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start [2021/08/02 18:23]
admin
start [2021/08/14 11:04]
ctacelli
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 +
  
  
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   * Davide Addona   * Davide Addona
 +  * Loredana Caso
   * Federica Gregorio   * Federica Gregorio
   * Cristian Tacelli   * Cristian Tacelli
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-$$\sigma \left(T(t)\right)\setminus\{0\}=e^{t\sigma(A)}$$+Starting from basic concepts of operator theory, the course concentrates on spectral theory for linear operators and strongly continuous semigroups on Hilbert and in general on Banach spaces. 
 + 
 +The main task of the spectral theory of linear operators is the following:\\ 
 + 
 +Given a linear operator $A$ with domain $D(A)$ on a Banach space $X$, determine the set, called 
 +//the resolvent set// 
 +of $A$,  
 +$$\rho(A):=\{\lambda \in \mathbb C : (\lambda I-A) \text{ has a bounded inverse in }X\},$$ 
 +its complement $\sigma(A)$ in $\mathbb C$ called //the spectrum// of $A$ and, consequently, investigate the relationship between the operator $A$ and the operators 
 +$(\lambda I-A)^{-1}$, if they exist. 
 + 
 + 
 + 
 + 
 +An important application of the spectral theory is the study of the long time behaviour  
 +of the solution $u(t)=e^{tA}u_0$ to the initial value problem 
 +$$ 
 +\left\{\begin{array}{ll} 
 +u'(t)=Au(t),\quad t\ge 0,\\ 
 +u(0)=u_0\in D(A)
 +\end{array} 
 +\right. 
 +$$ 
 +where the operators $e^{tA}$ form a so-called strongly continuous semigroup generated by $A$. 
 + 
 + 
 + 
 +In this course we shall develop the basic tools from spectral theory for linear operators and apply them to strongly continuous semigroups. 
 + 
 + 
 +Topics to be covered include: 
 + 
 + 
 +  * Compact operators and Riesz-Schauder theory; 
 + 
 +  * Spectral representation theorem for bounded and unbounded operators; 
 + 
 +  * Strongly continuous and analytic semigroups; 
 + 
 +  * Spectral mapping theorems for semigroups; 
 + 
 +  * Asymptotic behaviour of strongly continuous and analytic semigroups; 
 + 
 +  * Long time behaviour of solutions to nonhomogeneous abstract Cauchy problems. 
 + 
 +Some of these topics will be elaborated on in Phase 2, where the students will have the possibility to work on projects which are related to active research. 
  
 +You can download the program and the poster here:
  
-You can download the program here:+{{ :imagines:program-isem25.pdf | ISem25 Program}}
  
-{{ :imagines:isem25-program.pdf | ISem25 Program}}+{{ :imagines:poster.pdf | ISem25 Poster}}
  
  
start.txt · Last modified: 2022/04/29 17:13 by fede