Results 1 - 10 of 8059
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[en] Formulation of modular theory for weakly closed J-involutive algebras of bounded operators in Pontryagin spaces is continued. Spectrum of the modular operator of such an algebra is investigated in detail. The existence of strongly continuous J-unitary group is established and Tomita's fundamental theorem is proved under the assumption that the spectrum of Δ belongs to the right half-plane
[en] In this paper they discuss some topics related to the general theory of frames. In particular, they focus their attention to the existence of different ''reconstruction formulas'' for a given vector of a certain Hilbert space and to some refinement of the perturbative approach for the computation of the dual frame
[en] Consider an operator equation F(u) = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator F'(u) is not boundedly invertible, and well-posed otherwise. If F is monotone C2loc(H) operator, then we construct a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity and (3) the limit is the minimum norm solution to the equation F(u) = 0. An example of applications to linear ill-posed operator equation is given. (letter to the editor)
[en] In completely local settings, we establish that a dynamically evolving spherically symmetric black hole horizon can be assigned a Hawking temperature and with the emission of flux, radius of the horizon shrinks. (author)
[en] We introduce random-matrix theory to study the tomographic efficiency of a wide class of measurements constructed out of weighted 2-designs, including symmetric informationally complete (SIC) probability operator measurements (POMs). In particular, we derive analytic formulas for the mean Hilbert-Schmidt distance and the mean trace distance between the estimator and the true state, which clearly show the difference between the scaling behaviors of the two error measures with the dimension of the Hilbert space. We then prove that the product SIC POMs, the multipartite analog of the SIC POMs, are optimal among all product measurements in the same sense as the SIC POMs are optimal among all joint measurements. We further show that, for bipartite systems, there is only a marginal efficiency advantage of the joint SIC POMs over the product SIC POMs. In marked contrast, for multipartite systems, the efficiency advantage of the joint SIC POMs increases exponentially with the number of parties.
[en] The notation of generalized Bessel multipliers is obtained by a bounded operator on which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular, it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in a special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.
[en] This paper studies weaving properties of a family of operators which are analysis and synthesis systems with frame-like properties for closed subspaces of a separable Hilbert space , where the lower frame condition is controlled by a bounded operator on . In short, this family of operators is called a -g-frame, where is a bounded operator on . We present sufficient conditions for weaving -g-frames in separable Hilbert spaces. A characterization of weaving -g-frames in terms of an operator is given. It is shown that if frame bounds of frames associated with atomic spaces are positively confined, then -g-woven frames gives ordinary weaving -frames and vice-versa. We provide classes of operators for weaving -g-frames.
[en] In the present paper we establish several more accurate Hilbert-type inequalities in a difference form. Our main results rely on some recent improvements of the Young inequality. First, we give refinements and reverses of the Hölder inequality. Then, by virtue of the improved Hölder inequality, we give two classes of refinements and reverses for general Hilbert-type inequalities. As an application, we give strengthened forms of the classical Hilbert and Hardy inequalities.
[en] We propose a preconditioned alternating direction method of multipliers (ADMM) to solve linear inverse problems in Hilbert spaces with constraints, where the feature of the sought solution under a linear transformation is captured by a possibly non-smooth convex function. During each iteration step, our method avoids solving large linear systems by choosing a suitable preconditioning operator. In case the data is given exactly, we prove the convergence of our preconditioned ADMM without assuming the existence of a Lagrange multiplier. In case the data is corrupted by noise, we propose a stopping rule using information on noise level and show that our preconditioned ADMM is a regularization method; we also propose a heuristic rule when the information on noise level is unavailable or unreliable and give its detailed analysis. Numerical examples are presented to test the performance of the proposed method. (paper)