Algebraic operator

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)