Mathematics
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Bogdanović, Katarina (Beograd , 2025)[more][less]
Abstract: In the rst and the second chapter of dissertation we prove some new inequalities for the spectral radius, essential spectral radius, oper- ator norm, measure of non-compactness and numerical radius of Hadamard (Schur) weighted geometric means of positive kernel operators on Banach function and sequence spaces. The list of extensions and re nings of known inequalities has been expanded. Some new inequalities and equalities for the generalized and the joint spectral radius and their essential versions of Hadamard (Schur) geometric means of bounded sets of positive kernel op- erators on Banach function spaces have been proved. There are additional results in case of non-negative matrices that de ne operators on Banach sequence spaces. In the third part we present some inequalities for opera- tor monotone functions and (co)hyponormal operators and give relations of Schur multipliers to derivation like inequalities for operators. In particular, let Ψ, Φ be s.n. functions, p ⩾ 2 and φ be an operator monotone function on [0, ∞) such that φ(0) = 0. If A, B, X ∈ B(H) and A and B are strictly ac- cretive such that AX−XB ∈ CΨ(H), then also AXφ(B)−φ(A)XB ∈ CΨ(H) and ||AXφ(B) − φ(A)XB||Ψ ⩽ r φ A+A∗ 2 − A+A∗ 2 φ′ A+A∗ 2 A+A∗ 2 −1 A(AX − XB)B B+B∗ 2 −1 r φ B+B∗ 2 − B+B∗ 2 φ′ B+B∗ 2 Ψ . under any of the following conditions: (a) Both A and B are normal, (b) A is cohyponormal, B is hyponormal and at least one of them is normal, and Ψ := Φ(p)∗ , (c) A is cohyponormal, B is hyponormal and ||.||Ψ is the trace norm ||.||1. Alternative inequalities for ||.||Ψ(p) norms are also obtained. URI: http://hdl.handle.net/123456789/5780 Files in this item: 1
Katarina_Bogdanovic_disertacija.pdf ( 1.621Mb ) -
Krstić, Mihailo (Beograd , 2025)[more][less]
Abstract: This doctoral dissertation addresses the integration of functions taking values in spaces of bounded operators and in spaces of complex measures on a given σ-algebra. The mentioned integrability is considered in a more general sense than that required in the theory of weak integration of vector-valued functions. The first part of the dissertation deals with the integrability of families of operators. If (Ω, M, μ) is a space with a positive measure μ and (At)t∈Ω is a family of operators from B(X, Y ), where X and Y are Banach spaces, then μ-integrability of the function Ω ∋ t 7 → ⟨Atx, y∗⟩ ∈ C is required for every x ∈ X and y∗ ∈ Y ∗. In this case, we prove that the quantity sup∥x∥=∥y∗∥=1 R Ω ⟨Atx, y∗⟩ dμ(t) is finite. This expres- sion allows us to define a norm on the corresponding vector space of families of operators. Furthermore, for every E ∈ M, one obtains an operator R E At dμ(t) in B(X, Y ∗∗), whose defining property is ⟨y∗, R E At dμ(t) x⟩ = R E ⟨Atx, y∗⟩ dμ(t) for every x ∈ X and y∗ ∈ Y ∗. The second part of the dissertation deals with the integrability of families of measures. If (λx)x∈X is a family of complex measures on (Y, A), where (X, B, μ) is a space with a positive measure μ, and if for every A ∈ A the function X ∋ x 7 → λx(A) ∈ C is μ-integrable, then the quantity supA∈A R X |λx(A)| dμ(x) is finite. This allows us to define a norm on the corresponding vector space of families of measures. In this case, for every B ∈ B there exists a complex measure R B λx dμ(x) on A such that R B λx dμ(x) (A) = R B λx(A) dμ(x) for every A ∈ A. The dis- sertation is organized as follows. The first part (Chapters 2–4) deals with the integration of functions taking values in B(X, Y ). Chapter 2 provides a survey of the known results on the integration of functions in B(H), where H is a separable Hilbert space, and presents original results extending the existing theory. In Chapter 3, the developed theory is applied to the Laplace transform of B(H)-valued functions, which has been previously considered in the literature. Chapter 4 is significant because it generalizes the integrability of functions taking values in B(X, Y ). This type of integration was first defined in [8]. The second part of the dissertation (Chapter 5) deals with the integration of functions taking values in spaces of complex measures on a given σ-algebra. The introduced type of integration is more general than Pettis concept and has been considered in [6, 7]. These works represent a natural ex- tension and application of the experiences gained from working with functions taking values in operator spaces, including original results of the candidate with coauthors. Numerous concrete examples are included, making this abstract material much more illustrative. URI: http://hdl.handle.net/123456789/5779 Files in this item: 1
Disertacija_M_Krstic.pdf ( 3.184Mb ) -
Milović, Matija (Beograd , 2025)[more][less]
