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• W2080243363 abstract "A strongly continuous semigroup of bounded linear operators <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>T(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than-or-slanted-equals 0> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t geqslant 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the Banach space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has asynchronous exponential growth with intrinsic growth constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{lambda _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided that there is a nonzero finite rank operator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{P_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=limit Underscript t right-arrow normal infinity Endscripts e Superscript minus lamda 0 t Baseline upper T left-parenthesis t right-parenthesis equals upper P 0> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:munder> <mml:mo movablelimits=true form=prefix>lim</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>t</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{lim _{t to infty }}{e^{ - {lambda _0}t}}T(t) = {P_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Necessary and sufficient conditions are established for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>T(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than-or-slanted-equals 0> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t geqslant 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth." @default.
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• W2080243363 date "1987-01-01" @default.
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• W2080243363 title "An operator-theoretic formulation of asynchronous exponential growth" @default.
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