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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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TL;DR: In this paper, it was shown that the local supersymmetric extension of this action requires the algebra to be the maximal extension of the N = 1 super-Poincare algebra, and that a nontrivial propagator for the graviton exists only for d = 4 and positive cosmological constant.
59 citations
TL;DR: In this paper, a unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on B (H ) (or even more general algebras and rings).
59 citations
21 Oct 2004
TL;DR: In this article, the authors show that for effective dissipation terms essential properties of the solutions are lost in this way and show that the results transfer to estimates in the operator algebra in case of non-effective dissipation.
Abstract: problems of the form utt +Au+ b(t)ut = 0 for a function u(t) taking values in a Hilbert space H and with a positive closed operator A : H ⊇ D(A) → H can be treated by the same arguments in terms of a spectral calculus for the operator A. For the corresponding damped problem with b(t) ≡ 1 R. Ikehata and K. Nishihara investigated in [IN03] a corresponding diffusion phenomenon towards an abstract parabolic problem. A scattering theory for abstract Cauchy problems with time-dependent operator A(t) was developed by A. Arosio in [Aro84]. The treatment is closely related to our approach of Section 3.1. Coefficients depending on both variables seem to be a closely related problem. Nevertheless, there arise essential problems in dealing with utt −∆u+ b(t, x)ut = 0, u(0, ·) = u1, Dtu(0, ·) = u2.. The main point is that one has to control all frequencies in order to deduce sharp operator estimates. By means of the pseudo-differential calculus and a diagonalization/decoupling procedure J. Rauch and M. Taylor obtained in [RT75] estimates of the solution and the energy in the Calkin algebra L(L)/K(L2) of bounded modulo compact operators. The obtained pseudo-differential representations are closely related to our results restricted to the hyperbolic part. In case of non-effective dissipation their results transfer to estimates in the operator algebra. Our considerations show that for effective dissipation terms essential properties of the solutions are lost in this way. A different approach to handle coefficients depending on t and x are so-called weighted energy inequalities. By means of this technique the cited results of A. Matsumura, [Mat77], H. Uesaka, [Ues80], K. Mochizuki, [Moc77], [MN96] and F. Hirosawa / H. Nakazawa, [HN03], are obtained. All these results are estimates in L2-scale and provide no further structural information on the representation of solutions. For coefficients depending on x only and under the strong effectivity assumption b(x) ≥ c0 > 0 for large values of |x|, M. Nakao, [Nak01], has proven Lp–Lq estimates related to damped waves. His approach works on general exterior domains with further effectivity assumptions near parts of the boundary and is based on L2-estimates for the local energy. It is an interesting question to weaken the above effectivity assumption for large x and to consider coefficients estimated from below like b(x) ≥ c0 〈x〉−α for some α ∈ (0, 1). For the upper estimate |b(x)| ≤ 〈x〉−1−2 it is known from the scattering results of K. Mochizuki, [Moc77], that the solutions are asymptotically free. One may conjecture that in this case the same Lp–Lq estimates like for the free wave equation are valid.
59 citations
TL;DR: In this paper, the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and secondorder tensors were established.
Abstract: All kinds of physically possible material symmetry in two-dimensional space were investigated in a recent work of Q.-S. Zheng and J. P. Boehler. In this paper, we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and second-order tensors. These representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems.
58 citations
TL;DR: In this article, a generalization of a previous group manifold quantization formalism is proposed, in which discrete transformations in the group are allowed, and a nonabelian group replaces the ordinary (central)U(1) subgroup of the Heisenberg-Weyl-like quantum group.
Abstract: A generalization of a previous group manifold quantization formalism is proposed. In the new version the differential structure is circumvented, so that discrete transformations in the group are allowed, and a nonabelian group replaces the ordinary (central)U(1) subgroup of the Heisenberg-Weyl-like quantum group. As an example of the former we obtain the wave functions associated with the system of two identical particles, and the latter modification is used to account for the Virasoro constraints in string theory.
58 citations