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A [[Linear operator|linear operator]] with closed range. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677601.png" /> be a linear operator with dense domain in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677602.png" /> and with range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677603.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677604.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677605.png" /> is normally solvable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677606.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677607.png" /> is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677608.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n0677609.png" /> be the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776010.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776011.png" /> to be normally solvable it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776012.png" />, that is, that the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776013.png" /> is the orthogonal complement to the null space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776014.png" />.
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A [[Linear operator|linear operator]] with closed range. Let $  A $
 +
be a linear operator with dense domain in a Banach space $  X $
 +
and with range $  R ( A) $
 +
in a Banach space $  Y $.  
 +
Then $  A $
 +
is normally solvable if $  \overline{ {R( A) }}\; = R ( A) $,  
 +
that is, if $  R ( A) $
 +
is a closed subspace of $  Y $.  
 +
Let $  A  ^ {*} $
 +
be the adjoint of $  A $.  
 +
For $  A $
 +
to be normally solvable it is necessary and sufficient that $  R ( A) = ^  \perp  N ( A  ^ {*} ) $,  
 +
that is, that the range of $  A $
 +
is the orthogonal complement to the null space of $  A  ^ {*} $.
  
 
Suppose that
 
Suppose that
  
<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/n/n067/n067760/n06776015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
A x  = y
 +
$$
  
is an equation with a normally-solvable operator (a normally-solvable equation). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776016.png" />, that is, if the homogeneous adjoint equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776017.png" /> has only the trivial solution, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776018.png" />. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776019.png" />, then for (*) to be solvable it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776020.png" /> for all solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776021.png" />.
+
is an equation with a normally-solvable operator (a normally-solvable equation). If $  N ( A  ^ {*} ) = \{ 0 \} $,  
 +
that is, if the homogeneous adjoint equation $  A  ^ {*} \psi = 0 $
 +
has only the trivial solution, then $  R ( A) = Y $.  
 +
But if $  N ( A  ^ {*} ) \neq \{ 0 \} $,  
 +
then for (*) to be solvable it is necessary and sufficient that $  \langle  y , \psi \rangle = 0 $
 +
for all solutions of the equation $  A  ^ {*} \psi = 0 $.
  
From now on suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776022.png" /> is closed. A normally-solvable operator is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776024.png" />-normal if its null space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776025.png" /> is finite dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776026.png" />. A normally-solvable operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776027.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776029.png" />-normal if its [[Deficiency subspace|deficiency subspace]] is finite dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776030.png" />. Operators that are either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776031.png" />-normal or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776032.png" />-normal are sometimes called semi-Fredholm operators. For an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776033.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776034.png" />-normal it is necessary and sufficient that the pre-image of every compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776035.png" /> is locally compact.
+
From now on suppose that $  A $
 +
is closed. A normally-solvable operator is called n $-
 +
normal if its null space $  N ( A) $
 +
is finite dimensional $  ( n( A) =  \mathop{\rm dim}  N ( A) < + \infty ) $.  
 +
A normally-solvable operator $  A $
 +
is called $  d $-
 +
normal if its [[Deficiency subspace|deficiency subspace]] is finite dimensional $  ( d ( A) =  \mathop{\rm dim}  {}  ^  \perp  R ( A) < + \infty ) $.  
 +
Operators that are either n $-
 +
normal or $  d $-
 +
normal are sometimes called semi-Fredholm operators. For an operator $  A $
 +
to be n $-
 +
normal it is necessary and sufficient that the pre-image of every compact set in $  R ( A) $
 +
is locally compact.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776036.png" /> is compactly imbedded in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776037.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776038.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776039.png" />-normal it is necessary and sufficient that there is an a priori estimate
+
Suppose that $  X $
 +
is compactly imbedded in a Banach space $  X _ {0} $.  
 +
For $  A $
 +
to be n $-
 +
normal it is necessary and sufficient that there is an a priori estimate
  
