A good reference to begin Operator Semigroups I have finished a course in Functional Analysis and am currently taking a course in Operator Theory. I would like to read some Operator semigroups on my own. I'd like to know if there would be a book that I could find (or online notes) that would introduce me to the classical results.
 A: To give you some introductory insights, I'd suggest the texts


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*What is a semi-group? by Einar Hille, which can be found in the book Studies in Real and Complex Analysis edited by I. Hirschman.

*A Heuristic Survey of the Theory and Applications of Semigroups of Operators, which is the Chapter 0 of the book Semigroups of linear operators and applications written by J. Goldstein.
To introduce you the classical results with focus on the applications to (well posedness of) PDE, I'd suggest the book


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*C$\mathbf{_0}$-Semigroups and Applications by  Ioan I. Vrabie


which is similar (in content and purpose) to the classic Semigroups of Linear Operators and Applications to Partial Differential Equations written by A. Pazy but, in a sense, more complete (the first chapter presents some preliminaries) and easier to read (in my opinion).
If you want an approach without focus on the applications to PDE, I'd suggest the book


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*One-Parameter Semigroups for Linear Evolution Equations by K. Engel and R. Nagel


which has the shorter version


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*A Short Course on Operator Semigroups.

