UDC 517.941

MATHEMATICS

Z. PRESdorf

ON LINEAR EQUATIONS IN SPACES OF TEST AND GENERALIZED FUNCTIONS

(Presented by Academician V. I. Smirnov, 24 V 1965)

1°. Let \(\Psi_1, \Psi_2\) be countably normed (c.-n.) spaces; \(\Psi'_1, \Psi'_2\) their conjugates; \(A\) a linear continuous* operator mapping \(\Psi_1\) into \(\Psi_2\). We shall use the following notation: \(\mathfrak R(A)\) is the set of values of the operator \(A\); \(N(A)\) is the subspace of solutions of the equation \(Ax=0\); \(\alpha(A)=\dim N(A)\); \(\beta(A)=\dim \Psi_2/\mathfrak R(A)\).

The operator \(A\) is called normally solvable if the equation \(Ax=y\) is solvable if and only if \(\varphi(y)=0\) for all \(\varphi\in N(A^*)\), where \(A^*\) is the operator conjugate to \(A\). The property of normal solvability of the operator \(A\) is equivalent (²) to the closedness of the set \(\mathfrak R(A)\). Following (³), a normally solvable operator \(A\) will be called a \(\Phi_+\)-(\(\Phi_-\))-operator if the number \(\alpha(A)\) (\(\beta(A)\)) is finite, and a \(\Phi\)-operator if both numbers \(\alpha(A), \beta(A)\) are finite.

Theorem 1. In order that \(A\) be a \(\Phi_+\)-operator, it is necessary and sufficient that there exist a continuous operator \(B\) mapping \(\Psi_2\) into some c.-n. space \(\Psi_3\) and such that \(BA\) is a \(\Phi_+\)-operator.

Proof. The necessity is obvious. One may put \(B=I\), where \(I\) is the identity operator of the space \(\Psi_2\).

Sufficiency. Denote \(BA=S\). Clearly, \(\alpha(A)\leq \alpha(S)<\infty\). Let
\[ \|\varphi\|^{(i)}_1\leq \cdots \leq \|\varphi\|^{(i)}_p\leq \cdots \]
be a countable system of norms of the space \(\Psi_i\) \((i=1,2,3)\). Decompose \(\Psi_1\) into the direct (topological) sum
\[ \Psi_1=N(A)\dotplus N_1\dotplus \Psi_0, \]
where \(N_1\) is a direct complement of the subspace \(N(A)\) to \(N(S)\). Define the operator \(A_1\) as the restriction of the operator \(A\) to the subspace*** \(\Psi_0\dotplus N_1\), and prove that the operator \(A_1^{-1}\), defined on \(\mathfrak R(A)\), is bounded.

Let
\[ \mathfrak R(A)\supset F=\{f\in \mathfrak R(A):\ \|f\|^{(2)}_p\leq C_p\ (p=1,2,\ldots)\} \]
be a bounded set. For any element \(f\in F\) we have \(Bf=SA_1^{-1}f\), \(A_1^{-1}f=\xi_f+\eta_f\), where \(\xi_f\in\Psi_0,\ \eta_f\in N_1\). The restriction \(S_1\) of the operator \(S\) to \(\Psi_0\) maps one-to-one the c.-n. space \(\Psi_0\) onto the c.-n. space \(\mathfrak R(S)\). Hence (see (¹), § 7), the operator \(S_1^{-1}\) is bounded, and the set
\[ \{\xi_f=S_1^{-1}Bf,\ f\in F\} \]
is bounded in \(\Psi_1\).

Let \(r=\dim N_1\) (we may assume that \(r\geq 1\)) and let \(x_1,x_2,\ldots,x_r\) be a basis of the subspace \(N_1\). For any \(f\in F\) we have
\[ \eta_f=\sum_{i=1}^{r} a_i^f x_i, \]
where \(a_i^f\) are constants depending on \(f\in F\). We shall prove that \(|a_i^f|\leq C=\mathrm{const}\) \((i=1,\ldots,r)\), where \(C\) does not depend on \(f\). Fix an arbitrary number \(p_0\geq 1\). The elements \(y_i=A_1x_i\in\Psi_2\subset\Psi_{2p_0}\) (see (¹)) are linearly independent,

* By a countably normed space we shall mean a complete countably normed space.
* In a c.-n. space the notions of continuity and boundedness of an operator coincide (¹).
** By a subspace we shall mean a closed linear submanifold.

and therefore (4) for some \(M>0\)

\[ \sum_{i=1}^{r}|a_i^f|\leq M\left\|\sum_{i=1}^{r}a_i^f y_i\right\|_{p_0}^{(r)} = M\|f-A_1\xi_f\|_{p_0}^{(r)} \leq M(C_{p_0}+C'_{p_0})=C. \]

Hence we obtain \(\|\eta_f\|_{p}^{(1)}\leq C_p''\) for any \(p\), i.e., the set \(A_1^{-1}F\subset\Psi_1\) is bounded. Thus the operator \(A_1^{-1}\) is bounded and, consequently, continuous. It follows that the set \(\mathfrak{R}(A_1)=\mathfrak{R}(A)\) is closed.

