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A [[Linear operator|linear operator]] satisfying a [[Polynomial|polynomial]] identity with scalar coefficients.
 
A [[Linear operator|linear operator]] satisfying a [[Polynomial|polynomial]] identity with scalar coefficients.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202001.png" /> be a [[Linear space|linear space]] over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202002.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202003.png" /> be the set of all linear operators with domains and ranges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202004.png" /> and let
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Let $X$ be a [[Linear space|linear space]] over a [[Field|field]] $\mathbf{F}$. Let $L ( X )$ be the set of all linear operators with domains and ranges in $X$ and 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/a/a120/a120200/a1202005.png" /></td> </tr></table>
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\begin{equation*} L _ { 0 } ( X ) = \{ A \in L ( X ) : \operatorname { dom } A = X \}. \end{equation*}
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202006.png" /> the algebra of of all polynomials in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202007.png" /> and with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202008.png" />. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a1202009.png" /> is said to be algebraic if there exists a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020011.png" /> (cf. [[#References|[a2]]]). Note that I. Kaplansky in [[#References|[a1]]] considered rings with a polynomial identity (cf. also [[PI-algebra|PI-algebra]]).
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Denote by $\mathbf{F} [ t ]$ the algebra of of all polynomials in the variable $t$ and with coefficients in $\mathbf{F}$. A linear operator $T \in L _ { 0 } ( X )$ is said to be algebraic if there exists a non-zero $p ( t ) \in \mathbf{F} [ t ]$ such that $p ( T ) = 0$ (cf. [[#References|[a2]]]). Note that I. Kaplansky in [[#References|[a1]]] considered rings with a polynomial identity (cf. also [[PI-algebra|PI-algebra]]).
  
Usually, in applications it is assumed that the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020012.png" /> is of characteristic zero and algebraically closed (cf. also [[Algebraically closed field|Algebraically closed field]]; [[Characteristic of a field|Characteristic of a field]]).
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Usually, in applications it is assumed that the field $\mathbf{F}$ is of characteristic zero and algebraically closed (cf. also [[Algebraically closed field|Algebraically closed field]]; [[Characteristic of a field|Characteristic of a field]]).
  
An algebraic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020013.png" /> is said be of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020016.png" /> for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020017.png" /> such that the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020018.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020020.png" /> for any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020021.png" /> of degree less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020022.png" />. If this is the case, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020023.png" /> is also said to be the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020024.png" /> and its roots are called the characteristic roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020025.png" />. Here and in the sequel it is assumed that the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020026.png" /> is normalized or monic, i.e. the coefficient at the highest power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020027.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020028.png" />. Algebraic operators with characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020030.png" />) are said to be involutions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020032.png" />. Their characteristic (single) roots are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020033.png" />th roots of unity. Involutions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020034.png" /> are also briefly called involutions.
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An algebraic operator $T \in L _ { 0 } ( X )$ is said be of order $N$ if $p ( T ) = 0$ for a $p ( t ) \in \mathbf{F} [ t ]$ such that the degree of $p$ is $N$ and $q ( T ) \neq 0$ for any polynomial $q$ of degree less than $N$. If this is the case, then $p ( t )$ is also said to be the characteristic polynomial of $T$ and its roots are called the characteristic roots of $T$. Here and in the sequel it is assumed that the polynomial $p ( t )$ is normalized or monic, i.e. the coefficient at the highest power $t ^ { N }$ is equal to $1$. Algebraic operators with characteristic polynomial $p ( t ) = t ^ { N } - 1$ ($N \geq 2$) are said to be involutions of order $N$. Their characteristic (single) roots are $N$th roots of unity. Involutions of order $2$ are also briefly called involutions.
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020035.png" /> is algebraic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020036.png" /> if and only if
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An operator $T \in L _ { 0 } ( X )$ is algebraic of order $N$ if and only if
  
<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/a/a120/a120200/a12020037.png" /></td> </tr></table>
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\begin{equation*} \delta _ { T } = \operatorname { sup } _ { x \in X } \operatorname { dim } \operatorname { lin } \{ x , T x , T ^ { 2 } x , \ldots \} = N \end{equation*}
  
 
(cf. [[#References|[a2]]]).
 
