Mathematics

K. M. FISHMAN and Yu. N. VALITSKY

ON THE APPLICABILITY OF FREDHOLM’S THEORY TO CERTAIN LINEAR TOPOLOGICAL SPACES

(Presented by Academician V. I. Smirnov on 21 VI 1957)

1. Let \(\mathfrak{B}_r\) be a real or complex Banach space depending on a parameter \(r\) \((\alpha<r\leq\beta)\). Denote by \(\|\ \|_r\) the norm of \(\mathfrak{B}_r\). We shall assume the following concerning these spaces: 1) \(\mathfrak{B}_r\) is everywhere a dense linear manifold in \(\mathfrak{B}_{r'}\) \((r'<r)\) with respect to the norm \(\mathfrak{B}_{r'}\); 2) for every \(f\in\mathfrak{B}_r\) one has \(\|f\|_{r'}\leq\|f\|_r\) \((\alpha<r'<r\leq\beta)\).

Consider the sets

\[ \mathfrak{A}_r=\prod_{r'<r}\mathfrak{B}_{r'}\quad(\alpha<r\leq\beta). \]

By virtue of condition 1), \(\mathfrak{A}_r\) is everywhere dense in \(\mathfrak{B}_{r'}\) \((r'<r)\). The set \(\mathfrak{A}_r\) becomes a linear space if addition and multiplication by numbers are defined in it in the same way as in any \(\mathfrak{B}_{r'}\) \((r'<r)\). We define convergence in \(\mathfrak{A}_r\) in the following manner: we shall say that \(f_n\to0\) as \(n\to\infty\) in \(\mathfrak{A}_r\) if, for all \(r'<r\), \(\|f_n\|_{r'}\to0\) \((n\to\infty)\). Thus \(\mathfrak{A}_r\) becomes a complete linear topological space \(\left({}^{1}\right)\). Obviously, for any \(r\) and \(r''\), \(r<r''\), \(\mathfrak{B}_r\) forms a linear manifold in \(\mathfrak{A}_r\), and \(\mathfrak{A}_{r''}\) in \(\mathfrak{B}_r\), while the topology in \(\mathfrak{A}_{r''}\) is stronger than the topology in \(\mathfrak{A}_r\).

Let \(\mathfrak{B}_r^*\) be the space conjugate to \(\mathfrak{B}_r\). If \(F\in\mathfrak{B}_{r'}^*\) for some \(r'<r\), then, by virtue of 2), \(F\) is a linear functional also on \(\mathfrak{B}_r\), i.e. \(\mathfrak{B}_{r'}^*\subset\mathfrak{B}_r^*\) \((r'<r)\), and is there a linear subset. Introduce the linear space

\[ \mathfrak{A}_r^*=\sum_{r'<r}\mathfrak{B}_{r'}^*, \]

defining in it the same linear operations as in \(\mathfrak{B}_{r'}^*\), and call it the space conjugate to \(\mathfrak{A}_r\). Every functional \(F\in\mathfrak{A}_r^*\) is a linear functional on \(\mathfrak{A}_r\). On the other hand, taking into account that the same topology in \(\mathfrak{A}_r\) may be obtained by means of a countable system of norms \(\|\ \|_{r_n'}\), \((r>r_n'\to r)\) \(\left({}^{1}\right)\), one may assert also the converse, i.e. that every linear (continuous) functional on \(\mathfrak{A}_r\) belongs to \(\mathfrak{A}_r^*\). Obviously, for \(r<r''\), \(\mathfrak{A}_r^*\subset\mathfrak{A}_{r''}^*\), and \(\mathfrak{A}_r^*\subset\mathfrak{B}_r^*\); moreover the linear operations in \(\mathfrak{A}_r^*\) are the same as in \(\mathfrak{A}_{r''}^*\) and in \(\mathfrak{B}_r^*\).

\(\mathfrak{A}_r^*\) becomes a linear topological space if the notion of convergence is introduced in either of two ways: a) \(F_n\) converges (weakly) to zero \((F_n \xrightarrow{\mathrm{c}}0)\) if \(F_n(f)\to0\) for every \(f\in\mathfrak{A}_r\); b) \(F_n\) converges (strongly) to zero \((F_n\Rightarrow0)\) if \(F_n(f)\to0\) uniformly on every bounded set \(M\subset\mathfrak{A}_r\) (a set is called bounded if there exist constants \(K_{r'}<\infty\) \((r'<r)\) such that \(\|f\|_{r'}\leq K_{r'}\) for all \(f\in M\)).