Remark 1. In the particular case when \(\Psi_1,\Psi_2\) are Banach spaces, \(\Psi_3=\Psi_1\), and \(BA=I+T\) (\(T\) is a completely continuous operator), Theorem 1 was proved earlier by S. G. Mikhlin \((^5)\).

From Theorem 1 one easily obtains the following result.

Theorem 2. In order that \(A\) be a \(\Phi\)-operator, it is necessary and sufficient that there exist a continuous operator \(B\) such that \(\alpha(B)\) is finite and \(BA\) is a \(\Phi\)-operator.

Remark 2. Another characterization of the set of \(\Phi\)-operators is given in Theorem 12 of the paper \((^2)\). From the proof of necessity in that theorem it follows that if \(A\) is a \(\Phi\)-operator from \(\Psi_1\) into \(\Psi_2\), then there exists a continuous operator \(B\) from \(\Psi_2\) into \(\Psi_1\) such that \(\alpha(B)\) is finite and \(BA=I-L\), where \(L\) is a finite-dimensional operator in \(\Psi_1\). The converse assertion is true by virtue of Theorem 2.

\(2^\circ\). Let now \(\Psi\) be some basic space (in the sense of \((^1)\), Ch. II, § 1) such that it is a c.n. space and every function \(\varphi(x)\in\Psi\) determines a certain regular functional on \(\Psi\), in such a way that different functions correspond to different functionals. Examples of such spaces are the space \(C^\infty\), considered in \(3^\circ\), and also any space of type \(K\{M_p\}\) \((^1)\) possessing the property: for some \(p\) the function \(M_p^{-2}(x)\) is summable.

Let \(A\) be a linear continuous operator acting in the space \(\Psi\) and such that \(A^*\varphi\in\Psi\) for any function \(\varphi\in\Psi\). Elements \(\varphi\in\Psi\) are called basic functions, and elements \(f\in\Psi'\) are generalized functions (g.f.) (over the space \(\Psi\)). Let \(f\in\Psi'\). We shall say that \(u\in\Psi'\) is a generalized solution of the equation

\[ Au=f, \tag{1} \]

if \((u,A^*\psi)=(f,\psi)\) for any function \(\psi\in\Psi\). If \(f\in\Psi\) and \(u\in\Psi\) satisfies equation (1), then the function \(u\) will be called a classical solution of equation (1). For convenience we shall denote by \(A_0^*\) the operator \(A^*\) considered on the basic space \(\Psi\).

Definition. The operator \(A\) will be called normally solvable in g.f. if equation (1) is solvable (in g.f.) if and only if \((f,\varphi)=0\) for all \(\varphi\in N(A_0^*)\).

Theorem 3. If \(A_0^*\) is a \(\Phi\)-operator, then the operator \(A\) is normally solvable in g.f.

Proof. Let \(\{\varphi_j\}_{j=1}^{s}\) be a basis of the subspace \(N(A_0^*)\); let \(\{y_k\}_{k=1}^{q}\) be a complete system of generalized solutions of the homogeneous equation (1) (the numbers \(s,q\) are finite by assumption). Introduce into consideration the subspace \(\Psi_0=\mathfrak{R}(A_0^*)\). By the condition of Theorem 3, \(\varphi\in\Psi_0\) if and only if \((y_k,\varphi)=0\) \((k=1,\ldots,q)\). Construct elements \(\xi_i\in\Psi\) \((i=1,\ldots,q)\), biorthogonal to the g.f. \(y_k\), and let \(\Psi_1\) be the subspace with basis \(\xi_1,\ldots,\xi_q\). Then for the space \(\Psi\) we have the representation \(\Psi=\Psi_0+\Psi_1\). Denote by \(P_0\) the (continuous) projector of the space \(\Psi\) onto \(\Psi_0\).

Suppose the conditions are satisfied: \((f,\varphi_j)=0\) \((j=1,\ldots,s)\). Construct the functional \(u_0\), putting* for each \(\varphi\in\Psi\)

* We apply the well-known device (see \((^6)\)) for constructing a particular solution of an equation of the form (1).

\[ (u_0,\varphi)=(f,\varphi^{(0)}). \tag{2} \]

Here \(\varphi^{(0)}\in\Psi\) is a solution of the equation \(A_0^*\psi=P_0\varphi\). Let us prove that \(u_0\) is a uniquely determined linear continuous functional on \(\Psi\). Indeed, let \(A_1^*\) be the restriction of the operator \(A_0^*\) to the direct complement of the subspace \(N(A_0^*)\) in the space \(\Psi\). The operator \(B=(A_1^*)^{-1}\) is continuous and
\[ \varphi^{(0)}=BP_0\varphi+\sum_{j=1}^s a_j\varphi_j. \]
Consequently, \((u_0,\varphi)=(f,BP_0\varphi)\). It is easy to see that \(u_0\) is a solution of equation (1).

