Abstract
Full Text
MATHEMATICS
I. Ts. Gokhberg and M. G. Krein
ON A STABLE SYSTEM OF PARTIAL INDICES OF THE HILBERT PROBLEM FOR SEVERAL UNKNOWN FUNCTIONS
(Presented by Academician V. I. Smirnov, 3 XII 1957)
- Let \(\Gamma\) be a contour consisting of a finite number of simple smooth closed directed curves with continuous curvature, bounding on the left in the complex plane a connected finite domain \(D_{+}\). Its complement is denoted by \(D_{-}\). By \(H\) we shall denote the set of all functions defined on \(\Gamma\) and satisfying the Hölder condition. By \(H_{(n\times 1)}\) (respectively \(H_{(n\times n)}\)) we shall denote the set of all \(n\)-dimensional vector-functions (\(n\times n\) matrix functions) with coordinates (elements) from \(H\). The linear set \(H_{(n\times n)}\) will be regarded by us as an incomplete linear normed space with the norm defined by
\[ \|A\|=n\max |a_{jk}(t)|\qquad \left(A(t)=\|a_{jk}(t)\|_1^n\in H_{(n\times n)}\right), \]
where the maximum is taken over all \(t\in\Gamma\) and \(j,k=1,2,\ldots,n\).
Let \(A(t)\in H_{(n\times n)}\) be a nonsingular matrix function, and let
\(\varkappa_1(A)\geq\varkappa_2(A)\geq\cdots\geq\varkappa_n(A)\) be the partial indices of the corresponding Hilbert problem \((^1,^2)\)
\[ \Phi_{+}(t)=A(t)\Phi_{-}(t). \tag{1} \]
We shall call the system of partial indices \(\varkappa_j(A)\) \((j=1,2,\ldots,n)\) stable if to the matrix \(A(t)\in H_{(n\times n)}\) one can assign a \(\delta>0\) such that every matrix \(B(t)\in H_{(n\times n)}\) in the \(\delta\)-neighborhood of the matrix \(A(t)\): \(\|B-A\|<\delta\), has the same indices as \(A(t)\): \(\varkappa_j(B)=\varkappa_j(A)\) \((j=1,2,\ldots,n)\). This definition is justified by the following theorem.
Theorem 1*. Let the nonsingular matrix \(A(t)\in H_{(n\times n)}\), and let
\[ \varkappa=\varkappa(A)=\frac{1}{2\pi}\,[\arg\det A(t)]_{\Gamma}. \]
The system of partial indices of the matrix \(A(t)\) is stable if and only if
\[ \varkappa_1(A)=\cdots=\varkappa_r(A)=q+1;\qquad \varkappa_{r+1}(A)=\cdots=\varkappa_n(A)=q, \tag{2} \]
where the integers \(q,r\) are determined from the relation \(\varkappa=qn+r,\ 0\leq r<n\).
The necessity of the formulated assertion is easily proved by repeating the arguments given in the proof of Theorem 10.2 of the authors’ paper \((^3)\).
From the same arguments it follows, among other things, that in any neighborhood of an arbitrary nonsingular matrix function \(A(t)\in H_{(n\times n)}\) there will always be found matrices \(B(t)\) \((\in H_{(n\times n)})\) with a stable system of indices.
The sufficiency of the condition of Theorem 1 follows directly from the following more general propositions.
* After the present note had been submitted, the authors learned of a paper by G. F. Mandzhavidze \((^7)\), in which the stability of the system of partial indices is proved for any of its arithmetic structures; this result of \((^7)\) is erroneous.
- Theorem 2. Let the nonsingular matrix function \(A(t)\in H_{(n\times n)}\). Then there exists a number \(\delta(>0)\) such that any matrix function \(B(t)\) \((\in H_{(n\times n)})\) from the \(\delta\)-neighborhood of \(A\): \(\|B-A\|<\delta\), will be nonsingular and, for every integer \(p\), the inequalities
\[ \sum_{\chi_j(A)>p}\bigl(\chi_j(A)-p\bigr)\ge \sum_{\chi_j(B)>p}\bigl(\chi_j(B)-p\bigr) \tag{3} \]
will hold.
We shall precede the proof of the theorem with some remarks.