1. Let \(A\) be a linear operator mapping \(\mathfrak{B}_r\) into \(\mathfrak{B}_r\) \((\alpha<r\leq\beta)\), and suppose that its action does not depend on \(r\). The operator \(A\) induces in each \(\mathfrak{A}_r\) a certain linear continuous operator whose action likewise does not depend on \(r\). Denote by \(A^*\) the operator conjugate to \(A\), acting in \(\mathfrak{B}_r^*\). Its action also does not depend on \(r\). This permits one to consider the operator \(A^*\) on the spaces

\[ \mathfrak{A}_r^*=\sum_{r'<r}\mathfrak{B}_{r'}^*. \]

The distributivity

of this operator and the independence of its action from \(r\) are obvious. The operator \(A^*\) is continuous in both topologies in \(\mathfrak A_r^*\).

1. Let now Fredholm theory be applicable to the operator \(A\) in each space \(\mathfrak B_r\) \((^2)\). We shall show that in this case all Fredholm theorems also hold in the spaces \(\mathfrak A_r\) \((\alpha<r\leqslant \beta)\).

Theorem 1. The number of linearly independent solutions of the equations \(Af=0\) and \(A^*F=0\), respectively, in \(\mathfrak A_r\) and in \(\mathfrak A_r^*\) is finite and the same for both equations.

Proof. Let the equation \(Af=0\) have \(k\) \((k\leqslant\infty)\) linearly independent (l.i.) solutions in \(\mathfrak A_r\). Then in \(\mathfrak B_{r'}\), \((r'<r)\), the equation \(Af=0\) has \(k_1(r')\geqslant k\), \(k_1(r')<\infty\), l.i. solutions, and hence \(k<\infty\). The equation \(A^*F=0\) in \(\mathfrak B_{r'}^*\) also has \(k_1(r')\) l.i. solutions; therefore in \(\mathfrak A_r^*\) it has \(k_2\geqslant k_1(r')\) l.i. solutions. Since \(\mathfrak A_r^*\subset \mathfrak B_r^*\), the equation \(A^*F=0\) in \(\mathfrak B_r^*\) has \(k_3\geqslant k_2\) l.i. solutions, \(k_3<\infty\), and therefore \(k_2<\infty\). Then the equation \(Af=0\) in \(\mathfrak B_r\) likewise has \(k_3\) l.i. solutions, with \(k_3\leqslant k_1\) by virtue of the inclusion \(\mathfrak B_r\subset \mathfrak A_r\). Hence \(k_2=k(=k_1(r')\) for \(r'<r\), which indicates the independence of \(k\) and \(k_2\) from \(r\)).

Theorem 2. In order that the equation \(Af=g\) \([A^*F=G]\) have a solution in \(\mathfrak A_r\) \((\mathfrak A_r^*)\), it is necessary and sufficient that, for every \(\Phi\in\mathfrak A_r^*\) \((\varphi\in\mathfrak A_r)\) satisfying the equation \(A^*\Phi=0\) \((A\varphi=0)\), the equality \(\Phi(g)=0\) \((G(\varphi)=0)\) hold.

Necessity. Let \(Af=g\) be solvable in \(\mathfrak A_r\) and, consequently, in every \(\mathfrak B_{r'}\), \((r'<r)\). If \(\Phi\) is an arbitrary solution of the equation \(A^*\Phi=0\), \(\Phi\in\mathfrak A_r^*\), then \(\Phi\in\mathfrak B_{r-\varepsilon}^*\) \((\varepsilon\leqslant\varepsilon_0)\). By virtue of the normal solvability of the operator \(A\) in \(\mathfrak B_{r-\varepsilon}\), we obtain \(\Phi(g)=0\).

