MATHEMATICS

N. G. ASKEROV, S. G. KREIN, G. I. LAPTEV

ON A CLASS OF NON-SELF-ADJOINT BOUNDARY-VALUE PROBLEMS

(Presented by Academician I. N. Vekua on December 23, 1963)

A number of problems of mathematical physics lead to homogeneous boundary-value problems containing one and the same parameter \(\lambda\) in the differential equations and in the boundary conditions. Despite the fact that, for each fixed \(\lambda\), the differential operator and the boundary conditions are self-adjoint, the problem as a whole is often non-self-adjoint; the spectrum may be non-real. In the present article a general consideration of one class of such problems is carried out.

\(1^\circ\). Let, in a separable Hilbert space \(H\) with scalar product \((\, ,\,)\), a linear operator \(A\) with everywhere dense domain of definition \(D(A)\) be given. Suppose, moreover, that on \(D(A)\) two linear operators \(T\) and \(\Gamma\) are defined, mapping \(D(A)\) into some other separable Hilbert space \(H_1\) with scalar product \((\, ,\,)_1\). The operators \(A, T, \Gamma\) have the following properties:

1) The set of elements of \(D(A)\) satisfying the conditions \(\Gamma v = 0\) and \(Tv = 0\) is dense in \(H\).

2) The restriction \(A_0\) of the operator \(A\) to the set of all elements of \(D(A)\) for which \(Tu = 0\) is a self-adjoint, positive-definite operator having a completely continuous inverse.

We introduce into consideration the operator \(A_0^{1/2}\). Its domain of definition \(D(A_0^{1/2})\) will be regarded as the Hilbert space \(H_{1/2}\) with scalar product
\[ [uv] = (A_0^{1/2}u, A_0^{1/2}v)\quad (u, v \in D(A_0^{1/2})). \]
As is known, the space \(H_{1/2}\) can be obtained from \(D(A_0)\) by completing it with respect to the norm
\[ \sqrt{[u,u]} = \sqrt{(A_0u,u)}. \]

3) The operator \(\Gamma\) maps \(D(A_0)\) onto a set dense in \(H_1\), and at the same time is completely continuous as an operator from the space \(H_{1/2}\) into the space \(H_1\). We shall regard the operator \(\Gamma\) as extended by continuity to the whole space \(H_{1/2}\); moreover
\[ \|\Gamma u\|_1 \leq C\|A_0^{1/2}u\|\quad (u \in D(A_0^{1/2}) = H_{1/2}). \]

4) The “Green formula” holds:
\[ (Au,v) = A(u,v) - (Tu,\Gamma v)_1, \tag{1} \]
where \(A(u,v)\) is a bilinear functional such that \(A(u,u)\geq 0\). It is not difficult to see that the functional \(A(u,v)\) can be extended by continuity to all elements \(u, v \in D(A_0^{1/2})\), and moreover
\[ A(u,v) = (A_0^{1/2}u, A_0^{1/2}v). \]

Denote by \(N\) the set of all elements \(w\) in \(D(A_0^{1/2})\) for which the inequality
\[ |(A_0^{1/2}w,A_0^{1/2}z)| \leq C_w \|\Gamma z\|_1 \]
is valid for any \(z \in D(A_0^{1/2})\).

Lemma 1. For each \(\varphi \in H_1\) there exists a unique element \(w \in N\) satisfying the identity
\[ (A_0^{1/2}w,A_0^{1/2}z) = (\varphi,\Gamma z)_1 \tag{2} \]
for any \(z \in D(A_0^{1/2})\). Conversely, to each \(w \in N\) there corresponds a unique element \(\varphi \in H_1\) satisfying identity (2).

The lemma is proved by ordinary arguments with the aid of Riesz’s theorem on the general form of a linear functional in Hilbert space.