(cf. [[#References|[a2]]]).
  
In order to give another characterization of algebraic operators, more useful in applications, write for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020038.png" />-times differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020039.png" /> in an interval containing different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020040.png" />,
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In order to give another characterization of algebraic operators, more useful in applications, write for any $k$-times differentiable function $f$ in an interval containing different points $t _ { 1 } , \ldots , t _ { n }$,
  
<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/a/a120/a120200/a12020041.png" /></td> </tr></table>
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\begin{equation*} \{ f ( t ) \} _ { ( k ; t _ { i } ) } = \sum _ { m = 0 } ^ { k } \frac { ( t - t _ { i } ) ^ { m } } { m ! } \frac { d ^ { m } f ( t ) } { d t ^ { m } } | _ { t = t _ { i } }. \end{equation*}
  
 
Let
 
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/a/a120/a120200/a12020042.png" /></td> </tr></table>
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\begin{equation*} P ( t ) = \prod _ { m = 1 } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } ; \quad q _ { i } ( t ) = \left\{ \frac { ( t - t _ { i } ) ^ { r _ { i } } } { P ( t ) } \right\} _ { ( r _ { i } - 1 ; t _ { i } ) }; \end{equation*}
  
<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/a/a120/a120200/a12020043.png" /></td> </tr></table>
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\begin{equation*} \mathfrak { p } _ { i } ( t ) = q _ { i } ( t ) \prod _ { m = 1 , m \neq i } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } \quad ( i = 1 , \ldots , n ). \end{equation*}
  
 
Then, by the [[Hermite interpolation formula|Hermite interpolation formula]] with multiple knots, one obtains a partition of unity:
 
Then, by the [[Hermite interpolation formula|Hermite interpolation formula]] with multiple knots, one obtains a partition of unity:
  
<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/a/a120/a120200/a12020044.png" /></td> </tr></table>
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\begin{equation*} 1 = \sum _ { i = 1 } ^ { n } \mathfrak { p } _ { i } ( t ). \end{equation*}
  
This representation is unique, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020046.png" /> are fixed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020047.png" /> are single knots (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020048.png" />), then the Hermite formula yields the [[Lagrange interpolation formula|Lagrange interpolation formula]] and
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This representation is unique, provided that $t_i$ and $r_i$ are fixed. If $t_i$ are single knots (i.e. $r _ { 1 } = \ldots = r _ { n } = 1$), then the Hermite formula yields the [[Lagrange interpolation formula|Lagrange interpolation formula]] 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/a/a120/a120200/a12020049.png" /></td> </tr></table>
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\begin{equation*} \mathfrak { p } _ { i } ( t ) = \prod _ { m = 1 , m \neq i } ^ { n } \frac { t - t _ { m } } { t _ { i } - t _ { m } } \quad ( i = 1 , \ldots , n ). \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020050.png" />. Then the following conditions are equivalent (cf. [[#References|[a4]]]):
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Let $T \in L _ { 0 } ( X )$. Then the following conditions are equivalent (cf. [[#References|[a4]]]):
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020051.png" /> is an algebraic operator with characteristic polynomial
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i) $T$ is an algebraic operator with characteristic polynomial
  
<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/a/a120/a120200/a12020052.png" /></td> </tr></table>
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\begin{equation*} P ( t ) = \prod _ { m = 1 } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } \end{equation*}
  
of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020053.png" />;
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of order $N = r _1 + \ldots + r _ { n }$;
  
ii) the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020054.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020055.png" />) are disjoint projectors giving a partition of unity:
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ii) the operators $P _ { j } = \mathfrak { p } _ { j } ( T )$ ($j = 1 , \ldots , n$) are disjoint projectors giving a partition of unity:
  