Now let the equation \(A^*F=G\) be solvable in \(\mathfrak A_r^*\), and let \(\varphi\in\mathfrak A_r\) be an arbitrary solution of the equation \(A\varphi=0\). Then for some \(\varepsilon>0\), \(F\) and \(G\in\mathfrak B_{r-\varepsilon}^*\), \(\varphi\in\mathfrak B_{r-\varepsilon}\), and, by virtue of the normal solvability of the operator \(A\) in \(\mathfrak B_{r-\varepsilon}\), \(G(\varphi)=0\).

Sufficiency. Let \(\{\Phi_k\}_1^m\) \((m<\infty\) and independent of \(r\), by Theorem 1) be a maximal l.i. system of solutions of the equation \(A^*\Phi=0\) in \(\mathfrak A_r^*\) and \(\Phi_k(g)=0\) \((k=1,\ldots,m)\), \(g\in\mathfrak A_r\). For all \(\delta\), \(\delta\leqslant\delta_0\), \(\Phi_k\in\mathfrak B_{r-\delta}^*\), \(g\in\mathfrak B_{r-\delta}\). The system \(\{\Phi_k\}_1^m\) is a complete l.i. system of solutions of the equation \(A^*\Phi=0\) in \(\mathfrak B_{r-\delta}^*\). Thus, the equation \(Af=g\) is solvable in every \(\mathfrak B_{r-\delta}\) \((\delta\leqslant\delta_0)\). Then, taking into account the independence of the dimension \(m(<\infty)\) of the null subspace of the operator \(A\) from \(r\), we obtain its solvability in \(\mathfrak A_r\).

Let the functional \(G\in\mathfrak A_r^*\) satisfy the conditions \(G(\varphi_k)=0\) \((k=1,2,\ldots,m)\), where \(\{\varphi_k\}_1^m\) is a complete l.i. system of solutions of the equation \(A\varphi=0\) in \(\mathfrak A_r\). Then for \(\delta\leqslant\delta_0\), \(G\in\mathfrak B_{r-\delta}^*\), \(\varphi_k\in\mathfrak B_{r-\delta}\). The equation \(A\varphi=0\) has exactly \(m\) l.i. solutions in every \(\mathfrak B_{r-\delta}\); consequently, the equation \(A^*F=G\) is solvable in \(\mathfrak B_{r-\delta}^*\) and, thereby, in \(\mathfrak A_r^*\).

Theorem 3. In order that the equation \(Af=g\) \((A^*F=G)\) have, in \(\mathfrak A_r\) \((\mathfrak A_r^*)\), a solution for any \(g\) (for any \(G\)), it is necessary and sufficient that the homogeneous equation \(A\varphi=0\) \((A^*\Phi=0)\) have only the trivial solution. If these conditions are satisfied, the inverse operator \(A^{-1}\) is continuous in \(\mathfrak A_r\) (the operator \((A^*)^{-1}\) is weakly and strongly continuous in \(\mathfrak A_r^*\)).

The first assertion follows immediately from Theorems 1 and 2, if one takes into account that the equality \(G(\varphi)=0\) \((\varphi\in\mathfrak A_r)\) for every \(G\in\mathfrak A_r^*\) implies \(\varphi=0\).

We shall prove the continuity of the operators \(A^{-1}\) and \((A^*)^{-1}\) under the condition of their existence.

The range \(R(A)\) of the operator \(A\) in any \(\mathfrak{B}_{r'}\) \((r'<r)\) contains \(\mathfrak{A}_r\) and therefore is everywhere dense in \(\mathfrak{B}_{r'}\); by virtue of the normal solvability of the operator \(A\) in \(\mathfrak{B}_{r'}\), \(R(A)\) is closed \((^1)\), and consequently \(R(A)=\mathfrak{B}_{r'}\). Thus \(A^{-1}\) is continuous in each \(\mathfrak{B}_{r'}\) \((r'<r)\); since, moreover, it maps \(\mathfrak{A}_r\) into \(\mathfrak{A}_r\), it is continuous also in the topology of \(\mathfrak{A}_r\).

