Fredholm spectrum

A bounded operator $T$ on a complex Hilbert space is said to be essentially left invertible (respectively, essentially right invertible) if there exists a bounded operator $X$ such that $X T - I$ (respectively, $T X - I$) has finite rank. An operator that is essentially left or right invertible is also called a semi-Fredholm operator, and $T$ is a Fredholm operator if it is both left and right essentially invertible. F.V. Atkinson proved that an operator is Fredholm if and only if it has closed range and the spaces $\text{ker } T$ and $\operatorname{coker}T$ have finite dimension. For a semi-Fredholm operator $T$ one defines the index

\begin{equation*} \chi _ { T } = \operatorname { dim } \operatorname { ker } T - \operatorname { dim } \text { coker } T; \end{equation*}

this is an integer or $\pm \infty$. The set of semi-Fredholm operators is open, and the function $\chi _{ T }$ is continuous (thus locally constant). Moreover, $\chi_{ T + K} = \chi _{T}$ if $K$ is a compact operator; $\chi _{ T }$ is finite if $T$ is Fredholm. Let $\sigma _ { \text { lre } } ( T )$ be the set of complex scalars $\lambda$ such that $\lambda I - T$ is not semi-Fredholm; $\sigma _ { \text { lre } } ( T )$ is a closed subset of $\sigma ( T )$ (cf. Spectrum of an operator). The difference $\sigma ( T ) \backslash \sigma |_ { \text { lre } } ( T )$ is called the semi-Fredholm spectrum of $T$, and a value $\lambda$ in this set belongs to the Fredholm spectrum of $T$ if $\lambda I - T$ is Fredholm. If $G$ is one of the connected components of ${\bf C} \backslash \sigma _ { \text{lre} } ( T )$, then the number $\chi_{ \lambda I - T}$ is constant for $\lambda \in G$; call it $k_G$. If $k _ { G } \neq 0$, then $G$ is entirely contained in the semi-Fredholm spectrum of $T$ (in the Fredholm spectrum if $k _ { G } \notin \{ \pm \infty , 0 \}$). If $k _ { G } = 0$, then either $G$ is contained in the Fredholm spectrum of $T$, or $\sigma ( T ) \cap G$ has no accumulation points in $G$. More generally, I.C. Gohberg showed that $\dim \operatorname{ker}( \lambda I - T )$ is constant for $\lambda \in G$, with the possible exception of a countable set with no accumulation points in $G$; the dimension is greater if $\lambda$ is one of the exceptional points.

An interesting calculation of the Fredholm spectrum was done by Gohberg when $T$ is a Toeplitz operator (cf. also Toeplitz matrix; Calderón–Toeplitz operator) whose symbol $f$ is a continuous function on the unit circle. In this case $\sigma _ { \text { lre } } ( T )$ is the range of $f$, and $\chi_{ \lambda I - T}$ equals minus the winding number of $f$ about $\lambda$ if $\lambda \notin \sigma _ {\text { lre } } ( T )$.

The presence of a semi-Fredholm spectrum is related with the notion of quasi-triangularity introduced by P.R. Halmos. An operator $T$ is quasi-triangular if it can be written as $T = T _ { 1 } + K$ where $K$ is compact and $T _ { 1 }$ is triangular in some basis. R.G. Douglas and C.M. Pearcy showed that $T$ cannot be quasi-triangular if $\chi _ { \lambda I - T } < 0$ for some $\lambda$ in the semi-Fredholm spectrum of $T$. Quite surprisingly, the converse of this statement is also true, as shown by C. Apostol, C. Foiaş and D. Voiculescu. Subsequent developments involving the Fredholm spectrum include the approximation theory of Hilbert-space operators developed by Apostol, D. Herrero, and Voiculescu.

The notion of Fredholm spectrum can be extended to operators on other topological vector spaces, to unbounded operators, and to $n$-tuples of commuting operators. A different generalization is to define the Fredholm spectrum for elements in von Neumann algebras. The algebra of bounded operators on a Hilbert space is a factor of type $\mathrm{II} _ { \infty }$. An appropriate notion of finite-rank element exists in any factor of type $\mathrm{II} _ { \infty }$, and this leads to a corresponding notion of Fredholm operator and Fredholm spectrum. Analogues have been found in other Banach algebras as well.

The Fredholm property was also defined in a non-linear context by S. Smale. A differentiable mapping (cf. also Differentiation of a mapping) between two open sets in a Banach space is Fredholm if its derivative at every point is a linear Fredholm operator. This leads to the notion of a Fredholm mapping on an infinite-dimensional manifold. Smale used this notion to extend to infinite dimensions certain results of A. Sard and R. Thom.