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• W2055965844 abstract "One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices <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> under consideration. This so-called resolvent condition is known to imply, for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>nge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the upper bounds <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar upper A Superscript n Baseline double-vertical-bar less-than-or-equal-to e upper K left-parenthesis upper N plus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>e</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|A^n|le eK(N+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar upper A Superscript n Baseline double-vertical-bar less-than-or-equal-to e upper K left-parenthesis n plus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>e</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|A^n|le eK(n+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar dot double-vertical-bar> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|cdot |</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the spectral norm, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the constant occurring in the resolvent condition, and the order of <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> is equal to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N plus 1 greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N+1ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is a long-standing problem whether these upper bounds can be sharpened, for all fixed <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to bounds in which the right-hand members grow much slower than linearly with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N plus 1> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n plus 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>epsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there are fixed values <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C greater-than 0 comma upper K greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>C>0, K>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a sequence of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper N plus 1 right-parenthesis times left-parenthesis upper N plus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(N+1)times (N+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript upper N> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>A_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, satisfying the resolvent condition, 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 upper A Subscript upper N Baseline right-parenthesis Superscript n Baseline double-vertical-bar greater-than-or-equal-to upper C left-parenthesis upper N plus 1 right-parenthesis Superscript 1 minus epsilon> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>|(A_N)^n|ge C(N+ 1)^{1-epsilon }</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=equals upper C left-parenthesis n plus 1 right-parenthesis Superscript 1 minus epsilon> <mml:semantics> <mml:mrow> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>=C(n+1)^{1-epsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N equals n equals 1 comma 2 comma 3 comma ellipsis> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N=n=1,2,3,ldots</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The result proved in this paper is also relevant to matrices <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> whose <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:annotation encoding=application/x-tex>epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pseudospectra lie at a distance not exceeding <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K epsilon> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Kepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the unit disk for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>epsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2055965844 title "About the sharpness of the stability estimates in the Kreiss matrix theorem" @default.
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