If \(w \in N \cap D(A)\), then the conditions \(Aw=0\) and \(Tw=\varphi\) are satisfied. Consequently, the elements \(w \in N\) may be regarded as generalized solutions of these equations. In what follows, for any \(w \in N\) we shall denote \(\varphi=Tw\) and \(w=T^{-1}\varphi\).

Lemma 2. The operator \(R=A_0^{1/2}T^{-1}\Gamma A_0^{-1/2}\) is a self-adjoint nonnegative completely continuous operator in \(H\).

\(2^\circ\). Statement of the problem: it is required to find a solution of the equation

\[ Ay=\lambda y, \tag{3} \]

satisfying the condition

\[ \lambda Ty=\sigma \Gamma y, \tag{4} \]

where the given \(\sigma>0\).

By a generalized solution of problem (3)—(4) we shall mean an element \(y\in D(A_0^{1/2})\), representable in the form \(y=u+w\), where \(u\in D(A_0)\), \(w\in N\) satisfy the equations \(A_0u=\lambda y\) and \(\lambda Tw=\sigma \Gamma y\).

Theorem 1. The problem of finding generalized solutions of (3)—(4) is equivalent to the problem of solving, in the space \(H\), the equation

\[ x=\lambda A_0^{-1}x+\frac{\sigma}{\lambda}Rx. \tag{5} \]

If \(x\) is a solution of (5), then \(y=A_0^{1/2}x\) is a generalized solution of (3)—(4), and conversely.

\(3^\circ\). Let us investigate in a general form, in the Hilbert space \(H\), the equation

\[ f=\lambda Pf+\frac{1}{\lambda}Qf, \tag{6} \]

where \(P\) is a positive and \(Q\) a nonnegative completely continuous operator in \(H\).

From the results of I. Ts. Gokhberg \((^1)\) (see also \((^2)\)) it follows that the numbers \(\lambda\) corresponding to nonzero solutions of equation (6) (eigenvalues) can have only two points of accumulation: \(\lambda=0\) and \(\lambda=\infty\). In this connection it is natural to try to transform equation (6) to such a form that the eigenvalues of the new equation have only one point of accumulation, \(\lambda=0\).

It is verified directly that equation (6) is equivalent to the system of equations

\[ P^{1/2}BP^{1/2}g+P^{1/2}BQ^{1/2}h=\frac{1}{1+\lambda}g, \]

\[ -Q^{1/2}BP^{1/2}g+(I-Q^{1/2}BQ^{1/2})h=\frac{1}{1+\lambda}g, \tag{7} \]

where \(g=P^{1/2}f,\ h=\dfrac{1}{\lambda}Q^{1/2}f\), and \(B=(I+P+Q)^{-1}\).

System (7) may be regarded as the equation

\[ \mathfrak{A}X=\mu X \qquad \left( X=\binom{g}{h},\quad \mu=\frac{1}{1+\lambda} \right) \]

in the space \(H\times H\) with the operator

\[ \mathfrak{A}= \begin{pmatrix} P^{1/2}BP^{1/2} & P^{1/2}BQ^{1/2} \\ -Q^{1/2}BP^{1/2} & I-Q^{1/2}BQ^{1/2} \end{pmatrix}. \]

If \(\lambda=0\) and \(\lambda=\infty\) were points of accumulation of the eigenvalues of equation (6), then the points of accumulation of the spectrum of the operator \(\mathfrak{A}\) will be \(\mu=1\) and \(\mu=0\). Construct the operator \(\mathfrak{B}=\mathfrak{A}(I-\mathfrak{A})\). By direct calculation we obtain

\[ \mathfrak{B}= \begin{pmatrix} P^{1/2}B(I+2Q)BP^{1/2} & -P^{1/2}B(P-Q)BQ^{1/2} \\ Q^{1/2}B(P-Q)BP^{1/2} & Q^{1/2}B(I+2P)BQ^{1/2} \end{pmatrix}. \tag{8} \]

The operator \(\mathfrak{B}\) turns out to be completely continuous. Between the eigenvalues \(\lambda\) of equation (6) and the eigenvalues \(\nu\) of the operator \(\mathfrak{B}\) there is the relation \(\lambda/(1+\lambda)^2=\nu\).

