Full Text
UDC 513.88:513.83+517.948.35+517.948.5
MATHEMATICS
N. I. NAGNIBIDA
ON THE QUESTION OF ISOMORPHISMS OF AN ANALYTIC SPACE COMMUTING WITH A POWER OF THE DIFFERENTIATION OPERATOR
(Presented by Academician A. N. Kolmogorov on 26 VIII 1965)
The problem of finding the general form of isomorphisms \(Y\) of the space \(\mathfrak U_\infty\) of entire functions that commute with the operator \(D^n=d^n/dz^n\), \(n\geqslant 1\), was posed by Delsarte and Lions in \((^1)\). According to Theorem 2.1 of \((^1)\), every such isomorphism \(Y\) satisfies, in the case \(n\geqslant 2\), the relation
\[ Y\exp(\lambda z)=\sum_{j=0}^{n-1} M_j(\lambda)\exp(\lambda z\omega^j), \qquad \omega=\exp\frac{2\pi i}{n}, \tag{1} \]
where \(M_j(\lambda)\), \(0\leqslant j\leqslant n-1\), are entire functions of exponential type with determinant
\[ \Delta(\lambda)=\det \left\| \begin{array}{cccc} M_0(\lambda) & M_1(\lambda) & \cdots & M_{n-1}(\lambda)\\ M_{n-1}(\lambda\omega) & M_0(\lambda\omega) & \cdots & M_{n-2}(\lambda\omega)\\ \cdots & \cdots & \cdots & \cdots\\ M_1(\lambda\omega^{n-1}) & M_2(\lambda\omega^{n-1}) & \cdots & M_0(\lambda\omega^{n-1}) \end{array} \right\| \equiv \mathrm{const}\neq 0. \tag{2} \]
Recently the result of Delsarte and Lions was carried over by I. Ya. Viner \((^2)\) to the spaces \(\mathfrak U_R\), \(0<R<\infty\), of all single-valued analytic functions in the disk \(|z|<R\) with the topology of compact convergence \((^3)\). It turns out, however, that the results of Delsarte–Lions and Viner are erroneous, i.e., the full group of isomorphisms of the space \(\mathfrak U_R\) (respectively, \(\mathfrak U_\infty\)) commuting with \(D^n\), \(n\geqslant 2\), is not described by conditions (1), (2). Indeed, denoting by \(A_0\) and \(A_1\) the linear continuous operators in \(\mathfrak U_R\) (\(\mathfrak U_\infty\)) defined by the relations
\[ A_0 f(z)=\frac{f(z)+f(-z)}{2}, \qquad A_1 f(z)=\frac{f(z)-f(-z)}{2}, \]
let us consider the operators
\[ Y=(I+E)A_0+(D^3+D^2-D)A_1, \]
\[ Y_1=(D^2-I)A_0+(E+D-D^3)A_1; \]
where
\[ If(z)=\int_0^z f(\zeta)\,d\zeta \]
and \(Ef(z)=f(z)\). Since (as is easily verified on the basis elements \(z^k\), \(k=0,1,\ldots\)) \(YD^2=D^2Y\) and \(YY_1=Y_1Y=E\), it follows that \(Y\) is an isomorphism of the space \(\mathfrak U_R\) (\(\mathfrak U_\infty\)) commuting with \(D^2\). Further, we have \(Y1=1+z\), although if the theorems of Delsarte–Lions and Viner were valid, then from (1), as \(\lambda\to 0\), it would follow that \(Y1=\mathrm{const}\neq 0\). Moreover, writing relation (1) for this operator, we easily obtain
\[ M_0(\lambda)=\frac12\left(\frac1\lambda+1-\lambda+\lambda^2+\lambda^3\right), \]
\[ M_1(\lambda)=\frac12\left(-\frac1\lambda+1-\lambda-\lambda^2+\lambda^3\right), \qquad \Delta(\lambda)\equiv 1. \]
Thus, the operator \(Y\) constructed does not satisfy either of the conditions (1), (2).
The present paper is devoted to finding the general form of the isomorphisms of the spaces \(\mathfrak U_R\) and \(\mathfrak U_\infty\) considered here, relying on the matrix representation of linear continuous operators in these spaces \((^{4,5})\).
