UDC 517.43

MATHEMATICS

I. I. GOLICHEV

ON THE DISCRETENESS OF THE SPECTRUM OF NON-SELF-ADJOINT OPERATORS

(Presented by Academician L. S. Pontryagin, 25 IX 1969)

In the present note we give theorems that make it possible to reduce the question of the discreteness of the spectrum of certain non-self-adjoint operators to the question of the discreteness of the spectrum of self-adjoint operators.

1. Let \(D_0\) be a linear manifold dense in a Hilbert space \(H\); let \(L_1\) and \(L_2\) be operators symmetric on \(D_0\). Suppose that the operator \(L_0' = L_1 + iL_2\) has a closure \(L_0\). Then the following theorems are valid.

Theorem 1. If at least one of the operators \(L_1, L_2, L_1+L_2, L_1-L_2\) is semibounded and its semibounded self-adjoint extension has a discrete spectrum, then the kernel of the spectrum of the operator \(L_0\) is discrete.

Proof. Let, for example, \((L_1y,y)\ge (y,y)\) for \(y\in D_0\), and let \(\widetilde L_1\) be a semibounded self-adjoint extension of the operator \(L_1\). The set
\[ \widetilde E_1=[y:(\widetilde L_1y,y)\le 1,\ y\in D(\widetilde L_1)] \]
is compact in \(H\), since \(\widetilde L_1\) has a discrete spectrum. All the more compact is the set
\[ E_1=[y:(L_1y,y)\le 1,\ y\in D_0]. \]
The set
\[ E_0=[y: |(L_0y,y)|\le 1,\ y\in D_0]\subset E_1, \]
and therefore it is also compact. Suppose that \(L_0\) has a point \(\lambda_0\) of the continuous spectrum; then, for any \(\varepsilon>0\), there exists a noncompact sequence \(\{\varphi_i\}\), \(\varphi_i\in D_0\), such that \(1/2\le \|\varphi_i\|\le 2\) and
\[ \|L_0\varphi_i-\lambda_0\varphi_i\|\le \varepsilon . \tag{1} \]
Put \(\varepsilon=1\). From the last inequality we have
\[ \|L_0\varphi_i\|\le |\lambda_0|\|\varphi_i\|+1\le 2|\lambda_0|+1 . \tag{2} \]
Obviously, the sequence \(\{\varphi_i'\}\), where \(\varphi_i'=\varphi_i/2(2|\lambda_0|+1)\), is noncompact together with the sequence \(\{\varphi_i\}\), but, by virtue of inequality (2), it belongs to \(E_0\). We have obtained a contradiction to the compactness of the set \(E_0\). The remaining cases are easily reduced to the one proved.

The theorem just proved generalizes Theorem 4 of Ch. II of the paper \((^6)\).

Theorem 2. If
\[ (L_1y,y)\ge (y,y)\quad \text{for } y\in D_0;\qquad (L_2y,y)\ge (y,y)\quad \text{for } y\in D_0, \tag{3} \]
and the operator \(L_0\) has no residual spectrum, then the operator \(L_0\) has a discrete spectrum if and only if the set
\[ E_0=[y: |(L_0y,y)|\le 1,\ y\in D_0] \]
is compact in \(H\).

Proof of sufficiency. Suppose the set \(E_0\) is compact; then the set
\[ E=[y:(L_0y,y)\le 1,\ y\in D(L_0)]\subset \widetilde E_0 \]
is also compact, where \(\widetilde E_0\) is the closure of the set \(E_0\) in \(H\). The set
\[ E'=[y:\|L_0y\|\le 1,\ y\in D(L_0)]\subset E \]
is consequently also compact in \(H\); therefore the operator \(L_0^{-1}\) is completely continuous, and hence the spectrum of the operator \(L_0\) is discrete (the existence of the operator \(L_0^{-1}\) follows from the conditions (3) and from the absence of residual spectrum for the operator \(L_0\)).

