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YCor
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I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential EquationsSemigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.

For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:

An analytic semigroup of bounded linear operators on a sector $\Delta_\delta$ (see definition below) that is uniformly bounded on the non-negative real axis is uniformly bounded on every closed subsector of $\Delta_\delta$.

Does anyone know of a proof of this proposition? Or do you know a counterexample?

Thank you very much in advance!

For an analytic semigroup, I use the following definition:

Let $\Delta_\delta := \{ z \in \mathbb{C} : | \arg(z) | < \delta \}$. For $z \in \Delta_\delta$ let $T(z)$ be a bounded linear operator. The family $\{T(z)\}_{z \in \Delta_\delta}$ is an analytic semigroup on $\Delta_\delta$, if

  • $z \to T(z)$ is analytic on $\Delta_\delta$.
  • $T(0) = I$ and $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Delta_\delta$.
  • $\lim_{\substack{z \to 0 \\z \in \Delta_{\delta'}}} T(z)x = x$ for all $x \in X$ and $0< \delta' < \delta$.

I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.

For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:

An analytic semigroup of bounded linear operators on a sector $\Delta_\delta$ (see definition below) that is uniformly bounded on the non-negative real axis is uniformly bounded on every closed subsector of $\Delta_\delta$.

Does anyone know of a proof of this proposition? Or do you know a counterexample?

Thank you very much in advance!

For an analytic semigroup, I use the following definition:

Let $\Delta_\delta := \{ z \in \mathbb{C} : | \arg(z) | < \delta \}$. For $z \in \Delta_\delta$ let $T(z)$ be a bounded linear operator. The family $\{T(z)\}_{z \in \Delta_\delta}$ is an analytic semigroup on $\Delta_\delta$, if

  • $z \to T(z)$ is analytic on $\Delta_\delta$.
  • $T(0) = I$ and $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Delta_\delta$.
  • $\lim_{\substack{z \to 0 \\z \in \Delta_{\delta'}}} T(z)x = x$ for all $x \in X$ and $0< \delta' < \delta$.

I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.

For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:

An analytic semigroup of bounded linear operators on a sector $\Delta_\delta$ (see definition below) that is uniformly bounded on the non-negative real axis is uniformly bounded on every closed subsector of $\Delta_\delta$.

Does anyone know of a proof of this proposition? Or do you know a counterexample?

Thank you very much in advance!

For an analytic semigroup, I use the following definition:

Let $\Delta_\delta := \{ z \in \mathbb{C} : | \arg(z) | < \delta \}$. For $z \in \Delta_\delta$ let $T(z)$ be a bounded linear operator. The family $\{T(z)\}_{z \in \Delta_\delta}$ is an analytic semigroup on $\Delta_\delta$, if

  • $z \to T(z)$ is analytic on $\Delta_\delta$.
  • $T(0) = I$ and $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Delta_\delta$.
  • $\lim_{\substack{z \to 0 \\z \in \Delta_{\delta'}}} T(z)x = x$ for all $x \in X$ and $0< \delta' < \delta$.
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Kathy
Kathy
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Does uniform boundedness carry over from the non-negative real axis to closed sectors of $\mathbb{C}$ for analytic semigroups?

I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.

For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:

An analytic semigroup of bounded linear operators on a sector $\Delta_\delta$ (see definition below) that is uniformly bounded on the non-negative real axis is uniformly bounded on every closed subsector of $\Delta_\delta$.

Does anyone know of a proof of this proposition? Or do you know a counterexample?

Thank you very much in advance!

For an analytic semigroup, I use the following definition:

Let $\Delta_\delta := \{ z \in \mathbb{C} : | \arg(z) | < \delta \}$. For $z \in \Delta_\delta$ let $T(z)$ be a bounded linear operator. The family $\{T(z)\}_{z \in \Delta_\delta}$ is an analytic semigroup on $\Delta_\delta$, if

  • $z \to T(z)$ is analytic on $\Delta_\delta$.
  • $T(0) = I$ and $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Delta_\delta$.
  • $\lim_{\substack{z \to 0 \\z \in \Delta_{\delta'}}} T(z)x = x$ for all $x \in X$ and $0< \delta' < \delta$.