Theorem 2. All eigenvalues of equation (6) have nonnegative real part. If the condition

\[ 4\|P\|\,\|Q\|\leq 1, \tag{9} \]

is satisfied, then all eigenvalues are real. If condition (9) is not satisfied, then the nonreal eigenvalues of equation (6) can be located only in the annulus \(1/2\|P\|\leq |\lambda|\leq 2\|Q\|\) and, consequently, they form a finite set.

If we consider the sum of the root subspaces of the operator \(\mathfrak{B}\) corresponding to a given eigenvalue \(\nu\), then we obtain an invariant finite-dimensional subspace of the operator \(\mathfrak{A}\). In it one can choose a basis in which the operator \(\mathfrak{A}\) has normal Jordan form. In the subspace corresponding to one Jordan cell, it will be convenient for us to choose a basis consisting of elements \(X_0, X_1,\ldots,X_m\) such that

\[ \mathfrak{A}X_k=\sum_{l=0}^{k}\frac{(-1)^{k-l}}{(1+\lambda)^{k+1-l}}X_l \quad (k=0,1,\ldots,m). \]

If \(X_k=\binom{g_k}{h_k}\), then the elements \(f_k=P^{-1/2}g_k\) satisfy the equations

\[ f_k=\lambda P f_k+\frac{1}{\lambda}Q f_k+P f_{k-1} +\sum_{l=0}^{k-1}\frac{(-1)^{k-l}}{\lambda^{k+1-l}}Q f_l \quad (k=0,1,\ldots,m). \]

These equations can be obtained formally by differentiating equation (6) with respect to \(\lambda\), if one sets \(f_k=\frac{1}{k!}\frac{\partial^k f_0}{\partial\lambda^k}\). The subspace spanned by each chain of elements \(\{f_0,f_1,\ldots,f_m\}\) of maximal length is called the root subspace corresponding to the eigenvalue \(\lambda\).

The real part of the operator \(\mathfrak{B}\) is determined by the diagonal entries of the matrix (8) and is a nonnegative operator; therefore, for the study of the question of completeness of the system of root subspaces of equation (6), one can apply the theory of dissipative operators. If the operators \(P\) and \(Q\) have finite trace, then all operators of the matrix (8) also have finite trace. With the aid of results of V. B. Lidskii \((^3)\) and M. G. Krein \((^4)\) one can obtain:

Theorem 3. If the operators \(P\) and \(Q\) have finite trace, then the system of root subspaces of equation (6) is doubly complete in the space \(H\) in the following sense (see \((^5)\)): for every pair of elements \(x,y\in H\) there exists a sequence of linear combinations \(\sum c_{sk}^{(N)} f_k^{(s)}\) \((N=1,2,\ldots)\) of eigen and associated elements of equation (6) \((s\) is the number of the root subspace, \(k\) is the number of the associated element\()\) such that, as \(N\to\infty\),

\[ \sum_k c_{sk}^{(N)} f_k^{(s)} \to x \quad \text{in the norm } (Px,x), \]

\[ \sum c_{sk}^{(N)}\sum_{j=0}^{k}\frac{(-1)^{k-j}}{\lambda^{k+1-j}}f_j^{(s)} \to y \quad \text{in the seminorm } (Qx,x). \]

* Condition (9) can be replaced by the less restrictive condition \(4(Pf,f)(Qf,f)\leq (f,f)^2\) for all \(f\in H\).

** As M. G. Krein informed us, results obtained by him jointly with G. Langer make it possible in some cases to extend Theorem 3.