Proof of necessity. Suppose that the spectrum of the operator \(L_0\) is discrete; then the inverse operator \(L_0^{-1}=A\) is completely continuous. From

from condition (3) we obtain that \(A=A_1-iA_2\), where \(A_1\) and \(A_2\) are positive self-adjoint completely continuous operators.

Suppose that the set \(E_0\) is not compact; then there is a noncompact sequence \(\{y_n\}\subset E_0\). From (3) it follows that \(\|y_n\|\leq 1\). Denote \(L_0y_n=\varphi_n\); then

\[ (L_0y_n,y_n)=(\varphi_n,A\varphi_n)=(A_1\varphi_n,\varphi_n)-i(A_2\varphi_n,\varphi_n). \tag{4} \]

Since \(A_1\) and \(A_2\) are positive self-adjoint completely continuous operators, there exist orthonormal systems \(\{e_i\}\) and \(\{g_i\}\) such that

\[ A_1\varphi_n=\sum_1^\infty \lambda_i(A_1)(\varphi_n,e_i)e_i,\qquad A_2\varphi_n=\sum_1^\infty \lambda_i(A_2)(\varphi_n,g_i)g_i, \tag{5} \]

where \(\lambda_i(A_1)>0\), \(\lambda_i(A_2)>0\) and tend to zero as \(i\to\infty\). We shall prove that for every \(\nu>0\) there is an \(n\) such that at least one of the inequalities

\[ \left\|\sum_\nu^\infty \lambda_i(A_1)(\varphi_n,e_i)e_i\right\|\geq \varepsilon_0,\qquad \left\|\sum_\nu^\infty \lambda_i(A_2)(\varphi_n,g_i)g_i\right\|\geq \varepsilon_0, \tag{6} \]

holds, where \(\varepsilon_0>0\) is independent of both \(n\) and \(\nu\).

Suppose that the assertion is false; then for every \(\varepsilon>0\) there is a \(\nu\) such that

\[ \left\|\sum_\nu^\infty \lambda_i(A_1)(\varphi_n,e_i)e_i\right\|<\varepsilon,\qquad \left\|\sum_\nu^\infty \lambda_i(A_2)(\varphi_n,g_i)g_i\right\|<\varepsilon \tag{7} \]

for all \(n\). Then

\[ \begin{aligned} A\varphi_n &=A_1\varphi_n-iA_2\varphi_n \\ &=\left[\sum_1^\nu \lambda_i(A_1)(\varphi_n,e_i)e_i -i\sum_1^\nu \lambda_i(A_2)(\varphi_n,g_i)g_i\right]+ \\ &\quad+\left[\sum_\nu^\infty \lambda_i(A_1)(\varphi_n,e_i)e_i -i\sum_\nu^\infty \lambda_i(A_2)(\varphi_n,g_i)g_i\right] =g_n^\nu+\psi_n^\nu, \end{aligned} \tag{8} \]

where

\[ g_n^\nu=\sum_1^\nu \lambda_i(A_1)(\varphi_n,e_i)e_i -i\sum_1^\nu \lambda_i(A_2)(\varphi_n,g_i)g_i, \]

\[ \psi_n^\nu=\sum_\nu^\infty \lambda_i(A_1)(\varphi_n,e_i)e_i -i\sum_1^\nu \lambda_i(A_2)(\varphi_n,g_i)g_i. \]

