57 research outputs found
On partial isometries in C*-algebras
No full textWe study similarity to partial isometries in C*-algebras as well as their relationship with generalized inverses. Most of the results extend some recent results regarding partial isometries on Hilbert spaces. Moreover, we describe partial isometries by means of interpolation polynomials.Fil: Arias, Maria Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Mbekhta, Mostafa. Universite Lille; Franci
A-partial isometries and generalized inverses
No full textIn this work we study the relationship between A-partial isometries and generalized inverses.Fil: Arias, Maria Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Mbekhta, Mostafa. Centre National de la Recherche Scientifique. Unité Mixte de Recherche. Unité de Formation et de Recherche de Mathématiques. Université Lille; Franci
A note on the differentiable structure of generalized idempotents
No full textFor a fixed n > 2, we study the set of generalized idempotents, which are operators satisfying Tn+1 = T. Also the subsets † , of operators such that Tn−1 is the Moore–Penrose pseudo-inverse of T, and , of operators such that Tn−1 = T (known as generalized projections) are studied. The local smooth structure of these sets is examined.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Mbekhta, Mostafa. Univesité of Lillé 1; Franci
Metric and homogeneous structure of closed range operators
No full textLet CR be the set of all bounded linear operators between Hilbert spaces H, K with closed range. This paper is devoted to the study of the topological properties of CR if certain natural metrics are considered on it. We also define an action of the group GH × GK on CR and determine the orbits of this action. These orbits, which are strongly related to the connected components for the topology defined by the metrics mentioned above, determine a stratification of the set of Fredholm and semi-Fredholm operators. Finally, we calculate the distance, with respect to some of the metrics mentioned above, between different orbits of CR.Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Mbekhta, Mostafa. Universite Lille; Franci
Generalized inverses and similarity to partial isometries
No full textAbstractWe obtain some results related to the problems of Badea and Mbekhta (2005) [1] concerning the similarity to partial isometries using the generalized inverses. Especially, we involve the Moore–Penrose inverses. Also a characterization for such a similarity is given in the terms of dilations similar to unitary operators, which leads to a new criterion for the similarity to an isometry and to a quasinormal partial isometry
Split partial isometries
No full textA partial isometry V is said to be a split partial isometry if H = R(V) + N(V), with R(V)∩ N(V ) = {0} (R(V) = range of V, N(V ) = null-space of V).We study the topological properties of the set I0 of such partial isometries. Denote by I the set of all partial isometries of B(H), and by IN the set of normal partial isometries. Then IN ⊂ I0 ⊂ I, and the inclusions are proper. It is known that I is a C∞-submanifold of B(H). It is shown here that I0 is open in I, therefore is has also C∞-local structure. We characterize the set I0, in terms of metric properties, existence of special pseudoinverses, and a property of the spectrum and the resolvent of V. The connected components of I0 are characterized: V0, V1 ∈ I0 lie in the same connected component if and only if dim R(V0) = dim R(V1) and dim R(V0)⊥ = dim R(V1)⊥.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Mbekhta, Mostafa. Université de Lille 1; Franci
On the geometry of generalized inverses
No full textWe study the set S = {(a, b) ∈ A × A : aba = a, bab = b} which pairs the relatively regular elements of aBanach algebra A with their pseudoinverses, and prove that it is an analytic submanifold of A × A. If A is a C∗-algebra, inside S lies a copy the set I of partial isometries, we prove that this set is a C∞ submanifold of S (as well as a submanifold of A). These manifolds carry actions from, respectively, G_A × G_A and U_A ×U_A, where G_A is the group of invertibles of A and U_A is the subgroup of unitary elements. These actions define homogeneous reductive structures for S and I (in the differential geometric sense). Certain topological and homotopical properties of these sets are derived. In particular, it is shown that if A is a von Neumann algebra and p is a purely infinite projection of A, then the connected component I_p of p in I is simply connected. If 1 − p is also purely infinite, then I_p is contractible.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Mbekhta, Mostafa. Universite Lille; Franci
A geometry for split operators
No full textWe study the set X of split operators acting in the Hilbert space H: X = {T ∈ B(H) : N(T) ∩ R(T) = {0} and N(T) + R(T) = H}. Inside X , we consider the set Y: Y = {T ∈ X : N(T) ⊥ R(T)}. Several characterizations of these sets are given. For instance T ∈ X if and only if there exists an oblique projection Q whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. T S = ST,TST = T and STS = S). Analogous characterizations are given for Y. Two natural maps are considered: q : X → Q := {oblique projections in H}, q(T) = PR(T )//N(T ) and p : Y → P := {orthogonal projections in H}, p(T) = PR(T ), where PR(T )//N(T ) denotes the projection onto R(T) with nullspace N(T), and PR(T ) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets Xck ⊂ X of operators with rank k < ∞, and XFk ⊂ X of Fredholm operators with nullity k < ∞. For the map p there are analogous results. We show that the interior of X is XF0 ∪ XF1 , and that Xck and XFk are arc-wise connected differentiable manifolds.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Mbekhta, M.. Université de Lille; Franci
Linear maps preserving the generalized spectrum
No full textAbstract: Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA−1 for every T ∈ B(H), or φ(T) = AT trA−1 for every T ∈ B(H). Also, we prove that γ(φ(T)) = γ(T) for every T ∈ B(H) if and only if there is U ∈ B(H) unitary such that either φ(T) = UTU ∗ for every T ∈ B(H) or φ(T) = UT trU ∗ for every T ∈ B(H). Here γ(T) is the reduced minimum modulus of T. Key words: reduced minimum modulus, generalized spectrum, Jordan isomorphism,linear preserver problems. AMS Subject Class. (2000): 47B48, 47A10, 46H05. 1
Commuting maps with the Aluthge transform
No full textCette thèse se situe dans le cadre de l'analyse fonctionnelle et plus précisément dans le domaine de la théorie des opérateurs dans des espaces de Hilbert. Elle consiste à étudier les applications bijectives entre des algèbres d'opérateurs, qui commutent avec la transformation de Aluthge. Dans la première partie, nous allons étudier la transformation de Aluthge, qui joue un rôle important en théorie des opérateurs. Nous allons démontrer plusieurs résultats intéressants sur cette transformation. Ces résultats seront utilisés dans la suite de ce travail. Dans la deuxième partie, nous étudierons les bijections additives qui commutent avec la transformation de Aluthge. Nous donnerons également une forme complète des applications ω-additive qui commutent avec cette transformation. Ensuite, nous considérons les applications qui commutent avec la transformation de Aluthge sous le produit usuel et le produit de Jordan. Nous démontrerons que ces applications ont une forme simple. Dans la dernière partie, nous donnerons plusieurs expressions du rayon spectral via la transformation λ-Aluthge et ses itérées.Our aim in this thesis in function analysis is to study the bijective maps between the algebras of linear and bounded operators, which commute with the Aluthge transform in different way. In the first part, we study the Aluthge transformation which play an crucial role on operator theory in the recent years. We will establish some useful results and properties of the λ-Aluthge transform. These results are required to prove our main theorems in the next chapters. In the second part, we study the bijective and additive maps which commute with the λ-Aluthge transform. We also give a description of ω-additive commuting maps with this transformation. In the last part, we consider the problem of commuting maps with the λ-Aluthge transform, under the usual product and Jordan product, we show that these maps are a simple form. Finally, we give several expressions of the spectral radius via the λ-Aluthge transform and its iterates
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