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start [2021/08/04 10:55] ctacelli |
start [2022/04/29 17:13] fede |
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supervisor for each project. | supervisor for each project. | ||

- | **- Phase 3** (June 20 to June 25, 2022): There will be a final one-week | + | **- Phase 3** (June 20 to June 24, 2022): There will be a final one-week |

workshop in Salerno (Italy), where the project | workshop in Salerno (Italy), where the project | ||

teams of Phase 2 will present their projects and additional lectures | teams of Phase 2 will present their projects and additional lectures | ||

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==== ISem Team 2021/2022 ==== | ==== ISem Team 2021/2022 ==== | ||

- | === Virtual Lecture === | + | === Virtual Lecturers === |

* Angela Albanese | * Angela Albanese | ||

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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 $A$, | ||

+ | $$\rho(A):(\lambda I-A) \text{ has a bounded inverse in }X\},$$ | ||

+ | its complement $\sigma(A)$ in $\mathbb C$ called //the spectrum// of $A$ and, consequently, | ||

+ | $(\lambda I-A)^{-1}$, 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: |

- | {{ :imagines:isem25-program.pdf | ISem25 Program}} | + | {{ :imagines:poster.pdf | ISem25 Poster}} |

start.txt · Last modified: 2022/04/29 17:13 by fede

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