▫$A,B$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫ kongruenta. Najprej bomo relacijo temeljito raziskali, nato pa predstavili bijektivne linearne preslikave na ▫$\beta(\mathcal{H})$▫, ki ohranjajo relacijo kongruentnosti.

Ključne besede

disertacije;Banachov prostor;Hilbertov prostor;linearni operatorji;linearni ohranjevalci;ohranjevalci relacij;ekvivalentnost;involutivna ekvivalentnost;unitarna ekvivalentnost;kongruentnost;

Podatki

Jezik:	Slovenski jezik
Leto izida:	2019
Tipologija:	2.08 - Doktorska disertacija
Organizacija:	UM FNM - Fakulteta za naravoslovje in matematiko
Založnik:	G. Radić]
UDK:	517.983.2(043.3)
COBISS:	300346880
Št. ogledov:	904
Št. prenosov:	112
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Angleški jezik
Sekundarni naslov:	Preservers of relations of equivalence
Sekundarni povzetek:	Linear preserver problems concern the characterization of linear maps on spaces/algebras of matrices or operators that leave certain properties, functions, subsets or relations invariant. In this dissertation we restrict our attention to the linear maps on ▫$\beta(\chi)$▫ or ▫$\beta(\mathcal{H})$▫ preserving each of the given relations: equivalence, equivalence by unitaries, congruence. A unified approach by reducing the problem to the case of rank-one preserving maps is used. By this method we find a complete description of bijective linear maps ▫$\phi$▫ on ▫$\beta(\chi)$▫, the algebra of all bounded linear operators on a reflexive complex Banach space, which preserves equivalence. This means that ▫$\phi(A)$▫ and ▫$\phi(B)$▫ are equivalent whenever ▫$A,B$▫ ▫$\in$▫ ▫$\beta(\chi)$▫ are equivalent, i.e. there exist invertible such that▫$S,T$▫ ▫$\in$▫ ▫$\beta(\chi)$▫ ▫$A = SBT$▫. If ▫$S$▫ and ▫$T$▫ are, in addition, finite products of involutions on ▫$(\chi)$▫, ▫$A$▫ and ▫$B$▫ are said to be equivalent by products of involutions. By this newly defined relation we modified the stated problem and we determine those surjective linear maps where from equivalence by products of involutions of ▫$A$▫ and ▫$B$▫ ▫$\in$▫ ▫$\beta(\chi)$▫ it follows that ▫$\phi(A)$▫ and ▫$\phi(B)$▫ are equivalent. Moreover, we characterize surjective maps even without linearity, but this time with stronger assumption of ▫$A-B$▫ being equivalent to ▫$C$▫ if and only if ▫$\phi(A)$▫ and ▫$\phi(B)$▫ is equivalent to ▫$\phi(C)$▫, for every ▫$A,B,C$▫ ▫$\in$▫ ▫$\beta(\chi)$▫. In the special case when ▫$S,T$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫, where ▫$(\mathcal{H})$▫ is complex Hilbert space, being unitary, operators ▫$A,B$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫ are called to be equivalent by unitaries. In this thesis, we will also find the representation of bijective linear maps on ▫$\beta(\mathcal{H})$▫ which preserve equivalence by unitaries. To this aim we will apply a well-known result on the unitary group preserving maps. If ▫$A = SBS*$▫, for some invertible operator ▫$S$▫ ▫$\in$▫ ▫$\beta(\mathcal{H})$▫, we say that the operator ▫$A$▫ is congruent to the operator ▫$B$▫. Following the properties of this relation, we will be able to consider the structure of bijective linear maps on ▫$\beta(\mathcal{H})$▫ preserving congruence.
Sekundarne ključne besede:	dissertations;Banach space;Hilbert space;linear operator;linear preservers;relation preserving;equivalence relation;equivalence by product of involutions;equivalence by unitaries;congurence;
URN:	URN:SI:UM:
Vrsta dela (COBISS):	Doktorsko delo/naloga
Komentar na gradivo:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Strani:	X, 96 str.
ID:	11074417