Suppose that \((A^*)^{-1}\) is not a weakly continuous operator in \(\mathfrak{A}_r^*\). Then there exists a sequence \(\{F_n\}\subset \mathfrak{A}_r^*\) such that \(A^*F_n \xrightarrow{\mathrm{sl}}0\), but \(F_n\) does not tend weakly to zero, i.e. for some \(f\in \mathfrak{A}_r\), \(F_n(f)\) does not tend to zero. Since \(A\) is invertible in any \(\mathfrak{B}_{r'}\), \(r'<r\), we have \(A^{-1}f\in \mathfrak{A}_r\), and, by assumption, \(A^*F(A^{-1}f)\to0\), i.e. \(F_n(f)\to0\), which contradicts our supposition. The weak continuity of \((A^*)^{-1}\) is proved. By analogous arguments one proves the strong continuity of \((A^*)^{-1}\).

Consider the operator \(A_\lambda=E-\lambda B\), where \(B\) is a linear operator on any space \(\mathfrak{B}_r\), whose action does not depend on \(r\), \(\alpha<r\leq\beta\). Let \(\mathscr{G}\) be a domain of the complex \((\lambda)\)-plane in which \(A_\lambda\) is representable as the sum of a finite-dimensional and an invertible operator in any \(\mathfrak{B}_r\), and, in addition, suppose that there exists a point \(\lambda_0\in\mathscr{G}\) at which the operator \(A_\lambda\) is invertible in \(\mathfrak{B}_r\). Then the following theorem holds:

Theorem 4. The number of values \(\lambda\) belonging to \(\mathscr{G}\) at which the operator \(A_\lambda\) is not invertible in \(\mathfrak{A}_r\) is finite in every closed part of \(\mathscr{G}\); these values are the same for all \(r\), \(\alpha<r\leq\beta\).

Proof. Let \(\lambda\) be a point of noninvertibility for \(A_\lambda\); according to Theorem 3, there exists \(f\in\mathfrak{A}_r\) such that \(A_\lambda f=0\); since \(f\in\mathfrak{B}_{r'}\), \(r'<r\), \(\lambda\) is a singular point of the operator \(A_\lambda\) in \(\mathfrak{B}_{r'}\). The set of all such values \(\lambda\) for the operator \(A_\lambda\) in \(\mathfrak{B}_{r'}\) is discrete in every connected component of the set \(M_A\) \((^2)\), and hence, all the more, in \(\mathscr{G}\subset M_A\).

If \(\lambda\) is a point of noninvertibility of \(A_\lambda\) in \(\mathfrak{A}_r\), then the same is true in the spaces \(\mathfrak{A}_{r'}\), \(\alpha<r'\leq\beta\).

Example. Let
\[ E_r(z)=\sum_{0}^{\infty}\alpha_n^{-1}(r)z^n \qquad (0<\alpha_n<\infty;\ \alpha<r\leq\beta) \]
be an analytic function in the disk \(C_{r^p}\) \((|z|<r^p)\); let \(\mathfrak{B}_r\) be the Banach space \(Z_{E_n}^{p}\), i.e. the collection of all functions
\[ f(z)=\sum_n a_n z^n, \]
for which
\[ \|f\|_r=\left(\sum_n |a_n|^p\alpha_n(r)\right)^{1/p}<\infty, \]
where \(p\geq1\). Suppose that \(\alpha_n(r)\) increases monotonically with increasing \(r\). It is obvious that the spaces \(\mathfrak{B}_r\) satisfy conditions 1), 2) of item 1. To each functional \(F\in\mathfrak{B}_r^*\) there corresponds the function
\[ \sum_{n=0}^{\infty}\frac{b_n}{\zeta^{n+1}}, \]
where
\[ \sum_{0}^{\infty}|b_n|^q\alpha_n^{1-q}(r)<\infty \quad (p>1) \]
or
\[ \sup_n |b_n|\alpha_n^{-1}(r)<\infty \quad (p=1) \]
(here
\[ \frac1p+\frac1q=1 \]).
In this case
\[ F\left(\sum_n a_n z^n\right)=\sum_n a_n b_n. \]
The space \(\mathfrak{A}_r\) in our case is the space of all functions analytic in the disk \(C_r\), and the topology coincides with the one usually adopted for these spaces. The space \(\mathfrak{A}_r^*\) is isomorphic to the space of all functions analytic in the domain \(|z|\geq r\) \((^4)\).

