<p>We study the rate of growth of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Vert AT(t)\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mi>A</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t \downarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for an immediately differentiable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(T(t))_{t \ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> with generator <i>A</i>. We assume that the resolvent of the semigroup generator decays on the imaginary axis at rates described by functions of positive increase, which enable estimates on scales finer than polynomial ones. First, we present lower and upper bounds for the rates of growth of Banach space semigroups. Next, we improve the upper estimate for Hilbert space semigroups. Finally, for semigroups of normal operators on Hilbert spaces and multiplication <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroups on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spaces, we establish an estimate that exactly captures the asymptotic behavior of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Vert AT(t)\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mi>A</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t \downarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Relation between semigroup growth and resolvent decay for immediately differentiable semigroups

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2026年5月31日

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Masashi Wakaiki

摘要

We study the rate of growth of $\Vert AT(t)\Vert $ $‖ A T (t) ‖$ as $t \downarrow 0$ $t ↓ 0$ for an immediately differentiable $C_0$ $C_{0}$ -semigroup $(T(t))_{t \ge 0}$ ${(T (t))}_{t \geq 0}$ with generator A. We assume that the resolvent of the semigroup generator decays on the imaginary axis at rates described by functions of positive increase, which enable estimates on scales finer than polynomial ones. First, we present lower and upper bounds for the rates of growth of Banach space semigroups. Next, we improve the upper estimate for Hilbert space semigroups. Finally, for semigroups of normal operators on Hilbert spaces and multiplication $C_0$ $C_{0}$ -semigroups on $L^p$ $L^{p}$ -spaces, we establish an estimate that exactly captures the asymptotic behavior of $\Vert AT(t)\Vert $ $‖ A T (t) ‖$ as $t \downarrow 0$ $t ↓ 0$ .