<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/a/a120/a120200/a12020056.png" /></td> </tr></table>
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\begin{equation*} P _ { j } P _ { k } = \left\{ \left. \begin{array} { l l } { P _ { k } } &amp; { \text { for } j = k } \\ { 0 } &amp; { \text { for } j \neq k } \end{array} \right. ( j , k = 1 , \dots , n ) \right. ; \end{equation*}
  
<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/a/a120/a120200/a12020057.png" /></td> </tr></table>
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\begin{equation*} \sum _ { j = 1 } ^ { n } P _ { j } = I \end{equation*}
  
(where by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020058.png" /> is denotes the identity operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020059.png" />) and such that
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(where by $I$ is denotes the identity operator in $X$) and such 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/a/a120/a120200/a12020060.png" /></td> </tr></table>
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\begin{equation*} ( T - t _ { j } I ) ^ { r _ { j } } P _ { j } = 0 \quad ( j = 1 , \ldots , n ); \end{equation*}
  
iii) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020061.png" /> is the [[Direct sum|direct sum]] of the principal spaces of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020062.png" /> corresponding to the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020063.png" /> of multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020064.png" /> respectively (cf. also [[Eigen value|Eigen value]]), i.e.
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iii) the space $X$ is the [[Direct sum|direct sum]] of the principal spaces of the operator $T$ corresponding to the eigenvalues $t _ { 1 } , \ldots , t _ { n }$ of multiplicities $r_1 , \ldots , r_n$ respectively (cf. also [[Eigen value|Eigen value]]), i.e.
  
<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/a/a120/a120200/a12020065.png" /></td> </tr></table>
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\begin{equation*} X = X _ { 1 } \bigoplus \ldots \bigoplus X _ { n }, \end{equation*}
  
 
where
 
where
  
<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/a/a120/a120200/a12020066.png" /></td> </tr></table>
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\begin{equation*} X _ { j } = \operatorname { ker } ( T - t _ { j } I ) ^ { r _ { j } } , \quad ( j = 1 , \ldots , n ). \end{equation*}
  
If the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020067.png" /> are single, then iii) can be formulated as follows:
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If the roots $t _ { 1 } , \ldots , t _ { n }$ are single, then iii) can be formulated as follows:
  
iv) the space X is the direct sum of eigenspaces of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020068.png" /> corresponding to the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020069.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020070.png" />, where
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iv) the space X is the direct sum of eigenspaces of the operator $T$ corresponding to the eigenvalues $t _ { 1 } , \ldots , t _ { n }$: $X = X _ { 1 } \oplus \ldots \oplus X _ { n }$, where
  
<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/a/a120/a120200/a12020071.png" /></td> </tr></table>
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\begin{equation*} T x _ { j } = t _ { j } x _ { j }\, \text { for } x  _ { j } \in X _ { j } \quad (\, j = 1 , \dots , n ). \end{equation*}
  
An immediate consequence of these conditions is the classical Cayley–Hamilton theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020072.png" />, then every operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020073.png" /> is algebraic and its characteristic polynomial is a divisor of the polynomial
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An immediate consequence of these conditions is the classical Cayley–Hamilton theorem: If $\operatorname { dim } X &lt; + \infty$, then every operator $T \in L _ { 0 } ( X )$ is algebraic and its characteristic polynomial is a divisor of the polynomial
  
<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/a/a120/a120200/a12020074.png" /></td> </tr></table>
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\begin{equation*} Q ( \lambda ) = \operatorname { det } ( T - \lambda I ) \end{equation*}
  
(where to each operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020075.png" /> there corresponds a unique square matrix denoted by the same letter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020076.png" />). In that case the characteristic polynomial is said to be minimal.
+
(where to each operator $T \in L _ { 0 } ( X )$ there corresponds a unique square matrix denoted by the same letter $T$). In that case the characteristic polynomial is said to be minimal.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020077.png" /> is algebraic, then
+
If $T \in L _ { 0 } ( X )$ is algebraic, then
  
<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/a/a120/a120200/a12020078.png" /></td> </tr></table>
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\begin{equation*} T S - S T \neq \lambda I \end{equation*}
  
for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020080.png" /> (cf. [[Locally algebraic operator|Locally algebraic operator]]).
+
for every $S \in L _ { 0 } ( X )$ and $\lambda \in \mathbf{F} \backslash \{ 0 \}$ (cf. [[Locally algebraic operator|Locally algebraic operator]]).
  
