On the theory of remediability

Hassan Emamirad. "On the theory of remediability." Banach Center Publications 63.1 (2003): 165-176. <http://eudml.org/doc/282310>.

@article{HassanEmamirad2003,
abstract = {Suppose $\{G₁(t)\}_\{t≥0\}$ and $\{G₂(t)\}_\{t≥0\}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for G₁(t) and G₂(t), when the above map is well defined for all ρ ∈ (X) and satisfies the properties of a uniformly bounded semigroup on (X). In this paper we study some properties of the space (X) and we prove that when $A_i$ generate a regularized semigroup for i = 1,2, then the operator Δ defined on ℒ(X) by Δρ = A₂ρ + ρA₁ generates a tensor product regularized semigroup. Finally, we give two examples of remedial operators in radiotherapy and chemotherapy in proliferation of cancer cells.},
author = {Hassan Emamirad},
journal = {Banach Center Publications},
keywords = {remedial operator; regularized semigroups; tensor product semigroup},
language = {eng},
number = {1},
pages = {165-176},
title = {On the theory of remediability},
url = {http://eudml.org/doc/282310},
volume = {63},
year = {2003},
}

TY - JOUR
AU - Hassan Emamirad
TI - On the theory of remediability
JO - Banach Center Publications
PY - 2003
VL - 63
IS - 1
SP - 165
EP - 176
AB - Suppose ${G₁(t)}_{t≥0}$ and ${G₂(t)}_{t≥0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for G₁(t) and G₂(t), when the above map is well defined for all ρ ∈ (X) and satisfies the properties of a uniformly bounded semigroup on (X). In this paper we study some properties of the space (X) and we prove that when $A_i$ generate a regularized semigroup for i = 1,2, then the operator Δ defined on ℒ(X) by Δρ = A₂ρ + ρA₁ generates a tensor product regularized semigroup. Finally, we give two examples of remedial operators in radiotherapy and chemotherapy in proliferation of cancer cells.
LA - eng
KW - remedial operator; regularized semigroups; tensor product semigroup
UR - http://eudml.org/doc/282310
ER -