In a natural way, on the contour \(\Gamma\) one defines the Hilbert space \(L^{(2)}_{(n\times 1)}\) of \(n\)-dimensional vector functions whose coordinates have summable square on \(\Gamma\). For any \(\varphi\in L^{(2)}_{(n\times 1)}\) the singular integral
\[ S\varphi=\frac{1}{\pi i}\int_{\Gamma}\frac{\varphi(\tau)}{\tau-t}\,d\tau, \tag{4} \]
has meaning, defining a certain linear bounded operator \(S\) in the space \(L^{(2)}_{(n\times 1)}\) \((^4)\).
Every solution of the Hilbert problem (1) that vanishes at infinity generates a solution \(\varphi(t)=\Phi_+(t)-\Phi_-(t)\in H_{(n\times 1)}\) of the homogeneous singular integral equation
\[ (I+A(t))\varphi(t)+(I-A(t))S\varphi(t)=0, \tag{5} \]
and in this case
\[ \Phi(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\varphi(t)}{t-z}\,dt. \tag{6} \]
Conversely, to every solution \(\varphi\in H_{(n\times 1)}\) of equation (5), formula (6) corresponds a solution of the Hilbert problem (1) that vanishes at infinity.
It is proved without particular difficulty that if the nonsingular matrix function \(A(t)\in H_{(n\times n)}\), then all solutions of equation (5) belong to the set \(H_{(n\times 1)}\).
Proof of Theorem 2. Denote by \(U\) the operator standing on the left-hand side of equality (5). Note that the dimension \(\alpha(u)\) of the subspace \(\mathfrak Z(u)\) of all zeros of the operator \(U\) is always equal to the sum of the positive indices \(\chi_j(A)\) of the Hilbert problem (1), which easily follows from the general form of the solutions of this problem ((1), § 127). The sum of the absolute values of all negative indices of problem (1) gives the dimension of the orthogonal complement to the range of the operator \(U\).
Put \(A_1(t)=(t-c)^{-p}A(t)\), where \(c\) is an interior point of \(D_+\), and \(p\) is some integer satisfying the inequalities
\[ \chi_n(A)\le p\le \chi_1(A). \tag{7} \]
Obviously,
\[ \chi_j(A_1)=\chi_j(A)-p \qquad (j=1,2,\ldots,n). \]
Introduce for consideration the operator \(U_1\) acting in \(L^{(2)}_{(n\times 1)}\):
\[ U_1\varphi=(I+A_1(t))\varphi(t)+(I-A_1(t))S\varphi(t). \]
Then
\[ \alpha(U_1)=\sum_{\chi_j(A)>p}\bigl(\chi_j(A)-p\bigr),\qquad \beta(U_1)=-\sum_{\chi_j(A)<p}\bigl(\chi_j(A)-p\bigr). \]
By virtue of the general theorem from \((^5)\) (see also Theorems 2.4 in \((^6)\)), there exists a number \(\rho_p>0\) such that, for any linear operator \(V_1\) acting
in \(L^{(2)}_{(n\times 1)}\) and such that
\[ \|U_1-V_1\|<\rho_p, \tag{8} \]
we shall have
\[ \alpha(V_1)\leq \alpha(U_1). \tag{9} \]
Denote by \(\delta\;(>0)\) a number smaller than all the quantities
\[ \rho_p(1+\|S\|)^{-1}\min_{t\in\Gamma}|t-c|^p \]
and such that every matrix-function from the \(\delta\)-neighborhood of \(A(t)\) is nonsingular. We shall show that this \(\delta\) satisfies the requirements of the theorem.
Let, for \(B(t)\in H_{(n\times n)}\), the condition \(\|A-B\|<\delta\) be fulfilled. Then for any integer \(p\) from the interval (7) the matrix-function \(B_1(t)=(t-c)^{-p}B(t)\) will be nonsingular, and for it we shall have:
\[ \|A_1-B_1\|<\rho_p(1+\|S\|)^{-1} \qquad (\varkappa_n(A)\leq p\leq \varkappa_1(A)). \]
Consider the operator \(V_1\), acting in the space \(L^{(2)}_{(n\times 1)}\) according to the equality
\[ V_1\varphi=(I+B_1(t))\varphi(t)+(I-B_1(t))S\varphi(t). \]
Since
\[ \|U_1-V_1\|\leq \|A_1-B_1\|(1+\|S\|), \]
the operator \(V_1\) satisfies condition (8). Consequently, for the operator \(V_1\) the relations (9), equivalent to the relations (3), hold. Thus, the inequalities (3) will hold for all integers \(p\) from the interval (7). In particular, for \(p=\varkappa_1(A)\) and \(p=\varkappa_n(A)\) these inequalities give
\[ \varkappa_1(A)\geq \varkappa_1(B),\qquad \varkappa_n(A)\leq \varkappa_n(B). \tag{10} \]
Since the \(\delta\)-neighborhood of the matrix-function \(A(t)\) consists of nonsingular matrix-functions, for any matrix-function \(B(t)\) from this neighborhood \(\varkappa(B)=\varkappa(A)\), and, consequently:
\[ \sum_1^n \varkappa_i(B)=\sum_1^n \varkappa_i(A). \tag{11} \]
Hence, from (10) it follows that, for any integer \(p\) lying outside the interval \((\varkappa_n(A),\varkappa_1(A))\), the equality sign will occur in relation (3). The theorem is proved.