Abstract: The subject of this dissertation is the study of the belonging of weak operator in- tegrals in appropriate ideals of compact operators, as well as the investigation of perturbation inequalities. These questions were previously considered in [16], where Cauchy–Schwarz type inequalities were established. In addition to providing norm estimates, these inequalities also yield sufficient conditions for an operator integral to belong to a given ideal. In the first part of the dissertation, using these inequalities, perturbation norm inequalities are derived for elementary operators generated by analytic functions. Specially, for an analytic function f, trigonometric polynomials T, S : R → C and t ∈ R, if fT S,t, f¯T T,t and f¯SS,t are the associated analytic functions, and if X ∈ B(H) and the operator P∞ n=1(AnXBn − CnXDn) belongs to a symmetric norming (s.n.) ideal CΦ(H), for some s.n. function Φ, then the following inequality holds ∞X n=1 (A∗ nAn− C∗ nCn) 1 2 fT S,t ∞X n=1 An⊗Bn X − fT S,t ∞X n=1 Cn⊗Dn X ∞X n=1 (BnB∗ n − DnD∗ n) 1 2 Φ ⩽ f¯T T,t ∞X n=1 A∗ nAn − f¯T T,t ∞X n=1 C∗ nCn 1 2 ∞X n=1 (AnXBn − CnXDn) × f¯SS,t ∞X n=1 BnB∗ n − f¯SS,t ∞X n=1 DnD∗ n 1 2 Φ , under certain conditions on the families (An)∞ n=1, (Bn)∞ n=1, (Cn)∞ n=1 and (Dn)∞ n=1 in B(H). Next, the dissertation considers vector measures induced by weak∗ integrable operator- valued functions taking values in Shatten–von Neumann ideals. Furthermore, the criteria for the compactness and nuclearity of the Gel’fand integral are derived, with emphasis on positive operator-valued functions. Finally, depending on the properties of the symmetric norming function Φ, the conse- quences of the condition sup e,f ∈B Z Ω Φ((⟨Aten, fn⟩)∞ n=1)dμ(t) < +∞. are explored. More precisely, it is proved that the weak∗ integral belongs to the symmetric ideal CΦ(H), as well as the Gelfand and Pettis integrability of the CΦ(H)-valued function A . URI: http://hdl.handle.net/123456789/5768 Files in this item: 1
Matija_Milovic_doktorska_disertacija.pdf ( 2.652Mb ) -
Stefanović, Seđan (Beograd , 2025)[more][less]
Abstract: The subject of the dissertation is the investigation of the relation of strong BJ orthogonality in C∗-algebras. For two elements a and b of C∗-algebra A, we say that a is strong BJ orthogonal to b, if for all c ∈ A holds ‖a + bc‖ ⩾ ‖a‖ and we write a ⊥S b. If it is also true that b ⊥S a, then we say that a and b are mutual strong BJ orthogonal and write a ⊥⊥S b. To this relation, we associate an undirected graph Γ(A) (which we call an orthograf), where the vertices are the nonzero elements of the C∗-algebra A, with the identification of an element and its scalar multiple; while there is an edge between two vertices a and b if a ⊥⊥S b. We will show that for any C∗-algebra A, different from three simple C∗-algebras, and for any two non-isolated vertices a and b in the orthograph, we can find vertices c1, c2, c3 ∈ Γ(A) such that a ⊥⊥S c1 ⊥⊥S c2 ⊥⊥S c3 ⊥⊥S b. We will also describe the isolated vertices of the graph Γ(A) for any C∗-algebra A. Finally, in the case of finite-dimensional C -algebras, we will determine the diameter of Γ(A), i.e., the minimum number of elements required to connect any two vertices. URI: http://hdl.handle.net/123456789/5754 Files in this item: 1
Stefanovic_Srdjan_doktorska_disertacija.pdf ( 990.4Kb ) -
Jovanović, Milica (Beograd , 2024)[more][less]
Abstract: The analysis of Grassmann manifolds, which were first introduced in the 19th century, is one of the classical problems in the algebraic topology. When analyzing topological spaces, it is always useful to determine their cohomology algebra. The cohomology of Grassmann manifolds is already well known, but their covering spaces, so called oriented Grassmann manifolds, are far less examined. The oriented Grassmann manifold ˜Gn,k is defined to be the space of oriented k-dimensional subspaces of Rn. In this dissertation we analyze the cohomology algebra of oriented Grassmann manifolds ˜Gn,k with integer and modulo 2 coe!cients, predominantly the case k = 3. The dissertation comprises three chapters. The first chapter is an introduction where an overview of known results and necessary tools is given. In the second chapter we study the cohomology with the modulo 2 coe!cients. First of all, the known results in the case k = 2 are presented. Next, we move onto the case k = 3 where the partial description of the cohomology algebra is given. This section is based on papers published in the last several years. We give an overview of these results in the thesis, and we also present original results for n close to a power of two. In the last part of this chapter, we investigate the cohomology algebra of the manifold ˜G2t,4, and that is as far as we have come with the examination of modulo 2 cohomology. The third chapter is dedicated to the integral cohomology. This chapter, like the previous one, also splits in several sections, depending on the value of k. When k = 2, the integral cohomology is completely determined, and we present the proof for n odd. When k = 3, only the integral cohomology of ˜Gn,3, n → {6, 8, 10}, has been determined so far, while for k ↭ 4 only some partial results are known. In this segment we also analyze the connection between the integer and the modulo 2 cohomology algebra of these Grassmannians by analyzing the morphism between them induced by the modulo 2 reduction. URI: http://hdl.handle.net/123456789/5751 Files in this item: 1
Milica_Jovanovic_doktorat.pdf ( 1.723Mb )