In the same manner one can define algebraic elements in an algebra. In that case, the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020081.png" /> are idempotents giving a partition of unity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020082.png" /> is a two-sided [[Ideal|ideal]] in an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020083.png" /> and the coset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020084.png" /> corresponding to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020085.png" /> is an algebraic element in the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020086.png" /> (cf. also [[Rings and algebras|Rings and algebras]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020087.png" /> is said to be an almost algebraic element. By definition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020088.png" /> is the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020090.png" />.
+
In the same manner one can define algebraic elements in an algebra. In that case, the elements $\mathfrak{p} _ { j } ( T )$ are idempotents giving a partition of unity. If $J$ is a two-sided [[Ideal|ideal]] in an algebra $\mathcal{X}$ and the coset $T$ corresponding to an element $T \in \cal X$ is an algebraic element in the quotient algebra $\mathcal{X} / J$ (cf. also [[Rings and algebras|Rings and algebras]]), then $T$ is said to be an almost algebraic element. By definition, if $P ( t )$ is the characteristic polynomial of $T$, then $P ( T ) \in \mathcal{J}$.
  
In appropriate spaces, several integral transforms are algebraic; for instance, the [[Hilbert transform|Hilbert transform]], the [[Fourier transform|Fourier transform]], and the Cauchy singular integral on a closed curve (cf. also [[Cauchy integral|Cauchy integral]]). The cotangent Hilbert transform is an almost algebraic operator. The usual translation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020091.png" /> is an algebraic operator in the spaces of periodic, exponential-periodic and polynomial-exponential-periodic functions (i.e. linear combinations of products of polynomial, periodic and exponential functions, respectively), provided that the period of the functions under consideration is commensurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120200/a12020092.png" />. The so-called Carleman shift of the argument of a function is also an algebraic operator (cf. [[#References|[a5]]], [[#References|[a7]]]).
+
In appropriate spaces, several integral transforms are algebraic; for instance, the [[Hilbert transform|Hilbert transform]], the [[Fourier transform|Fourier transform]], and the Cauchy singular integral on a closed curve (cf. also [[Cauchy integral|Cauchy integral]]). The cotangent Hilbert transform is an almost algebraic operator. The usual translation by $r &gt; 0$ is an algebraic operator in the spaces of periodic, exponential-periodic and polynomial-exponential-periodic functions (i.e. linear combinations of products of polynomial, periodic and exponential functions, respectively), provided that the period of the functions under consideration is commensurable with $r$. The so-called Carleman shift of the argument of a function is also an algebraic operator (cf. [[#References|[a5]]], [[#References|[a7]]]).
  
 
Properties of algebraic and almost algebraic operators and elements are very useful in solving several problems involving these operators, in particular those involving the operators listed above, and in several kinds of integral and ordinary and partial differential equations with transformed argument. A particular advantage is that the equivalence of the conditions i)–ii) permits one to reduce a problem under consideration to a problem without any transformation of argument and, eventually, to determine solutions in closed form (cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]).
 
