Question

Operator semigroup: Notation Ambiguity

score 156 · Answer 1 · 2026-03-22 16:59:09

156

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Operator semigroup: Notation Ambiguity

Published on 22 Mar 2026 - 16:59

score 80 · Answer 2 · 2026-03-22 07:23:01

80

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$\{T(t):t\geq 0\}$ is a strongly continuous semi-group, suppose $x\in \mathbb{X}$, $w-\lim_{t\to 0^{+}}\frac{1}{t}(T(t)-I)x=y$, show that $y=Ax$.

Published on 22 Mar 2026 - 7:23

score 132 · Answer 3 · 2026-03-21 08:14:55

132

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Spectral gap of the generator of a contraction semi-group

Published on 21 Mar 2026 - 8:14

score 706 · Answer 4 · 2026-03-22 15:15:34

706

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Is every strongly continuous semi-group uniformly continuous?

Published on 22 Mar 2026 - 15:15

score 112 · Answer 5 · 2026-03-22 09:59:42

112

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Infinitesimal generator of $(T(t)u)(s) $ is $(Au)(s)=\lambda\Big(u(s-\mu)-u(s)\Big)$

Published on 22 Mar 2026 - 9:59

score 102 · Answer 6 · 2026-03-22 13:24:14

102

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When $L$ is the generator of an analytic semigroup and $\alpha, \beta >0$, $(-L)^{-\alpha}(-L)^{-\beta}= (-L)^{-\alpha-\beta}$

Published on 22 Mar 2026 - 13:24

score 227 · Answer 7 · 2026-03-22 08:47:50

227

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why $\lambda \Vert x \Vert \leq \Vert (\lambda-A)x\Vert$ imply $R(\lambda-A)$ is a closed subspace of $\mathscr{X}$ and $\lambda-A$ is 1-1 on $D(A)$?

Published on 22 Mar 2026 - 8:47

score 65 · Answer 8 · 2026-03-22 01:21:17

65

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If $R(\lambda-B)=\mathscr{X}=R(\lambda-A)$ , how to illustrate the $B=A$?

Published on 22 Mar 2026 - 1:21

score 79 · Answer 9 · 2026-03-22 00:30:05

79

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Why is $T(t+s)=T(t)T(s)$ called semigroup property?

Published on 22 Mar 2026 - 0:30

score 707 · Answer 10 · 2026-03-22 09:53:19

707

Views

semigroup property for Markov semigroup

Published on 22 Mar 2026 - 9:53

score 588 · Answer 11 · 2026-03-22 13:12:40

588

Views

Prove that the Ornstein Uhlenbeck semigroup is given by $E(f(X_t))$ where $X_t$ is the Ornstein Uhlenbeck process

Published on 22 Mar 2026 - 13:12

score 54 · Answer 12 · 2026-03-23 01:55:19

54

Views

Fractional calculus reference for hypercyclicity of fractional derivative.

Published on 23 Mar 2026 - 1:55

score 160 · Answer 13 · 2026-03-21 19:31:51

160

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Properties of a norm defined using the resolvent of an operator

Published on 21 Mar 2026 - 19:31

score 194 · Answer 14 · 2026-03-22 15:32:36

194

Views

Strong continuity and $C_0$ semigroups

Published on 22 Mar 2026 - 15:32

score 58 · Answer 15 · 2026-03-17 05:28:58

58

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isomorphism generalized semigroup

Published on 17 Mar 2026 - 5:28

List Question

Operator semigroup: Notation Ambiguity

$\{T(t):t\geq 0\}$ is a strongly continuous semi-group, suppose $x\in \mathbb{X}$, $w-\lim_{t\to 0^{+}}\frac{1}{t}(T(t)-I)x=y$, show that $y=Ax$.

Spectral gap of the generator of a contraction semi-group

Is every strongly continuous semi-group uniformly continuous?

Infinitesimal generator of $(T(t)u)(s) $ is $(Au)(s)=\lambda\Big(u(s-\mu)-u(s)\Big)$

When $L$ is the generator of an analytic semigroup and $\alpha, \beta >0$, $(-L)^{-\alpha}(-L)^{-\beta}= (-L)^{-\alpha-\beta}$

why $\lambda \Vert x \Vert \leq \Vert (\lambda-A)x\Vert$ imply $R(\lambda-A)$ is a closed subspace of $\mathscr{X}$ and $\lambda-A$ is 1-1 on $D(A)$?

If $R(\lambda-B)=\mathscr{X}=R(\lambda-A)$ , how to illustrate the $B=A$?

Why is $T(t+s)=T(t)T(s)$ called semigroup property?

semigroup property for Markov semigroup

Prove that the Ornstein Uhlenbeck semigroup is given by $E(f(X_t))$ where $X_t$ is the Ornstein Uhlenbeck process

Fractional calculus reference for hypercyclicity of fractional derivative.

Properties of a norm defined using the resolvent of an operator

Strong continuity and $C_0$ semigroups

isomorphism generalized semigroup

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