By inequalities (7) and equality (8), \(\|\psi_n^\nu\|\leq 2\varepsilon\), and \(\|g_n^\nu\|\leq 1+2\varepsilon\). The set \(\{g_n^\nu\}\) belongs to a finite-dimensional space and is bounded; therefore there exists a finite \(\varepsilon\)-net \(\{\eta_i\}\) for it, and since \(\|\psi_n^\nu\|\leq 2\varepsilon\), the set \(\{\eta_i\}\) forms a \(3\varepsilon\)-net of the set \(\{y_n\}=\{A\varphi_n\}\). But this contradicts the noncompactness of the sequence \(\{y_n\}\). Suppose, for example, that the first of inequalities (6) is satisfied; then

\[ 1\geq (A_1\varphi_n,\varphi_n) =\sum_1^\infty \lambda_i(A_1)|(\varphi_n,e_i)|^2 \geq \sum_\nu^\infty \lambda_i(A_1)|(\varphi_n,e_i)|^2 \geq \]

\[ \geq \frac{1}{\lambda_\nu(A_1)} \sum_\nu^\infty \lambda_i^2(A_1)|(\varphi_n,e_i)|^2 \geq \frac{1}{\lambda_\nu(A_1)}\varepsilon_0. \]

The last inequality contradicts the fact that \(\lambda_\nu(A_1)\to 0\) as \(\nu\to\infty\). The theorem is proved.

Remark. Under the conditions of Theorem 2 the inequalities

\[ \frac{1}{\sqrt{2}}\bigl((L_1+L_2)y,y\bigr)\le |(L_0y,y)|\le \bigl((L_1+L_2)y,y\bigr), \tag{9} \]

hold when \(y\in D_0\); therefore the set \(E_0\) is compact if and only if the set \(E_1=[y:\ ((L_1+L_2)y,y)\le 1,\ y\in D_0]\) is compact. Hence it is easy to obtain that the following assertion is valid: if the conditions of Theorem 2 are satisfied and the closure of the operator \(L_1+L_2\), the operator \(\overline{L_1+L_2}\), has finite defect indices, then the spectrum of the operator \(L_0\) is discrete if and only if the operator \(\overline{L_1+L_2}\) has a self-adjoint extension with discrete spectrum.

1. Let \(I\) be some conjugation operator in the Hilbert space \(H\). A linear operator \(A\) with domain of definition \(D(A)\) dense in \(H\) will be called \(I\)-symmetric if, for any \(\varphi\) and \(\psi\) in \(D(A)\), the equality \((A\varphi,I\psi)=(\varphi,IA\psi)\) holds, equivalent to the relation \(A\subset IA^*I\). From the latter relation it follows that an \(I\)-symmetric operator admits a closure. If \(A=IA^*I\), then the operator \(A\) is called \(I\)-self-adjoint \((^1)\). In \((^5)\) it is shown that an \(I\)-symmetric operator admits an \(I\)-self-adjoint extension.

Combining the methods of P. S. Ismagilov \((^3)\) and Levinson \((^2)\), it is easy to prove that the minimal operator \(L_0\), generated by the differential expression

\[ l[u]=-\frac{\partial}{\partial x}\left(a\frac{\partial u}{\partial x}\right) -\frac{\partial}{\partial y}\left(b\frac{\partial u}{\partial y}\right)+gu \tag{10} \]

in \(L_2(E_2)\), where \(a,b,g\) are complex-valued locally integrable functions of the variables \(x\) and \(y\), \(\operatorname{Re}a>0\), \(\operatorname{Re}b>0\), is \(I\)-self-adjoint if

\[ \iint_{E_2}\frac{dx\,dy}{\sqrt{|a|M}}=+\infty,\qquad \iint_{E_2}\frac{dx\,dy}{\sqrt{|a|M}}=+\infty; \tag{11} \]

here \(M(r)\) \(\bigl(r=\sqrt{x^2+y^2}\bigr)\) is a positive nondecreasing function on \((0,\infty)\);

\[ \frac{\sqrt{|a|}\,\partial M/\partial x}{\sqrt{M^3}}<+\infty,\qquad \frac{\sqrt{|b|}\,\partial M/\partial y}{\sqrt{M^3}}<+\infty,\qquad \operatorname{Re}g+\operatorname{Im}g\ge -kM(r). \tag{12} \]

