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Analysis on Locally Compact Groups
Published in Hugo D. Junghenn, Principles of Analysis, 2018
It is notable that continuity of π $ \pi $ in the strong operator topology is equivalent to continuity in the weak operator topology. Indeed, if (Uα) $ (U_\alpha ) $ is a net of unitary operators converging in the weak operator topology to a unitary operator U, then Uαx-Ux2=Uαx2-2ReUαx|Ux+Ux2=2x2-2ReUαx|Ux→2x2-2Ux|Ux=0. $$ \begin{aligned} \left\Vert U_\alpha x- U x\right\Vert^2&= \left\Vert U_\alpha x\right\Vert^2 - 2\mathrm{Re}\,\left(U_\alpha x \,\boldsymbol{|} \, U x\right) + \left\Vert U x\right\Vert^2 = 2\left\Vert x\right\Vert^2 - 2\mathrm{Re}\,\left(U_\alpha x \,\boldsymbol{|} \, U x\right) \\&\rightarrow 2\left\Vert x\right\Vert^2 - 2\left(U x \,\boldsymbol{|} \, U x\right) = 0. \end{aligned} $$
Fundamental solutions for semi-linear neutral retarded integro-differential systems and applications to control problems
Published in Optimization, 2022
In the sequel, we shall study the approximate controllability for System (4) by using the so-called resolvent operator condition. For this purpose, we introduce the following resolvent operator as follows. Let where and denote, respectively, the adjoint operators of B and G, then the resolvent operator for is given by Since the operator is clearly positive, is well defined. We will assume in this section that as in the strong operator topology.
Approximate controllability for semilinear second-order stochastic evolution systems with infinite delay
Published in International Journal of Control, 2020
In the sequel we shall study the approximate controllability for System (1) by assuming the approximate controllability of the linear deterministic system corresponding to (1). For this purpose, we introduce the following resolvent operator. Let where and denote respectively the adjoint operators of B and G, then the resolvent operator for is given by Since the operator is clearly positive, is well defined. We will always assume that as in the strong operator topology.
Optimal quantization via dynamics
Published in Dynamical Systems, 2020
Joseph Rosenblatt, Mrinal Kanti Roychowdhury
The above discussion show us that the following Baire category result gives us some important information about the limits of geometric distortion errors for the class of stationary dynamical systems. We take to be the group of invertible, measure-preserving transformations of the probability space . We consider the weak topology on , i.e. the strong operator topology on the group of continuous linear operators , with . A standard fact is that the weak topology on is given by a complete pseudo-metric. In particular, it is a Baire space, i.e. any intersection of a countable set of open, dense sets is also dense. We say that a set in a Baire space is residual if it contains such a dense set, and a set is meager if it is a complement of a residual set.