<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/n/n067/n067760/n06776040.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {X}  \leq  a \| x \| _ {X _ {0}  } + b \| A
 +
x \| _ {Y} ,
 +
\  x \in D ( A) .
 +
$$
  
It turns out that an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776041.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776042.png" />-normal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776044.png" />-normal. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776045.png" />. Consequently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776046.png" /> is compactly imbedded in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776049.png" />-normal if and only if there is an a priori estimate
+
It turns out that an operator $  A $
 +
is n $-
 +
normal if and only if $  A  ^ {*} $
 +
is $  d $-
 +
normal. Then $  n ( A) = d ( A  ^ {*} ) $.  
 +
Consequently, if $  X  ^ {*} $
 +
is compactly imbedded in a Banach space $  Z $,  
 +
then $  A $
 +
is $  d $-
 +
normal if and only if there is an a priori estimate
  
<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/n/n067/n067760/n06776050.png" /></td> </tr></table>
+
$$
 +
\| f \| _ {Y  ^ {*}  }  \leq  a \| f \| _ {Z} + b
 +
\| A  ^ {*} f \| _ {X  ^ {*}  } ,\ \
 +
f \in D ( A  ^ {*} ) .
 +
$$
  
The pair of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776051.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776053.png" />-characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776054.png" />. If a normally-solvable operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776056.png" />-normal or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776057.png" />-normal, the number
+
The pair of numbers $  ( n ( A) , d ( A) ) $
 +
is called the $  d $-
 +
characteristic of $  A $.  
 +
If a normally-solvable operator $  A $
 +
is n $-
 +
normal or $  d $-
 +
normal, the number
  
<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/n/n067/n067760/n06776058.png" /></td> </tr></table>
+
$$
 +
\chi ( a)  = n ( A) - d ( A)
 +
$$
  
is called the index of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776059.png" />. The properties of being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776060.png" />-normal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776061.png" />-normal are stable: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776062.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776063.png" />-normal (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776064.png" />-normal) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776065.png" /> is a linear operator of small norm or completely continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776066.png" /> is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776067.png" />-normal (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067760/n06776068.png" />-normal).
+
is called the index of the operator $  A $.  
 +
The properties of being n $-
 +
normal and $  d $-
 +
normal are stable: If $  A $
 +
is n $-
 +
normal (or $  d $-
 +
normal) and $  B $
 +
is a linear operator of small norm or completely continuous, then $  A + B $
 +
is also n $-
 +
normal (respectively, $  d $-
 +
normal).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Atkinson,  "Normal solvability of equations in Banach space"  ''Mat. Sb.'' , '''28''' :  1  (1951)  pp. 3–14  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Atkinson,  "Normal solvability of equations in Banach space"  ''Mat. Sb.'' , '''28''' :  1  (1951)  pp. 3–14  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  M.G. Krein,  "The basic propositions on defect numbers, root numbers and indices of linear operators"  ''Transl. Amer. Math. Soc. (2)'' , '''13'''  (1960)  pp. 185–264  ''Uspekhi Mat. Nauk'' , '''12'''  (1957)  pp. 43–118</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Goldberg,  "Unbounded linear operators" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for nullity, deficiency and other quantities of linear operators"  ''J. d'Anal. Math.'' , '''6'''  (1958)  pp. 261–322</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.G. Krein,  "Linear equations in Banach spaces" , Birkhäuser  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  M.G. Krein,  "The basic propositions on defect numbers, root numbers and indices of linear operators"  ''Transl. Amer. Math. Soc. (2)'' , '''13'''  (1960)  pp. 185–264  ''Uspekhi Mat. Nauk'' , '''12'''  (1957)  pp. 43–118</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Goldberg,  "Unbounded linear operators" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for nullity, deficiency and other quantities of linear operators"  ''J. d'Anal. Math.'' , '''6'''  (1958)  pp. 261–322</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.G. Krein,  "Linear equations in Banach spaces" , Birkhäuser  (1982)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


