427,873 research outputs found

    The Shilov boundary of an operator space - and the characterization theorems

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    We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C∗−^*-modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C∗−^*-algebras of an operator space, which generalize the algebras of adjointable operators on a C∗−^*-module, and the `imprimitivity C∗−^*-algebra'. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify, and strengthen several theorems characterizing operator algebras and modules, in a way that seems to give more information than other current proofs. We also include some general notes on the `commutative case' of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about `function modules'.Comment: This is the final revised versio

    A Rule-Based Logic for Quantum Information

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    In the present article, we explore a new approach for the study of orthomodular lattices, where we replace the problematic conjunction by a binary operator, called the Sasaki projection. We present a characterization of orthomodular lattices based on the use of an algebraic version of the Sasaki projection operator (together with orthocomplementation) rather than on the conjunction. We then define of a new logic, which we call Sasaki Orthologic, which is closely related to quantum logic, and provide a rule-based definition of this logic

    Reconstruction algorithms for a class of restricted ray transforms without added singularities

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    Let XX and X∗X^* denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data XfXf is usually based on a study of the composition X∗DXX^* D X, where DD is some local operator (usually a derivative). If X∗X^* is chosen appropriately, then X∗DXX^* D X is a Fourier Integral Operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing DD in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace DD with a nonlocal operator D~\tilde D that integrates XfXf along a curve in the data space. The result D~Xf\tilde D Xf resembles the generalized Radon transform RR of ff. The function D~Xf\tilde D Xf is defined on pairs (x0,Θ)∈U×S2(x_0,\Theta)\in U\times S^2, where U⊂R3U\subset\mathbb R^3 is an open set containing the support of ff, and S2S^2 is the unit sphere in R3\mathbb R^3. Second, we replace X∗X^* with a backprojection operator R∗R^* that integrates with respect to Θ\Theta over S2S^2. It turns out that if D~\tilde D and R∗R^* are appropriately selected, then the composition R∗D~XR^* \tilde D X is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting D~\tilde D we get access to the frequency variable Θ\Theta. In particular, we can incorporate suitable cut-offs in R∗R^* to eliminate bad directions Θ\Theta, which lead to added singularities

    Isoperimetric inequalities for the logarithmic potential operator

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    In this paper we prove that the disc is a maximiser of the Schatten pp-norm of the logarithmic potential operator among all domains of a given measure in R2\mathbb R^{2}, for all even integers 2≤p<∞2\leq p<\infty. We also show that the equilateral triangle has the largest Schatten pp-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{\'o}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.Comment: revised version with corrected formulations and arguments; to replace the previous versio

    On the H^1-L^1 boundedness of operators

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    We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H^1(R^n) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.Comment: This paper will appear in Proceedings of the American Mathematical Societ

    Koopman operator-based model reduction for switched-system control of PDEs

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    We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier--Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
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