- If \(T\) is a linear operator in \(\mathfrak U_R\) \((\mathfrak U_\infty)\), commuting with \(D^n\), \(n \geq 1\), then the elements of its matrix \(\{t_{i,k}\}_{i,k=0}^{\infty}\) in the power basis \(\{z^k\}_{k=0}^{\infty}\), i.e.
\[ Tz^k=\sum_{i=0}^{\infty} t_{i,k}z^i,\qquad k=0,1,\ldots, \]
are necessarily connected by the relations
\[ t_{sn+p,\,mn+q}= \begin{cases} 0, & m<s,\\[6pt] \dfrac{(mn+q)!\,p!}{(sn+p)!\,[(m-s)n+q]!}\,t_{p,\,(m-s)n+q}, & 0\leq s\leq m<\infty;\\ & 0\leq p,\ q\leq n-1. \end{cases} \tag{3} \]
Let us introduce the operators \(A_q\), \(0\leq q\leq n-1\), by putting
\[
A_q f(z)=A_q\left(\sum_{i=0}^{\infty} a_i z^i\right)
=\sum_{s=0}^{\infty} a_{sn+q}z^{sn+q},
\]
where \(f(z)\) is an arbitrary function from \(\mathfrak U_R\) \((\mathfrak U_\infty)\).
From the relations (3) and the description of linear continuous operators in \(\mathfrak U_R\) \((\mathfrak U_\infty)\) \((^{4,5})\) one may obtain
Theorem 1. In order that a linear operator \(T\) be a continuous operator in the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\), commuting with \(D^n\), \(n\geq 1\), it is necessary and sufficient that it have the form
\[
T=\sum_{s=0}^{\infty}\sum_{q=0}^{n-1}\sum_{p=0}^{n-1}
t_{p,\,sn+q}\frac{p!}{(sn+q)!}D^{sn+q-p}A_q^{*}
\tag{4}
\]
and satisfy the condition: for every \(\rho<R\) \((\rho<+\infty)\) there exists an \(r=r(\rho)<R\) \((r<+\infty)\) such that
\[
\sup_{0\leq mn+q<\infty}
\sum_{s=0}^{m}\sum_{p=0}^{n-1}
\frac{(mn+q)!\,p!}{(sn+q)!\,[(m-s)n+p]!}
\left|t_{p,\,sn+q}\right|
\frac{\rho^{(m-s)n+p}}{r^{mn+q}}
<+\infty.
\tag{5}
\]
Theorem 2. For condition (5) to hold in the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\), it is necessary and sufficient that for each (some) \(\varepsilon\), \(\varepsilon>0\), there exist a constant \(M(\varepsilon)>0\) such that
\[
|t_{p,j}|\leq M(\varepsilon)\varepsilon^j,\qquad
j\geq 0;\quad 0\leq p\leq n-1.
\tag{6}
\]
Corollary. In order that a linear operator \(T\) be a continuous operator in \(\mathfrak U_R\) \((\mathfrak U_\infty)\), commuting with \(D^n\), \(n\geq 1\), it is necessary and sufficient that it have the form (4) and satisfy the corresponding condition (6).
Finally, note that when condition (6) is fulfilled, the functions
\[
\psi_{p,q}(\lambda)=\sum_{m=0}^{\infty} t_{p,\,mn+q}\frac{\lambda^{mn}}{(mn+q)!},\qquad
0\leq p,\ q\leq n-1,
\]
are entire of class not exceeding \([1,0]\) (respectively, functions of finite degree).
\[ \text{* In this formula, for } s=0 \text{ and } q<p,\quad D^{q-p}=I^{p-q}. \]
- Let us proceed to find all isomorphisms \(T\) of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\) that commute with \(D^n\), \(n \geqslant 2\). Setting
\[ \varphi_q(\lambda,z)=\sum_{m=0}^{\infty}\frac{\lambda^{mn}z^{mn+q}}{(mn+q)!},\qquad 0\leqslant q\leqslant n-1, \]
we obtain
\[ T\varphi_q(\lambda,z)=\sum_{p=0}^{n-1}p!\psi_{p,q}(\lambda)\varphi_p(\lambda,z),\qquad 0\leqslant q\leqslant n-1. \]
Proceeding from these relations and from the consequence of Theorem 2, it is easy to prove the following theorem.
