Proper subspaces and compatibility

Let $\mathcal{E}$ be a Banach space contained in a Hilbert space $\mathcal{L}$. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambicki\v{\i}, we say that a bounded operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with respect to the inner product of $\mathcal{L}$. By a proper subspace $\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $\mathcal{L}$, then $\mathcal{S}$ belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition to ensure that a proper subspace is compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$ which is also a proper subspace. We give a characterization of the compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several examples are provided that illustrate different situations between proper and compatible subspaces.Comment: 18 page

Dugong stranding and suggestions for conservation: a tool to support endangered marine resource conservation

February 23-25, 2010, Phuket, ThailandThe study aimed to (a) provide record of dugong stranding and mortality, (b) conduct necropsy of dugong, (c) characterize the circumstantial causes of mortality and stranding and (d) determine the threats that impede dugong conservation.Secondary data available on strandings and mortality of dugong were utilized to confirm validity of data obtained from interview of fishfarmers. Record of logged and reported incidents of dugong strandings and mortality were also retrieved. Necropsy was done every time dugong mortality occurred. Information and data were kept in dugong data sheets. Preliminary results showed that from 2001 – 2008, six strandings of dugongs have been documented in Malita, Davao del Sur. All the six strandings occurred within an approximate 10 km stretch of coastline (from Brgy. Lacaron to Brgy. Tingolo) of Malita, Davao del Sur. One out of the six strandings was a female baby dugong. The other four were male adult dugong with a length of not less than 2m. The probable causes of death were mostly anthropogenic: victim of spear fishing, blasting, entangled in net and intentionally caught for food. Parasitism was also observed in one of the dugongs that was examined where the gut, stomach, intestine were heavily infested with round worms

Geometry of unitary orbits of pinching operators

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let {pi}1w(1≤w≤∞) be a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given by P:I→I,P(x)=∑i=1wpixpi. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study geometric properties of the orbit UI(P)={LuPLu*:u∈UI}, where Lu is the left representation of UI on the algebra B(I) of bounded operators acting on I. The results include necessary and sufficient conditions for UI(P) to be a submanifold of B(I). Special features arise in the case of the ideal K of compact operators. In general, UK(P) turns out to be a non complemented submanifold of B(K). We find a necessary and sufficient condition for UK(P) to have complemented tangent spaces in B(K). We also show that UI(P) is a covering space of another orbit of pinching operators.Facultad de Ciencias Exacta

Global solutions of approximation problems in Hilbert spaces

We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way and the solvability of the associated operator problem in a fixed p-Schatten norm.Fil: Contino, Maximiliano. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Di Iorio y Lucero, María Eugenia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Fongi, Guillermina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

The compatible Grassmannian

Let A be a positive injective operator in a Hilbert space View the MathML source, and denote by View the MathML source the inner product defined by A : [f,g]=〈Af,g〉. A closed subspace S⊂H is called A -compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product View the MathML source. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=S, which is symmetric for this inner product. The compatible Grassmannian GrA is the set of all A -compatible subspaces of H. By parametrizing it via the one to one correspondence S↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form View the MathML source. A Banach–Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form View the MathML source. Each connected component in GrA of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie–Poisson space. For 1⩽p⩽∞, in the presence of a fixed View the MathML source-orthogonal (direct sum) decomposition of H, H=S0+N0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p -Schatten operators which are symmetric for the form View the MathML source. It carries the locally transitive action of the Banach–Lie group of invertible operators which preserve View the MathML source, and are of the form G=1+K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; ArgentinaFil: Di Iorio y Lucero, María Eugenia. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentin

Essentially commuting projections

Let H=H+⊕H- be a fixed orthogonal decomposition of a Hilbert space, with both subspaces of infinite dimension, and let E+, E- be the projections onto H+ and H-. We study the set Pcc of orthogonal projections P in H which essentially commute with E+ (or equivalently with E-), i.e.[P,E+]=PE+-E+Pis compact. By means of the projection π onto the Calkin algebra, one sees that these projections P∈Pcc fall into nine classes. Four discrete classes, which correspond to π(P) being 0, 1, π(E+) or π(E-), and five essential classes which we describe below. The discrete classes are, respectively, the finite rank projections, finite co-rank projections, the Sato Grassmannian of H+ and the Sato Grassmannian of H-. Thus the connected components of each of these classes are parametrized by the integers (via de rank, the co-rank or the Fredholm index, respectively). The essential classes are shown to be connected.We are interested in the geometric structure of Pcc, being the set of selfadjoint projections of the C*-algebra Bcc of operators in B(H) which essentially commute with E+. In particular, we study the problem of existence of minimal geodesics joining two given projections in the same component. We show that the Hopf-Rinow Theorem holds in the discrete classes, but not in the essential classes. Conditions for the existence and uniqueness of geodesics in these latter classes are found.Facultad de Ciencias Exacta

Computing prime factors with a Josephson phase qubit quantum processor

A quantum processor (QuP) can be used to exploit quantum mechanics to find the prime factors of composite numbers[1]. Compiled versions of Shor's algorithm have been demonstrated on ensemble quantum systems[2] and photonic systems[3-5], however this has yet to be shown using solid state quantum bits (qubits). Two advantages of superconducting qubit architectures are the use of conventional microfabrication techniques, which allow straightforward scaling to large numbers of qubits, and a toolkit of circuit elements that can be used to engineer a variety of qubit types and interactions[6, 7]. Using a number of recent qubit control and hardware advances [7-13], here we demonstrate a nine-quantum-element solid-state QuP and show three experiments to highlight its capabilities. We begin by characterizing the device with spectroscopy. Next, we produces coherent interactions between five qubits and verify bi- and tripartite entanglement via quantum state tomography (QST) [8, 12, 14, 15]. In the final experiment, we run a three-qubit compiled version of Shor's algorithm to factor the number 15, and successfully find the prime factors 48% of the time. Improvements in the superconducting qubit coherence times and more complex circuits should provide the resources necessary to factor larger composite numbers and run more intricate quantum algorithms.Comment: 5 pages, 3 figure