scholarly journals Direct integrals of strongly continuous operator semigroups

2020 ◽  
Vol 489 (2) ◽  
pp. 124176
Author(s):  
Abraham C.S. Ng
Keyword(s):  
Continuous Operator ◽  
Operator Semigroups

scholarly journals Lower bounds and the asymptotic behaviour of positive operator semigroups

2017 ◽  
Vol 38 (8) ◽  
pp. 3012-3041 ◽  
Author(s):  
MORITZ GERLACH ◽  
JOCHEN GLÜCK
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Lower Bounds ◽  
Simple Proof ◽  
Continuous Operator ◽  
Banach Lattices ◽  
Operator Semigroups ◽  
Markov Operators ◽  
Discrete Semigroups ◽  
Analytical Tools

If $(T_{t})$ is a semigroup of Markov operators on an $L^{1}$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t\rightarrow \infty$. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius–Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

scholarly journals Hypercyclic and mixing operator semigroups

2011 ◽  
Vol 54 (3) ◽  
pp. 761-782 ◽  
Author(s):  
Stanislav Shkarin
Keyword(s):  
Continuous Operator ◽  
Vector Spaces ◽  
Operator Semigroups ◽  
Topological Vector Spaces ◽  
Operator Group ◽  
Uniformly Continuous

AbstractWe describe a class of topological vector spaces admitting a mixing uniformly continuous operator group$\smash{\{T_t\}_{t\in\mathbb{C}^n}}$with holomorphic dependence on the parametert. This result builds on those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups$\smash{\{T_t\}_{t\geq0}}$.

THE REPRESENTATION OF STRONGLY CONTINUOUS OPERATOR SEMIGROUPS ON VECTOR BUNDLES AS HIDA DISTRIBUTIONS

2009 ◽  
Vol 12 (03) ◽  
pp. 511-519
Author(s):  
BATU GÜNEYSU
Keyword(s):  
Vector Bundle ◽  
Compact Manifold ◽  
Vector Bundles ◽  
Short Note ◽  
Adjoint Operator ◽  
Continuous Operator ◽  
Time Parameter ◽  
Heat Semigroup ◽  
Operator Semigroups ◽  
Schrödinger Semigroup

Let E → M be a vector bundle over a compact manifold. In this short note we prove that the Schrödinger semigroup (complex heat semigroup) associated to a self-adjoint (non-negative self-adjoint) operator in ΓL2(M, E) corresponds to a ΓL2(M, E)*-valued Hida distribution, which depends continuously (holomorphically) on the time parameter.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

Fractional BVPs with strong time singularities and the limit properties of their solutions

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Svatoslav Staněk
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Existence Result ◽  
Continuous Operator ◽  
Nonlinear Alternative ◽  
Time Singularities

AbstractIn the first part, we investigate the singular BVP $$\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u$$, u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $$\mathcal{H}$$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems $$\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)$$, u(0) = A, u(1) = B, $$\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0$$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.

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