Theorem 3. In order that a linear operator \(T\) be an isomorphism of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\) that commutes with \(D^n\), \(n \geqslant 2\), it is necessary and sufficient that it have the form (4), satisfy the corresponding condition (6), and
\[ \det\|\psi_{p,q}(\lambda)\|_{p,q=0}^{n-1}\equiv \mathrm{const}\ne 0. \]
Hence, in particular, it follows that every isomorphism of the space \(\mathfrak U_R\) that commutes with \(D^n\), \(n \geqslant 2\), is at the same time an isomorphism of \(\mathfrak U_\infty\).
- Now let some isomorphism \(T\) of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\), commuting with \(D^n\), \(n \geqslant 2\), satisfy conditions (1), (2). Then from (1) and (4) it follows that
\[ \sum_{j=0}^{n-1} M_j(\lambda)\lambda^p\omega^{pj} = \sum_{q=0}^{n-1}p!\lambda^q\psi_{p,q}(\lambda), \qquad 0\leqslant p\leqslant n-1. \]
Therefore, taking into account relations (3), we must conclude that the following theorem is true.
Theorem 4. In order that an isomorphism of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\), commuting with \(D^n\), \(n \geqslant 2\), satisfy conditions (1), (2), it is necessary that its matrix have upper triangular form.
- Let us consider one application of the results presented above. Let
\[ A=\sum_{j=0}^{n}a_j(z)D^j, \]
where \(a_j(z)\in \mathfrak U_R(\mathfrak U_\infty)\) and \(a_n(z)\equiv 1\). As is known \((^{1,6,7})\), the operators \(A\) and \(D^n\) are equivalent to one another, i.e. there exists such an isomorphism \(T_0\) of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\) (in this case there are infinitely many of them) that \(T_0D^n=AT_0\). In the works \((^{1,6,7})\) it is indicated that the operator \(T_0\) can be chosen so that
\[ D^\nu T_0 f(z)\big|_{z=0}=D^\nu f(z)\big|_{z=0},\qquad 0\leqslant \nu\leqslant n-1, \]
and all other transformation operators \(T_1\), \(T_1D^n=AT_1\), can be obtained by the formula \(T_1=T_0T\), where \(T\) ranges over all possible isomorphisms of \(\mathfrak U_R(\mathfrak U_\infty)\) that commute with \(D^n\).
Let \(C\) be an operator in the space \(\mathfrak U_R(\mathfrak U_\infty)\), determined by the matrix \(\{c_{i,k}\}_{i,k=0}^{\infty}\); \(c_{i,k}=0\), if \(\min(i,k)\geqslant n\).
Theorem 5. In order that there exist an isomorphism \(T_1\) of the space \(\mathfrak U_R(\mathfrak U_\infty)\) such that
\[ T_1D^n=AT_1, \]
\[ D^\nu T_1 f(z)\big|_{z=0}=D^\nu C f(z)\big|_{z=0},\qquad 0\leqslant \nu\leqslant n-1;\qquad n\geqslant 2, \tag{7} \]
it is necessary and sufficient that
\[ \det \|c_{i,k}\|_{i,k=0}^{n-1} \ne 0. \tag{8} \]
Under these conditions the operator \(T_1\) is determined uniquely.
Remark. If
\[ B=\sum_{j=0}^{n} b_j(z)D^j,\quad b_j(z)\in \mathfrak U_R\;(\mathfrak U_\infty),\quad b_n(z)\equiv 1 \]
and (8) is satisfied, then one can also assert the existence of such an isomorphism \(T_2\) of the space \(\mathfrak U_R\) \((\mathfrak U_\infty)\) which satisfies relations (7) and \(T_2B=AT_2\).
In conclusion I express my sincere gratitude to K. M. Fishman for useful advice.
Chernivtsi State
University
Received
28 IV 1965
REFERENCES
- J. Delsarte, J. L. Lions, Comm. Math. Helv., 32, No. 2, 113 (1957).
- I. Ya. Viner, UMN, 20, issue 1 (121), 185 (1965).
- G. Köthe, J. reine u. angew. Math., 191, 30 (1953).
- M. G. Khaplanov, DAN, 80, No. 1, 21 (1951).
- K. M. Fishman, DAN, 127, No. 1, 40 (1959).
- M. K. Fage, in: Collected Papers: Studies on Modern Problems of the Theory of Functions of a Complex Variable, Moscow, 1961, p. 468.
- K. M. Fishman, UMN, 19, issue 5 (119), 143 (1964).