Locally algebraic operator
A 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.
Denote by the algebra of all polynomials in the variable and with coefficients in . Usually, in applications it is assumed that the field is of characteristic zero and algebraically closed (cf. also Algebraically closed field; Characteristic of a field).
Thus, a linear operator is said to be locally algebraic if for any there exists a non-zero polynomial such that (cf. [a1]). If there exists a non-zero polynomial such that for every , then is said to be algebraic (cf. Algebraic operator). Thus, an algebraic operator is locally algebraic, but not conversely.
A locally algebraic operator acting in a complete linear metric space (over the field of complex numbers) and that is right invertible (cf. Algebraic analysis) but not invertible, i.e.
is not continuous (cf. [a3]). The assumption about the completeness of is essential.
If satisfies, for any , the conditions:
iii) is locally algebraic; then there exists an operator such that
|[a1]||I. Kaplansky, "Infinite Abelian groups" , Univ. Michigan Press (1954)|
|[a2]||J. Mikusiński, "Extension de l'espace linéaire avec dérivation" Studia Math. , 16 (1958) pp. 156–172|
|[a3]||D. Przeworska-Rolewicz, "Algebraic analysis" , PWN&Reidel (1988)|
|[a4]||D. Przeworska–Rolewicz, S. Rolewicz, "Equations in linear spaces" , PWN (1968)|
|[a5]||R. Sikorski, "On Mikusiński's algebraic theory of differential equations" Studia Math. , 16 (1958) pp. 230–236|
Locally algebraic operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_algebraic_operator&oldid=13689