Fredholm mapping

For Banach spaces $X$, $Y$ (cf. Banach space), let $B ( X , Y )$ denote the set of bounded linear operators $T$ from $X$ to $Y$ with domain $D ( T ) = X$ (cf. also Linear operator). An operator $A \in B ( X , Y )$ is called a Fredholm mapping if

1) $\alpha ( A ) : = \operatorname { dim } N ( A ) < \infty$;

2) $R ( A )$ is closed in $Y$;

3) $\beta ( A ) : = \operatorname { codim } R ( A ) < \infty$. Here, $N ( A )$, $R ( A )$ denote the null space and range of $A$, respectively.

Properties.

Let $X$, $Y$, $Z$ be Banach spaces. If $A \in \Phi ( X , Y )$ and $B \in \Phi ( Y , Z )$, then $B A \in \Phi ( X , Z )$ and

\begin{equation} \tag{a1} i ( B A ) = i ( B ) + i ( A ), \end{equation}

where $i ( A ) = \alpha ( A ) - \beta ( A )$ (the index). If $A \in \Phi ( X , Y )$ and $K$ is a compact operator from $X$ to $Y$, then $A + K \in \Phi ( X , Y )$ and

\begin{equation} \tag{a2} i ( A + K ) = i ( A ). \end{equation}

Moreover, for each $A \in \Phi ( X , Y )$ there is a $\delta > 0$ such that $A + T \in \Phi ( X , Y )$ and

\begin{equation} \tag{a3} i ( A + T ) = i ( A ) , \quad \alpha ( A + T ) \leq \alpha ( A ) \end{equation}

for each bounded mapping from $X$ to $Y$ such that $\| T \| < \delta$. If $A \in B ( X , Y )$, $B \in B ( Y , Z )$ are such that $B A \in \Phi ( X , Z )$, then $\alpha ( B ) < \infty$ implies that $A \in \Phi ( X , Y )$ and $B \in \Phi ( Y , Z )$. The same is true if $\beta ( A ) < \infty$. If $A \in \Phi ( X , Y )$, then its adjoint operator $A ^ { \prime }$ is in $( Y ^ { \prime } , X ^ { \prime } )$ with $i ( A ^ { \prime } ) = - i ( A )$, where $X ^ { \prime }$, $Y ^ { \prime }$ denote the dual spaces of $X$, $Y$, respectively (cf. also Adjoint space).

If $Y = X$, it follows that $A ^ { n } \in \Phi ( X ) = \Phi ( X , X )$ for each positive integer $n$ if $A \in \Phi ( X )$. Let

\begin{equation*} r ( A ) = \operatorname { lim } _ { n \rightarrow \infty } \alpha ( A ^ { n } ) \end{equation*}

and

\begin{equation*} r ^ { \prime } ( A ) = \operatorname { lim } _ { n \rightarrow \infty } \beta ( A ^ { n } ). \end{equation*}

A necessary and sufficient condition for both $r ( A )$ and $r ^ { \prime } ( A )$ to be finite is that there exist an integer $n \geq 1$ and operators $E \in B ( X ) = B ( X , X )$ and $K$, compact on $X$, such that $E A ^ { n } = A ^ { n } E = I - K$, where $I$ denotes the identity operator.

Semi-Fredholm operators.

Let $\Phi _ { + } ( X , Y )$ denote the set of all $A \in B ( X , Y )$ such that $R ( A )$ is closed in $Y$ and $\alpha ( A ) < \infty$. Similarly, $\Phi _ { - } ( X , Y )$ is the set of all $A \in B ( X , Y )$ such that $R ( A )$ is closed in $Y$ and $\beta ( A ) < \infty$. If $A \in \Phi _ { + } ( X , Y ) \backslash \Phi ( X , Y )$, then $i ( A ) = - \infty$. If $A \in \Phi _ { - } ( X , Y ) \backslash \Phi ( X , Y ),$ then $i ( A ) = + \infty$. If $A \in \Phi _ { \pm } ( X , Y )$ and $K$ is compact from $X$ to $Y$, then $A + K \in \Phi _ { \pm } ( X , Y )$ and $i ( A + K ) = i ( A )$. If $A \in \Phi _ { \pm } ( X , Y )$, then there is a $\delta > 0$ such that $A + T \in \Phi _ { \pm } ( X , Y )$, $\alpha ( A + T ) \leq \alpha ( A )$, $\beta ( A + T ) \leq \beta ( A )$, and $i ( A + T ) = i ( A )$ for any $T \in B ( X , Y )$ such that $\| T \| < \delta$.

