MATHEMATICS

R. A. KORDZADZE

ON THE INDEX OF SINGULAR INTEGRO-FUNCTIONAL OPERATORS

(Presented by Academician I. N. Vekua on 29 VII 1968)

1. Let \(\Gamma=\sum \Gamma_j\) be a collection of a finite number of simple closed Lyapunov contours without common points, bounding a finite domain \(D\) of the complex plane \(z=x+iy\). We shall assume that the positive direction of the contour \(\Gamma\) leaves the domain \(D\) on the left. Let \(\alpha_1(t)\equiv \alpha(t)\) \((\alpha(t)\ne t,\ t\in \Gamma_j)\) be an orientation-preserving homeomorphism of the contour \(\Gamma\) (i.e., the function \(\alpha(t)\) maps each \(\Gamma_j\) homeomorphically onto itself with preservation of orientation), generating a cyclic group of finite order \(n\) with elements:
\[ \alpha_0(t)\equiv t,\ \alpha_1(t),\ldots,\alpha_{n-1}(t) \quad (\alpha_j(t)=\alpha[\alpha_{j-1}(t)]). \]
We shall suppose that the derivative \(\alpha'(t)=d\alpha(t)/dt\) exists, is different from zero, and satisfies a Hölder condition everywhere on \(\Gamma\).

Consider a pair of mutually adjoint singular integro-functional operators:
\[ T\varphi \equiv \sum_{p=0}^{n-1} \left\{ A_p(t_0)\varphi[\alpha_p(t_0)] + \left(\frac{1}{\pi i}\right) \int_\Gamma [B_p(t_0,t)\varphi(t)][t-\alpha_p(t_0)]^{-1}\,dt \right\}, \]
\[ T'\psi \equiv \sum_{p=0}^{n-1} \left\{ A'_{n-p}[\alpha_p(t_0)]\,\alpha'_p(t_0)\psi[\alpha_p(t_0)] - \right. \]
\[ \left. - \left(\frac{1}{\pi i}\right) \int_\Gamma [B'_p(t,t_0)\psi(t)][\alpha_p(t)-t_0]^{-1}\,dt \right\} \]
\[ (A'_n[\,]=A'_0[\,],\quad t_0\in\Gamma), \]
where \(A_p(t_0)\), \(B_p(t_0,t)\) are given square matrices of order \(m\ge 1\) on \(\Gamma\), satisfying a Hölder condition; \(\varphi,\psi\in H(m)\); \(H(m)\) is the complex space of \(m\)-dimensional vectors defined on \(\Gamma\) and satisfying a Hölder condition. A prime over matrices denotes passage to the transposed matrix.

The operator \(T\) in the space \(H(m)\) was studied in the works \((^{1-3})\). In them, in particular, the Noether condition and the index formula were established in the form
\[ \operatorname{ind} T=\operatorname{ind} D+\chi(T); \]
here \(D\) is the singular integral operator with Cauchy kernel (associated with the operator \(T\) \((^{1-3})\)), acting in the space \(H(mn)\), whose index is explicitly computed by the known formula of N. I. Muskhelishvili \((^4)\); \(\chi(T)\) is a fully determined integer possessing the property that
\[ \chi(T+V)=\chi(T) \]
for any completely continuous operator \(V\) acting in the space \(H(m)\).

Analogous results are also valid for the adjoint operators \(\Lambda\) and \(\Lambda'\) \((^5)\), where
\[ \Lambda\varphi = \sum_{p=0}^{n-1} \left\{ A_p^{(1)}(t_0)\varphi[\alpha_p(t_0)] + A_p^{(2)}(t_0)\varphi[\overline{\alpha_p(t_0)}] + S_{1,p}\varphi + \overline{S_{2,p}\varphi} \right\}, \]
\[ S_{\nu,p}\varphi \equiv \left(\frac{1}{\pi i}\right) \int_\Gamma [B_p^{(\nu)}(t_0,t)\varphi(t)][t-\alpha_p(t_0)]^{-1}\,dt. \]

Here \(A_p^{(\nu)}(t_0)\) and \(B_p^{(\nu)}(t_0,t)\) are square matrices of order \(m \geqslant 1\), given on \(\Gamma\), satisfying the Hölder condition; \(\varphi \in H(m)\). In this case, the associated operator is a singular integral operator with Cauchy kernel, acting both on the element \(\Phi(t)\in H(mn)\) and on its complex conjugate \(\overline{\Phi(t)}\); the theory of such operators and the explicit index formula are given in \({}^{(6)}\) (see also \({}^{(7)}\)).