A linear operator with closed range. Let $ A $ be a linear operator with dense domain in a Banach space $ X $ and with range $ R ( A) $ in a Banach space $ Y $. Then $ A $ is normally solvable if $ \overline{ {R( A) }}\; = R ( A) $, that is, if $ R ( A) $ is a closed subspace of $ Y $. Let $ A ^ {*} $ be the adjoint of $ A $. For $ A $ to be normally solvable it is necessary and sufficient that $ R ( A) = ^ \perp N ( A ^ {*} ) $, that is, that the range of $ A $ is the orthogonal complement to the null space of $ A ^ {*} $.

Suppose that

$$ \tag{* } A x = y $$

is an equation with a normally-solvable operator (a normally-solvable equation). If $ N ( A ^ {*} ) = \{ 0 \} $, that is, if the homogeneous adjoint equation $ A ^ {*} \psi = 0 $ has only the trivial solution, then $ R ( A) = Y $. But if $ N ( A ^ {*} ) \neq \{ 0 \} $, then for (*) to be solvable it is necessary and sufficient that $ \langle y , \psi \rangle = 0 $ for all solutions of the equation $ A ^ {*} \psi = 0 $.

From now on suppose that $ A $ is closed. A normally-solvable operator is called $ n $- normal if its null space $ N ( A) $ is finite dimensional $ ( n( A) = \mathop{\rm dim} N ( A) < + \infty ) $. A normally-solvable operator $ A $ is called $ d $- normal if its deficiency subspace is finite dimensional $ ( d ( A) = \mathop{\rm dim} {} ^ \perp R ( A) < + \infty ) $. Operators that are either $ n $- normal or $ d $- normal are sometimes called semi-Fredholm operators. For an operator $ A $ to be $ n $- normal it is necessary and sufficient that the pre-image of every compact set in $ R ( A) $ is locally compact.

Suppose that $ X $ is compactly imbedded in a Banach space $ X _ {0} $. For $ A $ to be $ n $- normal it is necessary and sufficient that there is an a priori estimate

$$ \| x \| _ {X} \leq a \| x \| _ {X _ {0} } + b \| A x \| _ {Y} , \ x \in D ( A) . $$

It turns out that an operator $ A $ is $ n $- normal if and only if $ A ^ {*} $ is $ d $- normal. Then $ n ( A) = d ( A ^ {*} ) $. Consequently, if $ X ^ {*} $ is compactly imbedded in a Banach space $ Z $, then $ A $ is $ d $- normal if and only if there is an a priori estimate

$$ \| f \| _ {Y ^ {*} } \leq a \| f \| _ {Z} + b \| A ^ {*} f \| _ {X ^ {*} } ,\ \ f \in D ( A ^ {*} ) . $$

The pair of numbers $ ( n ( A) , d ( A) ) $ is called the $ d $- characteristic of $ A $. If a normally-solvable operator $ A $ is $ n $- normal or $ d $- normal, the number

$$ \chi ( a) = n ( A) - d ( A) $$

is called the index of the operator $ A $. The properties of being $ n $- normal and $ d $- normal are stable: If $ A $ is $ n $- normal (or $ d $- normal) and $ B $ is a linear operator of small norm or completely continuous, then $ A + B $ is also $ n $- normal (respectively, $ d $- normal).

References

[1] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[2] F. Atkinson, "Normal solvability of equations in Banach space" Mat. Sb. , 28 : 1 (1951) pp. 3–14 (In Russian)
[3] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)

Comments

References

[a1] I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 185–264 Uspekhi Mat. Nauk , 12 (1957) pp. 43–118
[a2] S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a3] T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322
[a4] S.G. Krein, "Linear equations in Banach spaces" , Birkhäuser (1982) (Translated from Russian)
How to Cite This Entry:
Normally-solvable operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normally-solvable_operator&oldid=48021
This article was adapted from an original article by V.A. Trenogin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article