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Singular Sylvester Equation and Its Applications

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University of Niˇs Faculty of Sciences and Mathematics Bogdan D. Dord¯evi´c- Singular Sylvester equation and its applications DOCTORAL DISSERTATION Niˇs,2020. Univerzitet u Niˇsu Prirodno-matematiˇckifakultet Bogdan D. Dord¯evi´c- Singularna Silvesterova jednaˇcinai njene primene DOKTORSKA DISERTACIJA Niˇs,2020. Data on Doctoral Dissertation Doctoral Nebojša Dinčić, full professor at Faculty of Sciences and mathematics, University of Niš supervisor: Title: Singular Sylvester equation and its applications Abstract: This thesis concerns singular Sylvester operator equations, that is, equations of the form AX- XB=C, under the premise that they are either unsolvable or have infinitely many solutions. The equation is studied in different cases, first in the matrix case, then in the case when A, B and C are bounded linear operators on Banach spaces, and finally in the case when A and B are closed linear operators defined on Banach or Hilbert spaces. In each of these cases, solvability conditions are derived and then, under those conditions, the initial equation is solved. Exact solutions are obtained in their closed forms, and their classification is conducted. It is shown that all solutions are obtained in the manner illustrated in this thesis. Special attention is dedicated to approximation schemes of the solutions. Obtained results are illustrated on some contemporary problems from operator theory, among which are spectral problems of bounded and unbounded linear operators, Sturm-Liouville inverse problems and some operator equations from quantum mechanics. Scientific Mathematics field: Scientific Functional analysis discipline: Sylvester equation; operator equations; operator algebras; spectral theory of operators; closed Key words: operators; Fredholm theory; Banach algebras. UDC: 517.983.23; 517.983.24; 517.983.5; 517.986.3; 517.984; 517.984.3; 517.984.4 CERIF P140: Series, Fourier analysis, functional analysis classification: Creative CC BY-NC-ND common licence: Подаци о докторској дисертацији Ментор: Небојша Динчић, редовни професор Природно-математичког факултета Универзитета у Нишу Наслов: Сингуларна Силвестерова једначина и њене примене Резиме: У овој дисертацији се изучава сингуларна Силвестерова операторска једначина, односно, операторска једначина облика AX-XB=C, под претпоставком да је она или нерешива или да има бесконачно много решења. Једначина се посматра у више разчличитих случаја, најпре у матричном случају, затим у случају када су у питању ограничени линеарни оператори на Банаховим просторима и коначно у случају када су у питању затворени линеарни оператори на Банаховим или Хилбертовим просторима. У сваком од поменутих сценарија се прво изводе довољни услови решивости полазне једначине, а онда се под тим претпоставкама прелази на њено решавање. Долази се до егзактних решења у затвореној форми, те се прелази на њихову класификацију и карактеризацију, односно, показује се да су изведеним посупцима обухваћена сва могућа решења сингуларне Силвестерове једначине. Посебна пажња је посвећена апроксимацијама решења. Добијени резултати су илустровани на неким савременим проблемима из теорије оператора, као што су спектрални проблеми ограничених и неограничених линеарних оператора, инверзни проблеми Штурм-Лиувилове теорије и операторске једначине које се јављају у квантној механици. Научна Математичке науке област: Научна Функционална анализа дисциплина: Силвестерова једначина; операторске једначине; алгебре оператора; спектрална теорија Кључне оператора; затворени оператори; Фредхолмова теорија; Банахове алгебре. речи: УДК: 517.983.23; 517.983.24; 517.983.5; 517.986.3; 517.984; 517.984.3; 517.984.4 CERIF P140: Класе, Фуријеова анализа, функционална анализа класифи- кација: Тип CC BY-NC-ND лиценце креативне заједнице: Podaci o mentoru i ˇclanovima komisije Mentor: dr NebojˇsaDinˇci´c Redovni profesor Prirodno-matematiˇckifakultet Univerzitet u Niˇsu Clanoviˇ komisije: 1. dr Vladimir Rakoˇcevi´c Dopisni ˇclanSANU Redovni profesor Prirodno-matematiˇckifakultet Univerzitet u Niˇsu 2. Akademik dr Stevan Pilipovi´c Redovni profesor Prirodno-matematiˇckifakultet Univerzitet u Novom Sadu 3. dr Peter Semrlˇ Redovni profesor Fakultet za matematiku i fiziku Univerzitet u Ljubljani 4. dr Sneˇzana Zivkovi´c-Zlatanovi´cˇ Redovni profesor Prirodno-matematiˇckifakultet Univerzitet u Niˇsu 5. dr Dijana Mosi´c Redovni profesor Prirodno-matematiˇckifakultet Univerzitet u Niˇsu i Abstract The main goal of this doctoral dissertation is to investigate behavior of sin- gular Sylvester equations, i. e. behavior of operator equations AX − XB = C; under the assumption that they are either unsolvable or have infinitely many solutions. Once solvability conditions are derived, characterization and clas- sification of the solutions is conducted and an explicit general formula for those solutions is provided, thus forming the general solution of the given Sylvester equation. Standard techniques, such as the generalized inverses, are omitted, because the assumption that A and B have closed ranges which are complemented in the corresponding Banach spaces is dropped. Instead, new and original methods are developed for solving this problem, and they are the original scientific contribution of the author, published in papers [24]{[29]. The dissertation is broken down into several chapters. Chapter 1 is the in- troductory