Properties of algebraic and almost algebraic operators and elements are very useful in solving several problems involving these operators, in particular those involving the operators listed above, and in several kinds of integral and ordinary and partial differential equations with transformed argument. A particular advantage is that the equivalence of the conditions i)–ii) permits one to reduce a problem under consideration to a problem without any transformation of argument and, eventually, to determine solutions in closed form (cf. [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]).
Line 88: Line 96:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kaplansky,  "Rings with a polynomial identity"  ''Bull. Amer. Math. Soc.'' , '''54'''  (1948)  pp. 575–580</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Kaplansky,  "Infinite Abelian groups" , Univ. Michigan Press  (1954)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Nguyen Van Mau,  "Generalized algebraic elements and linear singular integral equations with transformed argument" , Warsaw Univ. Techn.  (1989)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Przeworska-Rolewicz,  "Équations avec opérations algébriques"  ''Studia Math.'' , '''22'''  (1963)  pp. 337–367</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Przeworska-Rolewicz,  "Equations with transformed argument. An algebraic approach" , PWN&amp;Elsevier  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Przeworska-Rolewicz,  "Algebraic analysis" , PWN&amp;Reidel  (1988)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Przeworska–Rolewicz,  "Logarithms and antilogarithms: An algebraic analysis approach" , Kluwer Acad. Publ.  (1998)  (Appendix by Z. Binderman)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Kaplansky,  "Rings with a polynomial identity"  ''Bull. Amer. Math. Soc.'' , '''54'''  (1948)  pp. 575–580</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Kaplansky,  "Infinite Abelian groups" , Univ. Michigan Press  (1954)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  Nguyen Van Mau,  "Generalized algebraic elements and linear singular integral equations with transformed argument" , Warsaw Univ. Techn.  (1989)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D. Przeworska-Rolewicz,  "Équations avec opérations algébriques"  ''Studia Math.'' , '''22'''  (1963)  pp. 337–367</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D. Przeworska-Rolewicz,  "Equations with transformed argument. An algebraic approach" , PWN&amp;Elsevier  (1973)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  D. Przeworska-Rolewicz,  "Algebraic analysis" , PWN&amp;Reidel  (1988)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Przeworska–Rolewicz,  "Logarithms and antilogarithms: An algebraic analysis approach" , Kluwer Acad. Publ.  (1998)  (Appendix by Z. Binderman)</td></tr></table>

Revision as of 15:30, 1 July 2020

A linear operator satisfying a polynomial identity with scalar coefficients.

Let $X$ be a linear space over a field $\mathbf{F}$. Let $L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let

\begin{equation*} L _ { 0 } ( X ) = \{ A \in L ( X ) : \operatorname { dom } A = X \}. \end{equation*}

Denote by $\mathbf{F} [ t ]$ the algebra of of all polynomials in the variable $t$ and with coefficients in $\mathbf{F}$. A linear operator $T \in L _ { 0 } ( X )$ is said to be algebraic if there exists a non-zero $p ( t ) \in \mathbf{F} [ t ]$ such that $p ( T ) = 0$ (cf. [a2]). Note that I. Kaplansky in [a1] considered rings with a polynomial identity (cf. also PI-algebra).

Usually, in applications it is assumed that the field $\mathbf{F}$ is of characteristic zero and algebraically closed (cf. also Algebraically closed field; Characteristic of a field).

An algebraic operator $T \in L _ { 0 } ( X )$ is said be of order $N$ if $p ( T ) = 0$ for a $p ( t ) \in \mathbf{F} [ t ]$ such that the degree of $p$ is $N$ and $q ( T ) \neq 0$ for any polynomial $q$ of degree less than $N$. If this is the case, then $p ( t )$ is also said to be the characteristic polynomial of $T$ and its roots are called the characteristic roots of $T$. Here and in the sequel it is assumed that the polynomial $p ( t )$ is normalized or monic, i.e. the coefficient at the highest power $t ^ { N }$ is equal to $1$. Algebraic operators with characteristic polynomial $p ( t ) = t ^ { N } - 1$ ($N \geq 2$) are said to be involutions of order $N$. Their characteristic (single) roots are $N$th roots of unity. Involutions of order $2$ are also briefly called involutions.

An operator $T \in L _ { 0 } ( X )$ is algebraic of order $N$ if and only if

\begin{equation*} \delta _ { T } = \operatorname { sup } _ { x \in X } \operatorname { dim } \operatorname { lin } \{ x , T x , T ^ { 2 } x , \ldots \} = N \end{equation*}

(cf. [a2]).