In particular, if the real and imaginary parts of \(a\) and \(b\) are positive, \(\operatorname{Re}g\ge -c\), \(\operatorname{Im}g\ge c\), and \(\alpha\le |a|\le M,\ \alpha\le |b|\le M\), then the operator \(L_0\) is \(I\)-self-adjoint and therefore has no residual spectrum \((^1)\). Now, using Theorem 2, it is easy to prove that the operator \(L_0\) has a discrete spectrum if and only if the minimal operator \(M_0\), generated by the differential expression

\[ m[u]=-(\partial^2 u/\partial x^2+\partial^2 u/\partial y^2)+(\operatorname{Re}g+\operatorname{Im}g)u, \]

has a discrete spectrum. The operator \(M_0\) has a discrete spectrum if and only if Molchanov’s condition \((^4)\) is fulfilled. In the one-dimensional case it has the form

\[ \int_x^{x+a}(\operatorname{Re}g+\operatorname{Im}g)\,dt\to\infty \]

as \(x\to\infty\) and for every \(a>0\).

Using Theorem 1, it is easy to prove that the following holds:

Theorem 3. Every \(I\)-self-adjoint extension of the minimal differential operator \(L_0\), generated by the differential expression

by the expression

\[ l[y]=-\bigl[(p_1(x)+ip_2(x))y'\bigr]'+(g_1(x)+ig_2(x))y \]

in \(L_2(0,\infty)\), has a discrete spectrum if at least one of conditions I–VI is satisfied:

I.
a) \(p_1(x)\ge \alpha>0\);
b) \(g_1(x)\to +\infty\) as \(x\to\infty\).

II.
a) \(p_1(x)\ge \alpha>0,\quad g_1(x)\ge -c\);

b)
\[ \int_x^{x+a} g_1(t)\,dt\to +\infty \quad \text{as } x\to\infty \text{ and for any } a>0. \]

III.
a) \(p_1(x)>0,\quad g_1(x)\ge -c\);

b)
\[ \lim_{x\to\infty} x\int_x^\infty \frac{dt}{p_1(t)}=0. \]

IV.
a) \(p_1(x)>0,\quad \displaystyle \int_0^\infty \frac{dt}{\sqrt{p_1(t)}}=c<\infty\);

b) \(g_1(x)-{}^{1}/_{16}\,[p_1'(x)]^2/p_1(x)\ge \alpha>0\).

V.
a) \(p_1(x)>0,\quad \displaystyle \int_0^\infty \frac{dt}{\sqrt{p_1(t)}}=\infty\);

b) \(g_1(x)-{}^{1}/_{16}\,[p_1'(x)]^2/p_1(x)\to +\infty\) as \(x\to\infty\).

VI.
a) \(p_1(x)>0,\quad g_1(x)\ge -c\);

b) \(g_1(x)+{}^{1}/_{4}\,[p''(x)-{}^{1}/_{4}[p'(x)]^2/p(x)]\to +\infty\) as \(x\to\infty\).

The assertion of Theorem 3 holds if conditions I–VI are imposed either on \(p_2,g_2\), or on \(p_1+p_2,\ g_1+g_2\), or on \(p_1-p_2,\ g_1-g_2\).

The author expresses his deep gratitude to M. A. Naimark for supervising the work and to R. S. Ismagilov for discussing the results.

Steklov Mathematical Institute
Academy of Sciences of the USSR
Moscow

Received
10 IX 1969

CITED LITERATURE

1. I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Moscow, 1963.
2. M. A. Naimark, Linear Differential Operators, Moscow, 1954.
3. R. S. Ismagilov, DAN, 142, No. 6, 1239 (1962).
4. A. M. Molchanov, Tr. Mosk. Mat. Obshch., 2, 169 (1953).
5. N. A. Zikhar’, Ukr. Mat. Zh., 11, No. 4, 352 (1959).
6. V. B. Lidskii, Tr. Mosk. Mat. Obshch., 8, 83 (1959).