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.
Let be a linear space over a field
. Let
be the set of all linear operators with domains and ranges in
and let
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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 continuous locally algebraic operator acting in a complete linear metric space is algebraic (cf. [a4]; for Banach spaces, see [a1]).
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.
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is not continuous (cf. [a3]). The assumption about the completeness of is essential.
If satisfies, for any
, the conditions:
i) ;
ii) ;
iii) is locally algebraic; then there exists an operator
such that
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(cf. [a2], [a5]). This means that is not an algebraic operator.
References
[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=50482