Non-linear Fredholm mappings.

Let $X$, $Y$ be Banach spaces, and let $\Omega$ be an open connected subset of $X$. A continuously Fréchet-differentiable mapping $F ( x )$ from $\Omega$ to $Y$ (cf. also Fréchet derivative) is Fredholm if $F ^ { \prime } ( x ) \in \Phi ( X , Y )$ for each $x \in \Omega$. Set $i ( F ( x ) ) = i ( F ^ { \prime } ( x ) )$. It is independent of $x$. If $F$ is a diffeomorphism, then $i ( F ( x ) ) = 0$. If $K ( x ) \in C ^ { 1 } ( \Omega , Y )$ is a compact operator, then $F ( x ) + K ( x )$ is Fredholm with $i ( F + K ) = i ( F )$. A useful extension of the Sard theorem due to S. Smale [a2] states that if $X$, $Y$ are separable (cf. also Separable space), $F ( x ) \in C ^ { k } ( \Omega , Y )$ with $k > \operatorname { max } ( i ( F ) , 0 )$, then the critical values of $F ( x )$ are nowhere dense in $Y$ (cf. also Nowhere-dense set). It follows from this that if $F ( x )$ has negative index, then $F ( \Omega )$ contains no interior points, i.e., if there is an $x _ { 0 } \in \Omega$ such that $F ( x _ { 0 } ) = y _ { 0 }$, then there are points $y$ arbitrarily close to $y _ { 0 }$ such that $F ( x ) = y$ has no solution in $\Omega$. Consequently, such equations are not considered well posed if $F$ has negative index.

Perturbation theory.

The classes $\Phi ( X , Y )$ and $\Phi _ { \pm } ( X , Y )$ are stable under various types of perturbations. The set $F ( X , Y )$ of Fredholm perturbations is the set of those $S \in B ( X , Y )$ such that $A - S \in \Phi ( X , Y )$ whenever $A \in \Phi ( X , Y )$. The sets $F _ { \pm } ( X , Y )$ of semi-Fredholm perturbations are defined similarly. As noted, compact operators from $X$ to $Y$ are in $F ( X , Y )$ and $F _ { \pm } ( X , Y )$. So are strictly singular operators [a3] (in some spaces they may be non-compact). An operator $S \in B ( X , Y )$ is in $F _ { + } ( X , Y )$ if and only if $\alpha ( A - S ) < \infty$ for all $A \in \Phi _ { + } ( X , Y )$. Similarly, it is in $F _ { - } ( X , Y )$ if and only if $\beta ( A - S ) < \infty$ for all $A \in \Phi _ { - } ( X , Y )$. But $S \in F ( X , Y )$ if and only if $\alpha ( A - S ) < \infty$ for all $A \in \Phi ( X , Y )$. On the other hand, $A \in \Phi _ { + } ( X , Y )$ if and only if $\alpha ( A - K ) < \infty$ for all compact operators $K$ from $X$ to $Y$. Also, $A \in \Phi _ { - } ( X , Y )$ if and only if $\beta ( A - K ) < \infty$ for all such $K$. Consequently, $A \in \Phi ( X , Y )$ if and only if $\alpha ( A - K ) < \infty$ and $\beta ( A - K ) < \infty$ for all compact operators $K$ from $X$ to $Y$.

Perturbation functions.