- Denote by \(\mathfrak S_n\) the totality of all possible ordered systems \(\{\varkappa_i\}_1^n\) of integers \(\varkappa_1\geq \varkappa_2\geq\cdots\geq \varkappa_n\). Let \(\{\varkappa_j\}_1^n\) and \(\{\varkappa'_j\}_1^n\) be two systems from \(\mathfrak S_n\); we agree to say that the second system is obtained from the first by means of an elementary operation if, for certain integers \(p\) and \(q\) \((1\leq p<q\leq n)\),
\[ \varkappa'_p=\varkappa_p-1,\qquad \varkappa'_q=\varkappa_q+1,\qquad \varkappa'_j=\varkappa_j\quad \text{for } j\ne p,q. \]
We shall further agree to write \(\{\varkappa_j\}_1^n>\{\varkappa'_j\}_1^n\) if the system \(\{\varkappa'_j\}\) either coincides with the system \(\{\varkappa_j\}\), or is obtained from it by successive application of a number of elementary operations. If a system \(\{\varkappa_j\}_1^n\in\mathfrak S_n\), then we shall call its averaging the system \(\{\hat{\varkappa}_j\}_1^n\) defined by the equalities
\[ \hat{\varkappa}_1=\hat{\varkappa}_2=\cdots=\hat{\varkappa}_r=q+1,\qquad \hat{\varkappa}_{r+1}=\hat{\varkappa}_{r+2}=\cdots=\hat{\varkappa}_n=q, \]
where the integers \(q\) and \(r\) are determined from the relations
\[ \sum_{1}^{n} x_j = qn + r, \qquad 0 \leq r < n. \]
It is easy to see that always
\[ \sum_{1}^{n} x_j = \sum_{1}^{n} \hat{x}_j, \qquad \{x_j\}_{1}^{n} > \{\hat{x}_j\}_{1}^{n}. \]
It can be shown that, when equality (11) is satisfied, the totality of all relations (3) is equivalent to the fact that
\[ \{x_j(A)\}_{1}^{n} > \{x_j(B)\}_{1}^{n} \quad \bigl(> \{\hat{x}_j(A)\}_{1}^{n}\bigr). \]
It turns out that, however a system of numbers \(\{x'_j\}_{1}^{n}\) satisfying the condition
\[ \{x_j(A)\}_{1}^{n} > \{x'_j\} \]
is chosen, in any \(\delta\)-neighborhood of the nonsingular matrix-function \(A(t) \in H_{(n\times n)}\) there exists a nonsingular matrix-function \(B(t) \in H_{(n\times n)}\) such that
\[ x_j(B) = x'_j \qquad (j = 1,2,\ldots,n). \]
Beltsy State Pedagogical Institute
Odessa Civil Engineering Institute
Received
29 XI 1957
References Cited
- N. I. Muskhelishvili, Singular Integral Equations, Moscow–Leningrad, 1946.
- N. P. Vekua, Systems of Singular Integral Equations, Moscow–Leningrad, 1950.
- I. Ts. Gokhberg, M. G. Krein, Uspekhi Mat. Nauk, 13, issue 2 (1958).
- S. G. Mikhlin, Uspekhi Mat. Nauk, 3, issue 3 (1948).
- M. G. Krein, M. A. Krasnosel’skii, Mat. Sb., 30 (72), 1 (1952).
- I. Ts. Gokhberg, M. G. Krein, Uspekhi Mat. Nauk, 12, issue 2 (1957).
- G. F. Mandzhavidze, Reports of the Academy of Sciences of the Georgian SSR, 14, No. 10 (1953).