MATHEMATICS

M. A. EVGRAFOV and A. D. SOLOV’EV

ON A CLASS OF INVERTIBLE OPERATORS IN THE RING OF ANALYTIC FUNCTIONS

(Presented by Academician M. V. Keldysh on 18 I 1957)

Let \(K_m(r_i, R_i)=K_m\) be the ring of analytic functions of \(m\) complex variables \(z_1,z_2,\ldots,z_m\), regular and single-valued for \(r_i<|z_i|<R_i\), \(i=1,2,\ldots,m\), in which a topology is defined by the notion of convergence as uniform convergence for \(r_i(1+\varepsilon)<|z_i|<R_i(1-\varepsilon)\) for any \(\varepsilon>0\). In complete analogy with how this was done in \((^{1,2})\), one can show that if \(K_m\) is considered only as a linear topological space, then the following assertion holds:

Theorem 1. Let \(A\) be a linear operator in \(K_m\), defined by the equalities

\[ Az_1^{n_1}\cdots z_m^{n_m} = z_1^{n_1}\cdots z_m^{n_m} \varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m), \qquad -\infty<n_1,\ldots,n_m<\infty, \tag{1} \]

where \(\varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m)\to0\) as \(\max_i |n_i|\to\infty\) (in the sense of the topology of \(K_m\)).

Then the operator \(E+\lambda A\) has an inverse, continuous in \(K_m\), for all \(\lambda\) except for a countable set of eigenvalues \(\lambda_n\), having no limit points in the finite part of the plane. Moreover, the multiplicity of each eigenvalue is finite and, under a suitable \((^2)\) definition of the adjoint operator, all the Fredholm alternatives hold.

This result easily admits the following inessential generalization:

Theorem 2. Let the linear operator \(A_\lambda\) be defined by the equalities

\[ A_\lambda z_1^{n_1}\cdots z_m^{n_m} = z_1^{n_1}\cdots z_m^{n_m} \varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda), \qquad -\infty<n_1,\ldots,n_m<\infty. \tag{2} \]

Here \(\varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda)\) are entire analytic functions of \(\lambda\), satisfying the conditions

\[ \lim_{\max_i |n_i|\to\infty} \varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda) =0 \tag{3} \]

uniformly in \(\lambda\) in any finite disk, and

\[ \lim_{\lambda\to0} \varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda) =0 \tag{4} \]

for fixed \(n_1,\ldots,n_m\).

Then the operator \(E+A_\lambda\) has in \(K_m\) a continuous inverse for all \(\lambda\), except for a countable set of eigenvalues \(\lambda_n\), having no limit points in the finite part of the plane. Moreover, the finite multiplicity of the eigenvalues and the Fredholm alternatives still hold.

If \(K_m\) is considered not only as a linear topological space, but also as a topological ring, then one can obtain a considerably stronger result.

Theorem 3. Let \(B_\lambda\) be a linear operator in \(K_m\), defined by the equalities

\[ B_\lambda z_1^{n_1}\cdots z_m^{n_m} = z_1^{n_1}\cdots z_m^{n_m} h_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda), \qquad -\infty<n_1,\ldots,n_m<\infty . \tag{5} \]

Here \(h_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda)\) are entire analytic functions of \(\lambda\), satisfying the conditions:

\[ h_{n_1,\ldots,n_m}\ne 0 \quad\text{for}\quad r_i(1+\varepsilon)<|z_i|<R_i(1-\varepsilon) \quad\text{for}\quad \max_i |n_i|>n_0(\varepsilon,\lambda), \tag{6} \]

\[ \lim_{\max_i |n_i|\to\infty} \frac{ h_{n_1+k_1,\ldots,n_m+k_m} }{ h_{n_1,\ldots,n_m} } =1 \tag{7} \]

uniformly with respect to \(\lambda\) in any finite circle for fixed \(k_1,\ldots,k_m\), and

\[ \lim_{\lambda\to 0} h_{n_1,\ldots,n_m}=1 \tag{8} \]

for fixed \(n_1,\ldots,n_m\).

Then the operator \(B_\lambda\) has an inverse, continuous in \(K_m\) for all \(\lambda\), except for a countable set \(\lambda_n\), having no limit points in the finite part of the plane. The finite multiplicity of eigenvalues and the Fredholm alternatives still hold.

To compare the strength of Theorems 2 and 3, let us note that the operator \(E+A_\lambda\), where \(A_\lambda\) is defined by the equalities (2), is easily represented in the form (5) by putting

\[ h_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda) = 1+\varepsilon_{n_1,\ldots,n_m}(z_1,\ldots,z_m;\lambda). \]

In this case, from condition (3) there follow not only conditions (6) and (7), but also the stronger assertion

\[ \lim_{\max_i |n_i|\to\infty} h_{n_1,\ldots,n_m}=1. \]

Theorem 3 essentially uses the fact that \(K_m\) is not only a linear topological space, but also a ring, and can be transferred (with these or other changes) to other topological or normed rings in which the powers of a finite number of elements form a basis under addition.

We do not give the proof because of lack of space. The main idea of the proof for the simplest case may be found in paper \({}^{(3)}\), a generalization of whose results is the present note.

Department of Applied Mathematics
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
17 I 1957

CITED LITERATURE

1. M. A. Evgrafov, Trudy Moskov. Mat. Obshch., 5, 89 (1956).
2. M. A. Evgrafov, Izv. AN SSSR, ser. matem., 21, No. 2, 223 (1957).
3. M. A. Evgrafov, A. D. Solov’ev, DAN, 113, No. 3 (1957).