chapter, where regular Sylvester equation (which has a unique solution) is introduced and solved. Some important applications of the equa- tion are mentioned. Chapter 2 concerns the singular case where A, B and C are matrices. The re- sults are obtained by the shared-eigenvalue discussion for matrices A and B, and by the analysis of the corresponding eigenspaces. Generalized commu- tators of matrices A and B are characterized, and the solutions are approxi- mated when possible. Perturbation analysis is conducted, using majorization theory for matrices. The main results in this chapter were obtained by the author and his PhD advisor in their joint works [28] and partially [29], and by the author in his individual paper [27]. Chapter 3 concerns the singular case when A, B and C are bounded linear operators on Banach spaces. Since the spectra of A and B do not necessarily consist of eigenvalues only, an alternative approach is required. First, a special operator algebra is introduced, which is not a Banach algebra per se, but still allows a functional calculus of its elements. This algebra gives a different form of the general solution to the given Sylvester equation, and solves every basic operator equation AX − XB = C; AXB = C; X − AXB = C; ii in the same manner, discarding regularity of the equations (only their solv- ability is required). The advantage of this method compared to the gener- alized inverses techniques (which are commonly used in singular equations) is that it does not require complementedness of the appropriate ranges and null-spaces, but rather solves each equation directly. This algebra is intro- duced and studied in detail by the author in [25]. Afterwards, this algebra is used to solve the initial Sylvester equation, with help from Fredholm theory. The author obtained these results in [26]. Applications to some contempo- rary problems in Banach spaces are illustrated as well. Chapter 4 concerns the singular case when A and B are densely defined closed operators on Banach spaces, and C is a densely defined linear operator. The initial premise is that the point spectra of A and B intersect, and in that case, weak solutions X are obtained, which are defined on appropriate eigenspaces of B. Techniques used involve decompositions of the given operators and spaces. Further, the results are extended to Schauder bases when possible, and are applied to Sturm-Liouville operators. These results were achieved by the author and his PhD advisor in the joint work [29]. Afterwards, a special case is analyzed, where A and B are densely defined self-adjoint operators on Hilbert spaces, while the point-spectrum-intersection assumption is dropped. In that case, the weak solutions X obtained in [29] are extended to the largest domains possible, which are constructed by the Spectral mapping theorem for self-adjoint operators and by the Berberian-Buoni-Harte-Wickstead con- struction. These results were obtained by the author in [24] and they are illustrated on an example which stems from quantum mechanics. iii Abstrakt Glavni cilj ove doktorske disertacije je ispitivanje prirode singularne Silves- terove jednaˇcine,odnosno, operatorske jednaˇcineoblika AX − XB = C; pod pretpostavkom da je ona ili nereˇsiva, ili da ima beskonaˇcnomnogo reˇsenja. Najpre bi se obezbedili dovoljni uslovi reˇsivosti jednaˇcine,a po- tom bi se sprovela karakterizacija i klasifikacija reˇsenja. Ta reˇsenjabi se zatim izvela analitˇckimi egzaktnim metodima, u zatvorenom obliku, ˇcimebi formirala opˇstereˇsenjepolazne jednaˇcine. Za razliku od standarndih metoda koriˇs´cenihza reˇsavanje singularnih op- eratorskih jednaˇcina,poput uopˇstenihinverza, u ovoj disertaciji se, izmed¯u ostalog, posmatraju i sluˇcajevi u kojima dati operatori nisu uopˇstenoin- vertibilni, odnonso, njihova jezgra i njihove slike ne moraju biti zatvoreni sa topoloˇskimkomplementima u odgovaraju´cimprostorima. Stoga se dati problem analizira na nov i originalan naˇcin,ˇstoje ujedno i nauˇcnidoprinos autora ovoj temi. Originalni rezultati autora, na kojima se i bazira ova dis- ertacija, publikovani su u radovima [24]{[29]. Sama disertacija je podeljena u nekoliko glava. Glava 1 je uvodnog karaktera,
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  • Affine Transformation

    Affine Transformation

    Affine transformation From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 3D projection 5 2.1 Orthographic projection ........................................ 5 2.2 Weak perspective projection ..................................... 5 2.3 Perspective projection ......................................... 6 2.4 Diagram ................................................ 8 2.5 See also ................................................ 8 2.6 References ............................................... 9 2.7 External links ............................................. 9 2.8 Further reading ............................................ 9 3 Affine coordinate system 10 3.1 See also ................................................ 10 4 Affine geometry 11 4.1 History ................................................. 12 4.2 Systems of axioms ........................................... 12 4.2.1 Pappus’ law .......................................... 12 4.2.2 Ordered structure ....................................... 13 4.2.3 Ternary rings ......................................... 13 4.3 Affine transformations ......................................... 14 4.4 Affine space .............................................