In order to give another characterization of algebraic operators, more useful in applications, write for any $k$-times differentiable function $f$ in an interval containing different points $t _ { 1 } , \ldots , t _ { n }$,

\begin{equation*} \{ f ( t ) \} _ { ( k ; t _ { i } ) } = \sum _ { m = 0 } ^ { k } \frac { ( t - t _ { i } ) ^ { m } } { m ! } \frac { d ^ { m } f ( t ) } { d t ^ { m } } | _ { t = t _ { i } }. \end{equation*}

Let

\begin{equation*} P ( t ) = \prod _ { m = 1 } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } ; \quad q _ { i } ( t ) = \left\{ \frac { ( t - t _ { i } ) ^ { r _ { i } } } { P ( t ) } \right\} _ { ( r _ { i } - 1 ; t _ { i } ) }; \end{equation*}

\begin{equation*} \mathfrak { p } _ { i } ( t ) = q _ { i } ( t ) \prod _ { m = 1 , m \neq i } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } \quad ( i = 1 , \ldots , n ). \end{equation*}

Then, by the Hermite interpolation formula with multiple knots, one obtains a partition of unity:

\begin{equation*} 1 = \sum _ { i = 1 } ^ { n } \mathfrak { p } _ { i } ( t ). \end{equation*}

This representation is unique, provided that $t_i$ and $r_i$ are fixed. If $t_i$ are single knots (i.e. $r _ { 1 } = \ldots = r _ { n } = 1$), then the Hermite formula yields the Lagrange interpolation formula and

\begin{equation*} \mathfrak { p } _ { i } ( t ) = \prod _ { m = 1 , m \neq i } ^ { n } \frac { t - t _ { m } } { t _ { i } - t _ { m } } \quad ( i = 1 , \ldots , n ). \end{equation*}

Let $T \in L _ { 0 } ( X )$. Then the following conditions are equivalent (cf. [a4]):

i) $T$ is an algebraic operator with characteristic polynomial

\begin{equation*} P ( t ) = \prod _ { m = 1 } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } \end{equation*}

of order $N = r _1 + \ldots + r _ { n }$;

ii) the operators $P _ { j } = \mathfrak { p } _ { j } ( T )$ ($j = 1 , \ldots , n$) are disjoint projectors giving a partition of unity:

\begin{equation*} P _ { j } P _ { k } = \left\{ \left. \begin{array} { l l } { P _ { k } } & { \text { for } j = k } \\ { 0 } & { \text { for } j \neq k } \end{array} \right. ( j , k = 1 , \dots , n ) \right. ; \end{equation*}

\begin{equation*} \sum _ { j = 1 } ^ { n } P _ { j } = I \end{equation*}

(where by $I$ is denotes the identity operator in $X$) and such that

\begin{equation*} ( T - t _ { j } I ) ^ { r _ { j } } P _ { j } = 0 \quad ( j = 1 , \ldots , n ); \end{equation*}

iii) the space $X$ is the direct sum of the principal spaces of the operator $T$ corresponding to the eigenvalues $t _ { 1 } , \ldots , t _ { n }$ of multiplicities $r_1 , \ldots , r_n$ respectively (cf. also Eigen value), i.e.

\begin{equation*} X = X _ { 1 } \bigoplus \ldots \bigoplus X _ { n }, \end{equation*}

where

\begin{equation*} X _ { j } = \operatorname { ker } ( T - t _ { j } I ) ^ { r _ { j } } , \quad ( j = 1 , \ldots , n ). \end{equation*}

If the roots $t _ { 1 } , \ldots , t _ { n }$ are single, then iii) can be formulated as follows:

iv) the space X is the direct sum of eigenspaces of the operator $T$ corresponding to the eigenvalues $t _ { 1 } , \ldots , t _ { n }$: $X = X _ { 1 } \oplus \ldots \oplus X _ { n }$, where

\begin{equation*} T x _ { j } = t _ { j } x _ { j }\, \text { for } x _ { j } \in X _ { j } \quad (\, j = 1 , \dots , n ). \end{equation*}

