Lindelöf's principle in infinite dimensions

1974 ◽  
pp. 41-57 ◽  
Author(s):  
Michel Hervé
Keyword(s):  
Infinite Dimensions

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

scholarly journals Correction to: Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions

2021 ◽  
Author(s):  
Nawaf Bou-Rabee ◽  
Andreas Eberle
Keyword(s):  
Monte Carlo ◽  
Infinite Dimensions ◽  
Hamiltonian Monte Carlo

A Correction to this paper has been published: 10.1007/s40072-020-00175-6

scholarly journals Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions

2019 ◽  
Vol 374 (2) ◽  
pp. 823-871 ◽  
Author(s):  
Simon Becker ◽  
Nilanjana Datta
Keyword(s):  
Convergence Rates ◽  
Lipschitz Continuity ◽  
Open Quantum Systems ◽  
High Energy ◽  
Quantum Systems ◽  
Quantum Evolution ◽  
Infinite Dimensions ◽  
Quantum Brownian Motion ◽  
Infinite Dimensional ◽  
Energy Constrained

Abstract By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law.

scholarly journals Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions

1986 ◽  
Vol 65 (3) ◽  
pp. 368-405 ◽  
Author(s):  
Michael G Crandall ◽  
Pierre-Louis Lions
Keyword(s):  
Viscosity Solutions ◽  
Infinite Dimensions ◽  
Jacobi Equations

scholarly journals Quantum critical local spin dynamics near the Mott metal-insulator transition in infinite dimensions

2017 ◽  
Vol 95 (16) ◽  
Author(s):  
Nagamalleswararao Dasari ◽  
N. S. Vidhyadhiraja ◽  
Mark Jarrell ◽  
Ross H. McKenzie
Keyword(s):  
Spin Dynamics ◽  
Insulator Transition ◽  
Infinite Dimensions ◽  
Metal Insulator Transition ◽  
Metal Insulator ◽  
Local Spin ◽  
Quantum Critical

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

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