One can directly establish the relation between the operator \(T\) and \(\Lambda\); namely, if linear independence is understood over the field of complex numbers in the case of the operator \(T\), and over the field of real numbers in the case of the operator \(\Lambda\) (as we shall do below), then the following holds:

Theorem 1. If it is assumed that the coefficients of the operator \(T\) are related to the coefficients of the operator \(\Lambda\) by the relations
\[ A_p(t_0)= \left\| \begin{array}{cc} A_p^{(1)}(t_0) & A_p^{(2)}(t_0)\\ \overline{A_p^{(2)}(t_0)} & \overline{A_p^{(1)}(t_0)} \end{array} \right\|,\qquad B_p(t_0,t)= \left\| \begin{array}{cc} B_p^{(1)}(t_0,t) & -\,l_p(t_0,t)\overline{B_p^{(2)}(t_0,t)}\\ B_p^{(2)}(t_0,t) & -\,l_p(t_0,t)\overline{B_p^{(1)}(t_0,t)} \end{array} \right\|, \]
\[ l_p(t_0,t)=(d\bar t/dt)[t-\alpha_p(t_0)]\,[\bar t-\overline{\alpha_p(t_0)}]^{-1}, \tag{1} \]
then the equations \(\Lambda\varphi=f\) \((f\in H(m))\) and \(T\Phi=\{f,\bar f\}\) are simultaneously Noetherian or not; moreover (in the Noetherian case) the formula
\[ \operatorname{ind}T=\operatorname{ind}\Lambda \]
is valid; further, if \(l,l',\lambda,\lambda'\) are the maximal numbers of linearly independent solutions of the equations \(T\Phi=0\), \(T'\Psi=0\), \(\Lambda\varphi=0\), and \(\Lambda'\psi=0\), respectively, then \(l=\lambda,\ l'=\lambda'\).

The proof of this theorem is not difficult to obtain by applying (with the appropriate additions) the method proposed by G. F. Mandzhavidze \({}^{(7)}\) in the study of Fredholm-type equations containing both unknown functions and their complex conjugate values.

The meaning of Theorem 1 is that it suffices to construct the theory of operators of the form \(T\)*.

The main purpose of the present paper is to obtain an explicit formula for the index of the operator \(T\) for \(n=2^k\) \((k=0,1,\ldots)\). We also obtain a formula explicitly expressing \(\operatorname{ind}T\) for even \(n\) in terms of \(\operatorname{ind}T\) for odd \(n\).

1. The number \(n\)—the order of the cyclic group generated by the mapping \(\alpha(t)\)—uniquely determines the number \(k\) \((k=0,1,\ldots)\) and the odd number \(r\) \((r=1,3,\ldots)\) from the condition \(n=r\cdot 2^k\). In what follows, everywhere, unless otherwise stated, by \(k\) and \(r\) we shall always denote the given numbers determined in the manner indicated above. For each fixed \(q\) \((q=0,1,\ldots,k)\), by
   \[ \theta_q(j)\equiv \theta_{k,r,q}^{(j)}(j) \]
   we shall denote the periodic integer-valued function with period \(r\cdot 2^{k-q}\), defined on the set \(j=0,1,\ldots\), which for \(0\leqslant j\leqslant r\cdot 2^{k-q}-1\) takes the value \(\theta_q(j)\equiv j\).

Put
\[ \alpha^{(0)}(t)=\alpha(t),\qquad \alpha^{(q)}(t)=\alpha_{2}^{(q-1)}(t),\qquad q=1,2,\ldots,k,\quad t\in\Gamma, \]
\[ (\alpha_0^{(q)}(t)\equiv t,\quad \alpha_1^{(q)}(t)\equiv \alpha^{(q)}(t),\quad \alpha_j^{(q)}(t)=\alpha^{(q)}[\alpha_{j-1}^{(q)}(t)], \]
\[ j=1,2,\ldots;\quad q=0,1,\ldots,k), \]
\[ \omega_p^{(q)}(t_0,t)=(d\alpha_p^{(q)}(t)/dt)\,[t-t_0]\,[\alpha_p^{(q)}(t)-\alpha_p^{(q)}(t_0)]^{-1} \]
\[ (q=0,1,\ldots,r\cdot 2^{k-q}-1,\quad t_0,t\in\Gamma) \]
and introduce into consideration square matrices \(A_p^{(q)}\) and \(\widetilde B_p^{(q)}\) of order \(m\cdot 2^q\), defined from the recurrent relations:

* In our paper \({}^{(5)}\) an example was given showing the erroneousness of certain results of the work of G. S. Litvinchuk \({}^{(8)}\), devoted to the study of an operator of the form \(\Lambda\). However, in the subsequent publication \({}^{(9)}\) the very same errors are repeated; from the results obtained by us below it follows that Lemma 1, Lemma 3, Theorem 2, and Theorem 5 of \({}^{(9)}\) are incorrect. It is also easy to give an example showing the erroneousness of Theorem \(2'\) of \({}^{(9)}\).

