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Infinitesimal generator why $limlimits_{sto 0}P_tleft(frac{P_sf-f}{s}right)=P_tLf$?

Does finiteness of $limlimits_{xtoinfty}f(x)$ and $limlimits_{xtoinfty}f'(x)$ imply $limlimits_{xtoinfty}f'(x)=0$?A confusion on Almost Everywhere ConvergenceShow $lim_{m to infty ,n to infty } f(frac{{leftlfloor {mx} rightrfloor }}{m},frac{{leftlfloor {ny} rightrfloor }}{n}) = f(x,y)$Showing that $ Pleft(limlimits_{n to infty}frac{S_n}{n} text{ exists in } mathbb Rright) = 0. $Why is $limlimits_{ntoinfty} sumlimits_{k=1}^n frac{1}{k} - log (n) = sumlimits_{k=1}^{infty} frac{1}{k}-log(frac{k+1}{k})$Why $left|int f f^{q-1}right|=|f|_q^q$?Show that $f$ is uni. cont. on $(a,b)$ if and only if it is continuous $(a,b)$ and $limlimits_{xto a^+}f(x)$ and $limlimits_{xto b^-}f(x)$ existShow there is $Cin [0,1)$ s.t. $|f(x)-f(y)|leq C|x-y|$ when $|x-y|geq a$.Why is $mu(E_{1}) + limlimits_{N rightarrow infty} sumlimits_{n=1}^{N} mu(E_{n+1} -E_{n}) = limlimits_{N rightarrow infty} mu(E_{N})$If $limlimits_{nto infty}|x_n|= 0$ then $suplimits_{ngeq 1}|x_n|<infty$

1

$begingroup$

Let $(P_t)_{tgeq 0}$ a semi group and $fin D(L)=left{fin mathcal C_0mid limlimits_{tto 0}frac{P_tf-f}{t}text{ exist}right}$ and $mathcal C_0$ is the set of continuous function s.t. $limlimits_{tto infty }f(t)=0$.

Set $$Lf=lim_{tto 0}frac{P_t f-f}{t}.$$

Suppose $fin D(L)$. We have $$lim_{sto 0}frac{P_sP_tf-P_tf}{t}=lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tLf.$$

I don't understand the last equality, i.e. why $$lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tlim_{sto 0}frac{P_sf-f}{s} ?$$

real-analysis

share|cite|improve this question

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are the $P_t$'s? Continuous linear transformations?
  $endgroup$
  – Berci
  Mar 12 at 14:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $P_t$ is s.t. $P_0=id$, $P_{t+s}=P_tcirc P_s$ and $lim_{tto 0}|P_tf-f|=0$ and $|P_t|leq 1$ for a certain norm $|cdot |$, but it's not specified. @Berci
  $endgroup$
  – Pierre
  Mar 12 at 14:37
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $(P_t)_{tgeq 0}$ a semi group and $fin D(L)=left{fin mathcal C_0mid limlimits_{tto 0}frac{P_tf-f}{t}text{ exist}right}$ and $mathcal C_0$ is the set of continuous function s.t. $limlimits_{tto infty }f(t)=0$.

Set $$Lf=lim_{tto 0}frac{P_t f-f}{t}.$$

Suppose $fin D(L)$. We have $$lim_{sto 0}frac{P_sP_tf-P_tf}{t}=lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tLf.$$

I don't understand the last equality, i.e. why $$lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tlim_{sto 0}frac{P_sf-f}{s} ?$$

real-analysis

share|cite|improve this question

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are the $P_t$'s? Continuous linear transformations?
  $endgroup$
  – Berci
  Mar 12 at 14:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $P_t$ is s.t. $P_0=id$, $P_{t+s}=P_tcirc P_s$ and $lim_{tto 0}|P_tf-f|=0$ and $|P_t|leq 1$ for a certain norm $|cdot |$, but it's not specified. @Berci
  $endgroup$
  – Pierre
  Mar 12 at 14:37
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $(P_t)_{tgeq 0}$ a semi group and $fin D(L)=left{fin mathcal C_0mid limlimits_{tto 0}frac{P_tf-f}{t}text{ exist}right}$ and $mathcal C_0$ is the set of continuous function s.t. $limlimits_{tto infty }f(t)=0$.

Set $$Lf=lim_{tto 0}frac{P_t f-f}{t}.$$

Suppose $fin D(L)$. We have $$lim_{sto 0}frac{P_sP_tf-P_tf}{t}=lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tLf.$$

I don't understand the last equality, i.e. why $$lim_{sto 0}P_tfrac{P_sf-f}{s}=P_tlim_{sto 0}frac{P_sf-f}{s} ?$$

real-analysis

share|cite|improve this question

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

$endgroup$

share|cite|improve this question

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

share|cite|improve this question

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

edited Mar 12 at 18:56

rtybase

11.5k31534

edited Mar 12 at 18:56

rtybase

11.5k31534

asked Mar 12 at 14:27

PierrePierre

7611

asked Mar 12 at 14:27

PierrePierre

7611