4°. The elements \(y_1, y_2, \ldots, y_m\) will be called associated with the generalized solution \(y_0\) of problem (3)—(4) if \(y_k=u_k+w_k\), where \(u_k\in D(A_0)\), \(w_k\in N\), and they satisfy the equations

\[ A_0u_k=\lambda y_k+y_{k-1},\qquad \lambda T w_k+T w_{k-1}=\sigma\Gamma y_k \quad (k=1,2,\ldots,m). \]

The subspace spanned by \(y_0,y_1,\ldots,y_m\) is called the root subspace of problem (3)—(4).

Denote by \(\{e_k\}\) the complete orthonormal system of eigenvectors of the operator \(A_0\), and by \(\{\mu_k\}\) the system of corresponding eigenvalues. Then, in the basis \(\{e_k\}\), the operator \(R\) is given by the matrix

\[ R=\left(\frac{(\Gamma e_i,\Gamma e_j)}{\sqrt{\mu_i}\sqrt{\mu_j}}\right). \]

Its trace is equal to \(\displaystyle \sum \frac{\|\Gamma e_n\|_1^2}{\mu_n}\). From Theorems 1–3 there follows

Theorem 4. Starting from some index, all eigenvalues of problem (3)—(4) are real. If the conditions

\[ \sum \frac{1}{\mu_n}<\infty,\qquad \sum \frac{\|\Gamma e_n\|_1^2}{\mu_n}<\infty, \tag{10} \]

are satisfied, then the system \(\{y_k^{(s)}\}\) of generalized and associated solutions of problem (3)—(4) is doubly complete in the following sense: for each pair of elements \(x\in H\) and \(\varphi\in H_1\) there are linear combinations of solutions \(y_k^{(s)}\) such that, as \(N\to\infty\),

\[ \sum c_{sk}^{(N)}y_k^{(s)}\longrightarrow x \quad \text{in } H;\qquad \sum c_{sk}^{(N)}\sum_{j=0}^{k} \frac{(-1)^{k-j}\Gamma y_j^{(s)}}{\lambda^{k+1-j}} \longrightarrow \varphi \quad \text{in } H_1. \tag{11} \]

Remark 1. Theorem 4 remains valid if the multiplier \(\sigma\) is replaced by a bounded nonnegative operator in \(H_1\). In the last part of the theorem one can then assert that the linear combinations (11) converge in the seminorm \((\sigma\varphi,\varphi)_1\).

Remark 2. If \(\sigma\) is negative, then problem (3)—(4) is also reduced to the investigation of equation (5), or to the equation

\[ f=\lambda Pf-\frac{1}{\lambda}Qf, \]

where \(P\) and \(Q\) are the same as in (6). The last equation, by the substitution \(g=P^{1/2}f\) and \(h=\dfrac{1}{\lambda}Q^{1/2}f\), is reduced to the equation \(\mathfrak A X=\lambda X\), where \(\mathfrak A\) is a self-adjoint operator in the space \(H\times H\). Thus, for \(\sigma<0\) the problem is sharply simplified. For any pair of elements \(x\in H\) and \(\varphi\in H_1\), one can write expansions in series in the generalized solutions of problem (3)—(4).

Remark 3. The conditions under which the first part of Theorem 4 has been proved are verified in various problems for partial differential equations with the help of embedding theorems and energy inequalities. In this case the operator \(A\) is defined by a differential expression, while the operators \(T\) and \(\Gamma\) are boundary operators. Conditions (10) are considerably more restrictive. To broaden the range of possible applications, the conditions on the operators \(P\) and \(Q\) in Theorem 3 should be weakened.

The authors express their sincere gratitude to M. G. Krein for valuable discussions.

Received
16 XII 1963

CITED LITERATURE

1. I. Ts. Gokhberg, DAN, 78, 629 (1951).
2. A. E. Taylor, Ann. Math., ser. 2, 39, No. 3 (1938).
3. V. B. Lidskii, Tr. Mosk. matem. obshch., 8, 83 (1959).
4. M. G. Krein, DAN, 130, No. 2 (1960).
5. M. V. Keldysh, DAN, 77, 11 (1951).