\[ A_p^{(0)}(t_0)=A_p(t_0),\quad A_p^{(q)}(t_0)=\left\|A_{\theta_{q-1}}^{(q-1)}{}_{[r\cdot 2^{k-q+1}+\nu-j]}\bigl[\alpha_j^{(q-1)}(t_0)\bigr]\right\|_{\substack{j=0,1\\ \nu=2p,\,2p+1}}, \]
\[ B_p^{(0)}(t_0,t)=\omega_p^{(0)}(t_0,t)B_p\bigl[t_0,\alpha_p^{(0)}(t)\bigr], \]
\[ B_p^{(q)}(t_0,t)= \left\|\omega_j^{(q-1)}(t_0,t)B_{q-1}^{(q-1)}{}_{[r\cdot 2^{k-q+1}+\nu-j]}\times \right. \]
\[ \left. {}\times \bigl[\alpha_j^{(q-1)}(t_0),\alpha_j^{(q-1)}(t)\bigr]\right\|_{\substack{j=0,1\\ \nu=2p,\,2p+1}} \]
\[ (q=1,2,\ldots,k;\quad p=0,1,\ldots,r\cdot 2^{k-q}-1;\quad t,t_0\in\Gamma). \tag{2} \]

Here \(j\) is the row number of the matrix, and \(\nu\) is the column number.

The Noetherian condition for the operator \(T\) has the form \((1\text{–}3)\)

\[ \det\left\|A_\theta^{(k)}{}_{[r+\nu-j]}\bigl[\alpha_j^{(k)}(t)\bigr] \pm B_\theta^{(k)}{}_{[r+\nu-j]}\times \right. \]
\[ \left. {}\times \bigl[\alpha_j^{(k)}(t),\alpha_j^{(k)}(t)\bigr]\right\|_{\substack{j=0,\,r-1\\ \nu=0,\,r-1}}\ne 0 \quad (t\in\Gamma), \tag{3} \]

and we shall assume that it is satisfied.

Consider a pair of mutually adjoint singular integro-functional operators

\[ h_k\Phi_k= \sum_{p=0}^{r-1} \left\{ A_p^{(k)}(t_0)\Phi_k\bigl[\alpha_p^{(k)}(t_0)\bigr] + \left(\frac{1}{\pi i}\right) \int_\Gamma \bigl[B_p^{(k)}(t_0,t)\Phi_k[\alpha_p^{(k)}(t)]\bigr](t-t_0)^{-1}\,dt \right\}, \]

\[ h_k'\psi_k= \sum_{p=0}^{r-1} \alpha_{r-p}'{}^{(k)}(t_0) \left\{ A_p'{}^{(k)}\bigl[\alpha_{r-p}^{(k)}(t_0)\bigr]\psi_k\bigl[\alpha_{r-p}^{(k)}(t_0)\bigr] -\right. \]
\[ \left. {}- \left(\frac{1}{\pi i}\right) \int_\Gamma \bigl[B_p'{}^{(k)}(t,\alpha_{r-p}^{(k)}(t_0))\psi_k(t)\bigr] \bigl[t-\alpha_{r-p}^{(k)}(t_0)\bigr]^{-1}\,dt \right\} \]
\[ \bigl(\alpha_{r-p}'{}^{(k)}(t)=d\alpha_{r-p}^{(k)}(t)/dt,\quad \Phi_k,\psi_k\in H(m\cdot 2^k),\quad t_0\in\Gamma\bigr). \]

The Noetherian condition for the operator \(h_k\) also has the form (3). We note that the order of the cyclic group generated by the mapping \(\alpha^{(k)}(t)\) is equal to the odd number \(r\). It turns out that it is enough to know \(\operatorname{ind} h_k\) in order to find \(\operatorname{ind} T\); namely, the following is true:

Theorem 2. The number \(2^k \operatorname{ind} T\) coincides with the index of the operator \(h_k\).

For \(r=1\) (i.e., when \(n=2^k\)) condition (3) takes the form

\[ \det\left\|A_0^{(k)}(t)\pm B_0^{(k)}(t,t)\right\|\ne 0 \quad (t\in\Gamma), \tag{4} \]

and from Theorem 2 there follows directly

Theorem 3. If the order of the cyclic group generated by the mapping \(\alpha(t)\) is equal to \(2^k\) \((k=0,1,\ldots)\), then the number \(2^{k+1}\pi i\,\operatorname{ind}T\) coincides with the increment of the function

\[ \arg\frac{\det\left\|A_0^{(k)}(t)-B_0^{(k)}(t,t)\right\|} {\det\left\|A_0^{(k)}(t)+B_0^{(k)}(t,t)\right\|} \tag{5} \]

under one traversal of the contour \(\Gamma\) in the positive direction.