Consider an operator \(B\), acting in any \(\mathfrak{B}_r\) and given by the matrix \([\gamma_{mn}]_{m,n=0}^{\infty}\), on which we impose the following requirement:
\[ \gamma_{mn}=\gamma'_{mn}(r)+\gamma''_{mn}(r)\quad(\alpha<r\leq\beta), \]
where:

1) for any finite sequence of complex numbers \(M_1,M_2,\ldots,M_k\) the inequality holds
\[ \sum_{n=0}^{\infty}\left|\sum_{m=0}^{k}\gamma''_{mn}M_m\right|^p\alpha_n(r) \leq [\theta(r)]^p\sum_{n=0}^{k}|M_n|^p\alpha_n(r), \qquad \theta(r)<1,\quad \alpha<r\leq\beta; \]

2) there exist two matrices \([c_n^{(i)}(r)]_{\substack{n=0,1,\ldots\\ i=1,2,\ldots,s<\infty}}\) and \([d_n^{(i)}(r)]_{\substack{n=0,1,\ldots\\ i=1,2,\ldots,s<\infty}}\),

\[ \sum_{n=0}^{\infty} |c_n^{(i)}(r)|^p \alpha_n(r) < \infty \qquad (i=1,2,\ldots,s), \]

\[ \sum_{n=0}^{\infty} |d_n^{(i)}|^q \alpha_n^{1-q}(r) < \infty \qquad (i=1,2,\ldots,s), \quad \text{if } p>1 \]

(in the case \(p=1\), the latter condition must be replaced by the following: \(\sup |d_n^{(i)}(r)|\alpha_n^{-1}(r)<\infty\)), \(\alpha<r\leqslant\beta\), so that the relation

\[ \gamma'_{mn}=\sum_{i=1}^{s} d_m^{(i)}(r)c_n^{(i)}(r). \]

holds.

Define the action of the operator \(B\) in \(\mathfrak{B}_r\) as follows:

\[ B\left(\sum_n a_n z^n\right)=\sum_n\left(\sum_m \gamma_{mn}a_m\right)z^n. \]

The operator \(B\) is an integral operator with kernel

\[ \sum_{m,n=0}^{\infty}\frac{\gamma_{nm} z^m}{\zeta^{n+1}}. \]

To the operator \(B^*\) there corresponds the matrix \([\gamma_{nm}]\), transposed to \([\gamma_{mn}]\). By virtue of our assumptions, the operator \(B\) is decomposed into the sum of operators \(B'+B''\), to which correspond the matrices \([\gamma'_{mn}]\) and \([\gamma''_{mn}]\), \(\|B''\|<1\), and \(B'\) is a finite-dimensional operator. Therefore the operator

\[ A=E-B=(E-B'')-B' \]

is the sum of an invertible and a finite-dimensional operator. By Theorems 1–3, for the operator \(A\) acting in the analytic spaces \(\mathfrak{A}_r\), Fredholm theory holds.

If one discards the condition \(\theta(r)<1\) and considers, instead of \(A\), the operator

\[ A_\lambda=E-\lambda B=(E-\lambda B'')-\lambda B', \]

taking into account that for \(|\lambda|<\left[\sup_{\alpha<r\leqslant\beta}\theta(r)\right]^{-1}\) the conditions of Theorem 4 are satisfied, then the assertion of this theorem is applicable to the indicated domain.

The conditions set out here for \(A_\lambda\) are satisfied, in particular, if we assume \(p=1\), \(\alpha_n(r)=r^n\), and

\[ \overline{\lim}_{n\to\infty}\sum_{m=0}^{\infty}\gamma_{nm}r^{m-n}\leqslant \theta(r). \]

In this case, Fredholm theory for the operator \(A_\lambda\) in the analytic spaces \(\mathfrak{A}_r\) was constructed by M. A. Evgrafov by another method \({}^{3}\).

It is not hard to find other examples fitting the proposed scheme.

Chernivtsi State University

Received
20 VI 1957

References

\({}^{1}\) I. M. Gel'fand, Uspekhi Mat. Nauk, 11, no. 6 (1956).
\({}^{2}\) S. M. Nikol'skii, Izv. Akad. Nauk SSSR, Ser. Mat., 7, no. 3 (1943).
\({}^{3}\) M. A. Evgrafov, Trudy Moskov. Mat. Obshch., 5 (1956).
\({}^{4}\) A. I. Markushevich, Matem. Sbornik, 17 (59), no. 2 (1945).