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For Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201502.png" /> (cf. [[Banach space|Banach space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201503.png" /> denote the set of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201504.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201505.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201506.png" /> with domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201507.png" /> (cf. also [[Linear operator|Linear operator]]). An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201508.png" /> is called a Fredholm mapping if
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f1201509.png" />;
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2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015010.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015011.png" />;
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For Banach spaces $X$, $Y$ (cf. [[Banach space|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|Linear operator]]). An operator $A \in B ( X , Y )$ is called a Fredholm mapping if
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015012.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015014.png" /> denote the null space and range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015015.png" />, respectively.
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1) $\alpha ( A ) : = \operatorname { dim } N ( A ) < \infty$;
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2) $R ( A )$ is closed in $Y$;
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3) $\beta ( A ) : = \operatorname { codim } R ( A ) < \infty$. Here, $N ( A )$, $R ( A )$ denote the null space and range of $A$, respectively.
  
 
==Properties.==
 
==Properties.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015018.png" /> be Banach spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015021.png" /> and
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} i ( B A ) = i ( B ) + i ( A ), \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015023.png" /> (the index). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015025.png" /> is a [[Compact operator|compact operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015026.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015028.png" /> and
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where $i ( A ) = \alpha ( A ) - \beta ( A )$ (the index). If $A \in \Phi ( X , Y )$ and $K$ is a [[Compact operator|compact operator]] from $X$ to $Y$, then $A + K \in \Phi ( X , Y )$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} i ( A + K ) = i ( A ). \end{equation}
  
Moreover, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015030.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015032.png" /> and
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Moreover, for each $A \in \Phi ( X , Y )$ there is a $\delta > 0$ such that $A + T \in \Phi ( X , Y )$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} i ( A + T ) = i ( A ) , \quad \alpha ( A + T ) \leq \alpha ( A ) \end{equation}
  
for each bounded mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015034.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015038.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015040.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015042.png" />. The same is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015044.png" />, then its [[Adjoint operator|adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015045.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015049.png" /> denote the dual spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015051.png" />, respectively (cf. also [[Adjoint space|Adjoint space]]).
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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|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|Adjoint space]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015052.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015053.png" /> for each positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015054.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015055.png" />. Let
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015056.png" /></td> </tr></table>
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\begin{equation*} r ( A ) = \operatorname { lim } _ { n \rightarrow \infty } \alpha ( A ^ { n } ) \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015057.png" /></td> </tr></table>
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\begin{equation*} r ^ { \prime } ( A ) = \operatorname { lim } _ { n \rightarrow \infty } \beta ( A ^ { n } ). \end{equation*}
  
A necessary and sufficient condition for both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015059.png" /> to be finite is that there exist an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015060.png" /> and operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015062.png" />, compact on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015063.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015065.png" /> denotes the identity operator.
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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.==
 
==Semi-Fredholm operators.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015066.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015068.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015070.png" />. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015071.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015073.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015075.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015076.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015078.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015079.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015081.png" /> is compact from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015082.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015083.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015085.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015086.png" />, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015090.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015091.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015092.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015093.png" />.
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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.==
 
==Non-linear Fredholm mappings.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015095.png" /> be Banach spaces, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015096.png" /> be an open connected subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015097.png" />. A continuously Fréchet-differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015098.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f12015099.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150100.png" /> (cf. also [[Fréchet derivative|Fréchet derivative]]) is Fredholm if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150101.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150102.png" />. Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150103.png" />. It is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150105.png" /> is a [[Diffeomorphism|diffeomorphism]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150106.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150107.png" /> is a compact operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150108.png" /> is Fredholm with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150109.png" />. A useful extension of the [[Sard theorem|Sard theorem]] due to S. Smale [[#References|[a2]]] states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150111.png" /> are separable (cf. also [[Separable space|Separable space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150112.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150113.png" />, then the critical values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150114.png" /> are nowhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150115.png" /> (cf. also [[Nowhere-dense set|Nowhere-dense set]]). It follows from this that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150116.png" /> has negative index, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150117.png" /> contains no interior points, i.e., if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150118.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150119.png" />, then there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150120.png" /> arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150121.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150122.png" /> has no solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150123.png" />. Consequently, such equations are not considered well posed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150124.png" /> has negative index.
+
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|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|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|Sard theorem]] due to S. Smale [[#References|[a2]]] states that if $X$, $Y$ are separable (cf. also [[Separable space|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|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.==
 