There are several known "constants" that determine either the fact that a mapping is Fredholm or limit the size of arbitrary perturbations to keep the sum Fredholm. A well-known constant is due to T. Kato [a4]:

\begin{equation*} \gamma ( T ) = \operatorname { inf } \frac { \| Tx \| } { d ( x , N ( T ) ) }, \end{equation*}

where the infimum is taken over those $x \in X$ such that $d ( x , N ( T ) ) > 0$. If $\| T \| < \gamma ( A )$ and $A \in \Phi _ { + } ( X , Y )$, then $A + T \in \Phi_+ ( X , Y )$ with (a3) holding. Other constants are:

$\mu ( A ) = \operatorname { inf } \{ \| T \| : \alpha ( A - T ) = \infty \}$. A mapping $A \in B ( X , Y )$ is in $\Phi _ { + } ( X , Y )$ if and only if $\mu ( A ) > 0$. Moreover, if $A \in \Phi _ { + } ( X , Y )$ and $\| T \| < \mu ( A )$, then $A + T \in \Phi_+ ( X , Y )$ with (a3) holding.

$\Gamma ( A ) = \operatorname { inf } _ { M } \| A |_M \|$, where the infimum is taken over all infinite-dimensional subspaces $M$ of $X$. A mapping $A \in B ( X , Y )$ is in $\Phi _ { + } ( X , Y )$ if and only if $\Gamma ( A ) > 0$. Moreover, $A \in \Phi _ { + } ( X , Y )$ and $\| T \| < \Gamma ( A )$ imply that $A + T \in \Phi_+ ( X , Y )$ with (a3) holding.

$\nu ( A ) = \operatorname { sup } _ { M } \text { inf } \{ \| A x \| : x \in M , \| x \| = 1 \}$, where the supremum is taken over all subspaces $M$ having finite codimension. If $A \in \Phi _ { + } ( X , Y )$ and $\| T \| < \nu ( A )$, then $A + T \in \Phi_+ ( X , Y )$ with (a3) holding as well.

Unbounded Fredholm operators.

A linear operator $A$ from $X$ to $Y$ is called Fredholm if it is closed, $D ( A )$ is dense in $X$ and $A \in \Phi ( D ( A ) , Y )$, where $D ( A )$ is considered a Banach space with norm $\| x \| _ { A } = \| x \| + \| A x \|$. Many of the facts that are true for bounded Fredholm mappings are true for such operators. In particular, the perturbation theorems hold. In fact, one can generalize them to include unbounded perturbations. A linear operator $B$ from $X$ to $Y$ is called $A$-compact if $D ( B ) \subset D ( A )$ and for every sequence $\{ x _ { n } \} \subset D ( A )$ such that $\|x_n \| < C$, $\{ B x _ { n } \}$ has a convergent subsequence. If $A$ is Fredholm and $B$ is $A$-compact, then $A + B$ is Fredholm with the same index. A similar result holds when $B$ is $A$-bounded. Thus, if $A$ is Fredholm, then there is a $\delta > 0$ such that $\| B \| _ { A } < \delta$ implies that $A + B$ is Fredholm with (a3) holding for $B = T$. If $A \in \Phi _ { - } ( D ( A ) , Y )$ and $B$ is a densely-defined closed operator from $Y$ to $Z$, then $( B A ) ^ { \prime } = A ^ { \prime } B ^ { \prime }$, where $A ^ { \prime }$, $B ^ { \prime }$ denote the conjugates of $A$, $B$, respectively (cf. also Adjoint operator).

References

[a1]	I.C. Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" , Transl. Ser. 2 , 13 , Amer. Math. Soc. (1960) pp. 185–264
[a2]	S. Smale, "An infinite dimensional version of Sard's theorem" Amer. J. Math. , 87 (1965) pp. 861–867
[a3]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a4]	T. Kato, "Perturbation theory for linear operators" , Springer (1966)
[a5]	M. Schechter, "Basic theory of Fredholm operators" Ann. Scuola Norm. Sup. Pisa , 21 (1967) pp. 361–380
[a6]	M. Schechter, "Riesz operators and Fredholm perturbations" Bull. Amer. Math. Soc. , 74 (1968) pp. 1139–1144
[a7]	M. Schechter, "Principles of functional analysis" , Acad. Press (1971)
[a8]	M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)