Merge branch 'mli' into 'master'

Mli

See merge request fabiangabel/research-topics-mat-tuhh!21

topics/fredholm_and_spectral_theory.md

+# Fredholm and Spectral theory
+
+### Working Groups: aa
+
+### Collaborators (MAT): mlindner
+
+We are interested in equations $Ax = b$, where $A$ is a linear operator acting boundedly between Banach spaces. 
+$A$ is *invertible* (i.e. it has an inverse, $A^1$, that is again bounded and linear) if and only if $A$ is *injective* (its nullspace, the set of solutions $x$ to the homogeneous equation $Ax = 0$, consists of $0$ only) and *surjective* (its image/range is the whole space). 
+Deviation from both properties, and hence from invertibility, can be measured in terms of the dimension of the nullspace and the codimension of the image. 
+One says that $A$ is a *Fredholm operator* if both numbers are finite — so injectivity and surjectivity might be violated but only to some (finite dimensional) extent. 
+In particular, $Ax = b$ is solvable for all $b$ in a space of finite codimension and the solution $x$ is unique up to perturbations from a finite-dimensional space.
+The Fredholm property is robust under certain perturbations which makes it more accessible than (but still in touch with) invertibility. 
+If you want to prove that your operator $A$ is invertible (i.e. $Ax = b$ has a unique solution $x$ for all b) it is often a good idea (and a big step in the right direction) to start with its Fredholm property.
+
+Typical decay properties of infinite matrices under investigation.
+
+
+<div align="center">
+<img src="img/BO_BDO.jpg" height="200">
+</div>
+
+left: band matrices, right: norm-limits of such
+
+Our operators $A$ usually appear as infinite matrices $(A_{ij})$ — possibly with $i,j \in \mathbb{Z}^d$.
+
+and $A_{ij}$ operator-valued — usually coming from integral or differential equations (and their discretizations
+). Our studies on invertibility and Fredholm property give us (in general lower) bounds on spectrum and essential spectrum of $A$.
+
+<div align="center">
+<img src="img/sm_J=14.png" width="200">
+<img src="img/sm_n=25_zoom1.png" width="200">
+<img src="img/sm_random_full_50K.png" width="200">
+</div>
+
+Figure: Spectra of non-selfadjoint Hamiltonian of randomly hopping particle in $1D$ 
\ No newline at end of file