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Operator Theory
Published in Hugo D. Junghenn, Principles of Analysis, 2018
An operator T is said to be positive, written T≥0 $ T \ge 0 $ , if Tx|x≥0 $ \left(T x \,\boldsymbol{|} \, x\right) \ge 0 $ for all x∈H $ x\in \mathcal{H} $ . Thus a positive operator is self-adjoint; the converse is trivially false.
An Introductory Review of Quantum Mechanics
Published in Ramaswamy Jagannathan, Sameen Ahmed Khan, Quantum Mechanics of Charged Particle Beam Optics, 2019
Ramaswamy Jagannathan, Sameen Ahmed Khan
An operator for which the expectation value in any state is nonnegative is called a positive operator. Now, a^†a^ is seen to be a positive operator because
An implicit iterative algorithm with a tuning parameter for Itô Lyapunov matrix equations
Published in International Journal of Systems Science, 2018
Ying Zhang, Ai-Guo Wu, Hui-Jie Sun
It can be seen from the preceding literature that the It Lyapunov matrix equations are of great importance for stochastic systems. Some algorithms have been proposed to solve such a type of Lyapunov matrix equations. In Deng, Feng, and Liu (1996), the Kronecker product was used to transform this type of matrix equations into matrix–vector equations. Obviously, some higher-dimensional matrices appear if such a treatment is applied. In addition, an iterative algorithm was developed in Deng et al. (1996) to solve this class of Lyapunov matrix equations. In this algorithm, a standard continuous Lyapunov matrix equation needs to be solved at each iteration step. A class of coupled Lyapunov matrix equations was investigated in the literature (Li, Zhou, Lam, and Wang, 2011), and some iterative algorithms were established for these equations. The convergence properties of the proposed algorithms were proven by using positive operator theory. By specialising these algorithms, two iterative algorithms were given to solve It Lyapunov matrix equations by Li et al. (2011).
Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems
Published in International Journal of Systems Science, 2021
Recall that a self-adjoint operator is called positive, if for all and all A positive operator is called coercive, if there exists such that We are left with giving the following theorem.