An immediate consequence of these conditions is the classical Cayley–Hamilton theorem: If $\operatorname { dim } X < + \infty$, then every operator $T \in L _ { 0 } ( X )$ is algebraic and its characteristic polynomial is a divisor of the polynomial

\begin{equation*} Q ( \lambda ) = \operatorname { det } ( T - \lambda I ) \end{equation*}

(where to each operator $T \in L _ { 0 } ( X )$ there corresponds a unique square matrix denoted by the same letter $T$). In that case the characteristic polynomial is said to be minimal.

If $T \in L _ { 0 } ( X )$ is algebraic, then

\begin{equation*} T S - S T \neq \lambda I \end{equation*}

for every $S \in L _ { 0 } ( X )$ and $\lambda \in \mathbf{F} \backslash \{ 0 \}$ (cf. Locally algebraic operator).

In the same manner one can define algebraic elements in an algebra. In that case, the elements $\mathfrak{p} _ { j } ( T )$ are idempotents giving a partition of unity. If $J$ is a two-sided ideal in an algebra $\mathcal{X}$ and the coset $T$ corresponding to an element $T \in \cal X$ is an algebraic element in the quotient algebra $\mathcal{X} / J$ (cf. also Rings and algebras), then $T$ is said to be an almost algebraic element. By definition, if $P ( t )$ is the characteristic polynomial of $T$, then $P ( T ) \in \mathcal{J}$.

In appropriate spaces, several integral transforms are algebraic; for instance, the Hilbert transform, the Fourier transform, and the Cauchy singular integral on a closed curve (cf. also Cauchy integral). The cotangent Hilbert transform is an almost algebraic operator. The usual translation by $r > 0$ is an algebraic operator in the spaces of periodic, exponential-periodic and polynomial-exponential-periodic functions (i.e. linear combinations of products of polynomial, periodic and exponential functions, respectively), provided that the period of the functions under consideration is commensurable with $r$. The so-called Carleman shift of the argument of a function is also an algebraic operator (cf. [a5], [a7]).

Properties of algebraic and almost algebraic operators and elements are very useful in solving several problems involving these operators, in particular those involving the operators listed above, and in several kinds of integral and ordinary and partial differential equations with transformed argument. A particular advantage is that the equivalence of the conditions i)–ii) permits one to reduce a problem under consideration to a problem without any transformation of argument and, eventually, to determine solutions in closed form (cf. [a5], [a6], [a7]).

Other generalizations (for instance, operators satisfying a polynomial identity with non-scalar coefficients) and their applications are examined in [a3].

References

[a1] I. Kaplansky, "Rings with a polynomial identity" Bull. Amer. Math. Soc. , 54 (1948) pp. 575–580
[a2] I. Kaplansky, "Infinite Abelian groups" , Univ. Michigan Press (1954)
[a3] Nguyen Van Mau, "Generalized algebraic elements and linear singular integral equations with transformed argument" , Warsaw Univ. Techn. (1989)
[a4] D. Przeworska-Rolewicz, "Équations avec opérations algébriques" Studia Math. , 22 (1963) pp. 337–367
[a5] D. Przeworska-Rolewicz, "Equations with transformed argument. An algebraic approach" , PWN&Elsevier (1973)
[a6] D. Przeworska-Rolewicz, "Algebraic analysis" , PWN&Reidel (1988)
[a7] D. Przeworska–Rolewicz, "Logarithms and antilogarithms: An algebraic analysis approach" , Kluwer Acad. Publ. (1998) (Appendix by Z. Binderman)
How to Cite This Entry:
Algebraic operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_operator&oldid=19155
This article was adapted from an original article by Danuta Przeworska-Rolewicz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article