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• W2111571651 abstract "Given the infinitesimal generator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 0> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>C_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-semigroup on 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> which satisfies the Kreiss resolvent condition, i.e., there exists an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>M>0</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=double-vertical-bar left-parenthesis s upper I minus upper A right-parenthesis Superscript negative 1 Baseline double-vertical-bar less-than-or-equal-to StartFraction upper M Over normal upper R normal e left-parenthesis s right-parenthesis EndFraction> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mi>I</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mi>M</mml:mi> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>R</mml:mi> <mml:mi mathvariant=normal>e</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>| (sI-A)^{-1}| leq frac {M}{mathrm {Re}(s)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all complex <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 0> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>C_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Furthermore, we show that for every <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of left-parenthesis 0 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Superscript gamma> <mml:semantics> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>γ<!-- γ --></mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>t^gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a consequence, we find that for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript upper N> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>{mathbb R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the standard Euclidian norm the estimate <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar exp left-parenthesis upper A t right-parenthesis double-vertical-bar less-than-or-equal-to upper M 1 min left-parenthesis upper N comma t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo movablelimits=true form=prefix>min</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|exp (At)| leq M_1 min (N,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cannot be replaced by a lower power of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2111571651 date "2006-07-10" @default.
• W2111571651 modified "2023-10-16" @default.
• W2111571651 title "Continuous-time Kreiss resolvent condition on infinite-dimensional spaces" @default.
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