We note that if \(r=1,\ k=0\) (i.e., \(n=1\)), then the operator \(T\) becomes a singular integral operator with Cauchy kernel, and the index formula given in Theorem 3 becomes the formula of N. I. Muskhelishvili \((^4)\).

The method of proof of Theorem 2 is as follows. We include the operator \(T\) in a finite family of singular integro-functional operators

\[ T_q\Phi_q= \sum_{p=0}^{r\cdot 2^{k-q}-1} \left\{ A_p^{(q)}(t_0)\Phi_q\bigl[\alpha_p^{(q)}(t_0)\bigr]+ \right. \]
\[ \left. {}+\frac{1}{\pi i} \int_\Gamma \bigl[B_p^{(q)}(t_0,t)\Phi_q[\alpha_p^{(q)}(t)]\bigr](t-t_0)^{-1}\,dt \right\} \]

\[ (q=0,1,\ldots,k,\quad \Phi_q\in H(m\cdot 2^q),\quad t_0\in\Gamma); \]

It is easy to see that \(T_0 \equiv T,\ T_k=h_k\). Further, the proof of Theorem 2 is completed by the following lemma:

Lemma 1. Let \(q\) \((q=1,2,\ldots,k)\) be an arbitrary fixed number; then the numbers \(l(T_{q-1})\), \(l(T'_{q-1})\), \(l(T_q)\), and \(l(T'_q)\) are finite, and the formula \(2\,\operatorname{ind} T_{q-1}=\operatorname{ind} T_q\) holds; here \(l(E)\) denotes the dimension of the null subspace of the operator \(E\).

One can give formulas relating the numbers \(l(T)\), \(l(h_k)\), \(l(T')\), \(l(h'_k)\), respectively. Denote by \(S_q\) \((q=0,1,\ldots,k-1)\) the operator obtained from \(T_q\) \((q=0,1,\ldots,k-1)\) if in it \(A_p^{(q)}(t_0)\) and \(B_p^{(q)}(t_0,t)\) are replaced respectively by \((-1)^p A_p^{(q)}(t_0)\) and \((-1)^p B_p^{(q)}(t_0,t)\). Then the following lemma is valid:

Lemma 2. The numbers \(l(S_q)\) and \(l(S'_q)\) \((q=0,1,\ldots,k-1)\) are finite, and the formulas
\[ l(T_k)=l(T)+\sum_{q=0}^{k-1} l(S_q),\qquad l(T'_k)=l(T')+\sum_{q=0}^{k-1} l(S'_q). \]

1. With the aid of Theorem 1, all the results obtained above for the operator \(T\) are rephrased also for the operator \(\Lambda\). We note only that the condition of nondegeneracy of the operator \(\Lambda\) has the form
   \[ \det \left\| A_{\theta_k}^{(k)}[r+\nu-j]\bigl[\alpha_j^{(k)}(t)\bigr] - B_{\theta_k}^{(k)}[r+\nu-j]\bigl[\alpha_j^{(k)}(t),\alpha_j^{(k)}(t)\bigr] \right\|_{j=0,\overline{r-1},\ \nu=\overline{0,r-1}}\ne 0 \]
   \[ (t\in \Gamma), \tag{6} \]
   where \(A_p(t_0)\) and \(B_p(t_0,t)\) have the form (1), and from Theorem 3 there follows

Theorem 4. If the order of the cyclic group generated by the mapping \(\alpha(t)\) is equal to \(2^k\) \((k=0,1,\ldots)\), then the number \(2^k\pi\,\operatorname{ind}\Lambda\) coincides with the increment of the function (6) under one traversal of the contour \(\Gamma\) in the positive direction.

Tbilisi State University

Received
22 VII 1968

CITED LITERATURE

1. R. A. Kordzadze, DAN, 154, No. 6 (1964).
2. R. A. Kordzadze, DAN, 160, No. 6 (1965).
3. R. A. Kordzadze, Singular integro-functional equations and some of their applications in the theory of boundary-value problems, Candidate’s dissertation, Novosibirsk, 1964.
4. N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
5. R. A. Kordzadze, DAN, 168, No. 6 (1966).
6. L. G. Mikhailov, A new class of special integral equations and its applications to differential equations with singular coefficients, Dushanbe, 1963.
7. G. F. Mandzhavidze, PMM, 15, issue 3 (1951).
8. G. S. Litvinchuk, DAN, 162, No. 1 (1965).
9. G. S. Litvinchuk, Izv. AN SSSR, ser. matem., 31, No. 3 (1967).