==Perturbation theory.==
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150126.png" /> are stable under various types of perturbations. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150127.png" /> of Fredholm perturbations is the set of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150128.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150129.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150130.png" />. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150131.png" /> of semi-Fredholm perturbations are defined similarly. As noted, compact operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150132.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150133.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150135.png" />. So are strictly singular operators [[#References|[a3]]] (in some spaces they may be non-compact). An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150136.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150137.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150138.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150139.png" />. Similarly, it is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150140.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150141.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150142.png" />. But <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150143.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150144.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150145.png" />. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150146.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150147.png" /> for all compact operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150148.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150149.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150150.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150151.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150152.png" /> for all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150153.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150154.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150156.png" /> for all compact operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150157.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150158.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150159.png" />.
+
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 [[#References|[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.==
 
==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 [[#References|[a4]]]:
 
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 [[#References|[a4]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150160.png" /></td> </tr></table>
+
\begin{equation*} \gamma ( T ) = \operatorname { inf } \frac { \| Tx \| } { d ( x , N ( T ) ) },  \end{equation*}
  
where the infimum is taken over those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150161.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150162.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150164.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150165.png" /> with (a3) holding. Other constants are:
+
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:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150166.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150167.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150168.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150169.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150170.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150171.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150172.png" /> with (a3) holding.
+
$\mu ( A ) = \operatorname { inf } \{ \| T \| : \alpha ( A - T ) = \infty \}$. 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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150173.png" />, where the infimum is taken over all infinite-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150174.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150175.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150176.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150177.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150178.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150179.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150180.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150181.png" /> 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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150182.png" />, where the supremum is taken over all subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150183.png" /> having finite codimension. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150185.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150186.png" /> with (a3) holding as well.
+
$\nu ( A ) = \operatorname { sup } _ { M } \text { inf } \{ \| A x \| : x \in M , \| x \| = 1 \}$, 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.==
 
==Unbounded Fredholm operators.==
A [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150187.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150188.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150189.png" /> is called Fredholm if it is closed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150190.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150191.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150192.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150193.png" /> is considered a Banach space with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150194.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150195.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150196.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150197.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150199.png" />-compact if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150200.png" /> and for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150201.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150203.png" /> has a convergent subsequence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150204.png" /> is Fredholm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150205.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150206.png" />-compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150207.png" /> is Fredholm with the same index. A similar result holds when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150208.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150209.png" />-bounded. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150210.png" /> is Fredholm, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150211.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150212.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150213.png" /> is Fredholm with (a3) holding for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150214.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150215.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150216.png" /> is a densely-defined [[Closed operator|closed operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150217.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150218.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150219.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150220.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150221.png" /> denote the conjugates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150222.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120150/f120150223.png" />, respectively (cf. also [[Adjoint operator|Adjoint operator]]).
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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|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|Adjoint operator]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Smale,  "An infinite dimensional version of Sard's theorem"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 861–867</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Goldberg,  "Unbounded linear operators" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Schechter,  "Basic theory of Fredholm operators"  ''Ann. Scuola Norm. Sup. Pisa'' , '''21'''  (1967)  pp. 361–380</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Schechter,  "Riesz operators and Fredholm perturbations"  ''Bull. Amer. Math. Soc.'' , '''74'''  (1968)  pp. 1139–1144</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Schechter,  "Principles of functional analysis" , Acad. Press  (1971)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M.S. Berger,  "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>
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<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Smale,  "An infinite dimensional version of Sard's theorem"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 861–867</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Goldberg,  "Unbounded linear operators" , McGraw-Hill  (1966)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1966)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Schechter,  "Basic theory of Fredholm operators"  ''Ann. Scuola Norm. Sup. Pisa'' , '''21'''  (1967)  pp. 361–380</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M. Schechter,  "Riesz operators and Fredholm perturbations"  ''Bull. Amer. Math. Soc.'' , '''74'''  (1968)  pp. 1139–1144</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. Schechter,  "Principles of functional analysis" , Acad. Press  (1971)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M.S. Berger,  "Nonlinearity and functional analysis" , Acad. Press  (1977)</td></tr>
 +
</table>

Latest revision as of 16:12, 11 February 2024

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)
How to Cite This Entry:
Fredholm mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_mapping&oldid=18618
This article was adapted from an original article by Martin Schechter (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article