View source for Normally-solvable operator

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" />.

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>

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" />.

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.

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

<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>

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

<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>

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

<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>

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).

====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>



====Comments====


====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>