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A [[Linear operator|linear operator]] such that for each element of the space under consideration there exists a polynomial in this operator (with scalar coefficients) annihilating this element.

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

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

Denote by $\mathbf{F} [ t ]$ the algebra of all polynomials in the variable $t$ and with coefficients in $\mathbf{F}$. 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]]).

Thus, a linear operator $T \in L _ { 0 } ( X )$ is said to be locally algebraic if for any $x \in X$ there exists a non-zero polynomial $p ( t ) \in \mathbf{F} [ t ]$ such that $p ( T ) x = 0$ (cf. [[#References|[a1]]]). If there exists a non-zero polynomial $p ( t ) \in \mathbf{F} [ t ]$ such that $p ( T ) x = 0$ for every $x \in X$, then $T$ is said to be algebraic (cf. [[Algebraic operator|Algebraic operator]]). Thus, an algebraic operator is locally algebraic, but not conversely.

A continuous locally algebraic operator acting in a complete linear [[Metric space|metric space]] $X$ is algebraic (cf. [[#References|[a4]]]; for Banach spaces, see [[#References|[a1]]]).

A locally algebraic operator $T$ acting in a complete linear metric space $X$ (over the field $\mathbf{C}$ of complex numbers) and that is right invertible (cf. [[Algebraic analysis|Algebraic analysis]]) but not invertible, i.e.

\begin{equation*} \ker T = \{ x \in X : T x = 0 \} \neq \{ 0 \}, \end{equation*}

is not continuous (cf. [[#References|[a3]]]). The assumption about the completeness of $X$ is essential.

If $T \in L _ { 0 } ( X )$ satisfies, for any $p ( t ) , q ( t ) \in \mathbf{F} [ t ]$, the conditions:

i) $\operatorname{dim}\operatorname {ker}p(T)=\operatorname{dim}\operatorname{ker}T\cdot\operatorname{degree}p ( T ) $;

ii) $\operatorname { dim } \operatorname { ker }q ( T ) p ( T ) \leq \operatorname { dim } \operatorname { ker } q ( T ) + \operatorname { dim } \operatorname { ker } p ( T )$;

iii) $T$ is locally algebraic; then there exists an operator $A \in L _ { 0 } ( X )$ such that

\begin{equation*} T A - A T = I \end{equation*}

(cf. [[#References|[a2]]], [[#References|[a5]]]). This means that $T$ is not an [[Algebraic operator|algebraic operator]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Kaplansky,   "Infinite Abelian groups" , Univ. Michigan Press  (1954)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Mikusiński,   "Extension de l'espace linéaire avec dérivation"  ''Studia Math.'' , '''16'''  (1958)  pp. 156–172</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Przeworska-Rolewicz,   "Algebraic analysis" , PWN&amp;Reidel  (1988)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D. Przeworska–Rolewicz,   S. Rolewicz,   "Equations in linear spaces" , PWN  (1968)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R. Sikorski,   "On Mikusiński's algebraic theory of differential equations"  ''Studia Math.'' , '''16'''  (1958)  pp. 230–236</td></tr></table>