\(3^\circ\). We now apply the results from \(1^\circ\) and \(2^\circ\) to the investigation of a system of singular integral equations with Cauchy kernel

\[ \mathfrak A\varphi\equiv A(t)\varphi(t)+B(t)S\varphi+\int_\Gamma K(t,\tau)\varphi(\tau)\,d\tau=f(t) \tag{3} \]

(where by \(S\) we shall denote the singular operator
\[ \frac{1}{\pi i}\int_\Gamma \frac{\varphi(\tau)\,d\tau}{\tau-t}, \]
the symbolic matrices \(C(t)=A(t)+B(t)\), \(D(t)=A(t)-B(t)\) of which degenerate at a finite number of points of the contour \(\Gamma\). As the basic space we take the space \(C^\infty\) of functions given and infinitely differentiable on \(\Gamma\).* We shall assume that \(A(t),B(t)\in C^\infty\), \(K(t,\tau)\) is infinitely differentiable on \(\Gamma\times\Gamma\), and \(\Gamma\) is a closed smooth, infinitely differentiable contour bounding a finite simply connected domain.

It is known \((^{7-9})\) that in the spaces \(H\) (see \((^7)\)) and \(L_2(\Gamma)\) the Noether theorems are valid if and only if \(\Delta_1(t)\equiv \det D(t)\ne0\), \(\Delta_2(t)\equiv \det C(t)\ne0\) on \(\Gamma\). Nevertheless, the following is valid.

Theorem 4. If each of the functions \(\det(A-B)\), \(\det(A+B)\) has no more than a finite number of zeros of integral multiplicity on the contour, then for equation (3) the Noether theorems are valid in the space \(C^\infty\).

Proof. It is clear that the operator \(\mathfrak A\) maps \(C^\infty\) continuously into itself. For simplicity of notation we shall assume that \(\Delta_l(t)\) \((l=1,2)\) has one root \(t=\alpha_l\) of multiplicity \(m_l\). Consider the operator \(\mathfrak B=\mathfrak B_1+\mathfrak B_2\), where**

\[ \mathfrak B_l\equiv \frac{I-P_l}{2\Delta_l}H_l[I+(-1)^lS],\qquad P_l\psi=\sum_{\nu=0}^{m_l-1}\frac{1}{\nu!}\psi^{(\nu)}(\alpha_l)(t-\alpha_l)^\nu. \]

Here \(H_1,H_2\) are matrices adjugate to \(D,C\), respectively; the operators \(\mathfrak B_1,\mathfrak B_2\) are closed and defined on all of \(C^\infty\), hence \((^{10})\) they are continuous. It is easy to see that
\[ \mathfrak B\mathfrak A\varphi=\varphi(t)+\int_\Gamma T(t,\tau)\varphi(\tau)\,d\tau, \]
where \(T(t,\tau)\) is a matrix of the type \(K(t,\tau)\). \(\mathfrak B\mathfrak A\) is a \(\Phi\)-operator (with index equal to zero) in the space \(\widetilde H\) (see \((^7)\)) and, consequently, in the space \(C^\infty\), since every solution of the equation \(\mathfrak B\mathfrak A\varphi=\mathfrak Bf\) \((f\in C^\infty)\) from the class \(\widetilde H\) belongs to \(C^\infty\) and \(N((\mathfrak B\mathfrak A)^*)\subset C^\infty\). The conditions of Theorem 2 are fulfilled, and therefore \(\mathfrak A\) is a \(\Phi\)-operator in the space \(C^\infty\).

From what was said above, on the basis of Theorem 2 and remark \((^{11})\), we obtain the following.

Corollary. If the conditions of Theorem 4 are fulfilled, then the operator \(\mathfrak A\) admits a bounded regularization in the space \(C^\infty\).

Theorem 5. If the conditions of Theorem 4 are fulfilled, then the operator \(\mathfrak A\) is normally solvable in o.f. over the space \(C^\infty\).

* We introduce the topology in \(C^\infty\) in the usual way:
\[ \|\varphi\|_p=\max_{\substack{0\le k\le p\\ t\in\Gamma}}|\varphi^{(k)}(t)|\qquad (p=0,1,\ldots). \]

** In the case of several zeros of the function \(\Delta_l(t)\) \((l=1,2)\), \(P_l\) is replaced by the corresponding Hermite interpolation polynomial.

Proof. It is easy to see that \(\mathfrak A^*\varphi=\mathfrak M_1\mathfrak A'\mathfrak M_2\varphi\) for any \(\varphi\in C^\infty\), where \(\mathfrak M_1,\mathfrak M_2\) are operators that map \(C^\infty\) onto itself one-to-one and continuously in both directions, and \(\mathfrak A'\) is an operator of the form (3) with coefficients \(A'(t),-B'(t)\) (the prime denotes the transposition operation). Thus, \(\mathfrak M_1,\mathfrak M_2,\mathfrak A'\), and together with them (see (2), Satz 13) also the operator \(\mathfrak A_0^*=\mathfrak A^*\), considered on \(C^\infty\), are \(\Phi\)-operators. Consequently, the condition of Theorem 3 is satisfied.

The author expresses his sincere gratitude to Prof. S. G. Mikhlin for his attention to the present work.

Leningrad State University
named after A. A. Zhdanov

Received
19 V 1965

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