Continuous-time Kreiss resolvent condition on infinite-dimensional spaces
2006-07-10

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>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">M&gt;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>.