Reports of the Academy of Sciences of the USSR

1964, Vol. 157, No. 2

MATHEMATICS

KARL-PETER HADELER

AN ESTIMATE FOR THE SPECTRUM OF NORMAL OPERATORS

(Presented by Academician P. S. Novikov on 14 II 1964)

I. In 1963 H. Erman \((^{5})\) generalized a long-known theorem \((^{3})\):
If \(A\) is a completely continuous, symmetric operator in the space \(L_2(a,b)\) (or \(l_2, R_n\)) and \(f_0 \in L_2(a,b)\), \(f_0 \ne 0\), is an arbitrary function, then the eigenvalues of the operator \(A\) can be estimated by means of the values of the function \(g = Af_0/f_0\).

He succeeded in dropping certain assumptions on the representation of the operator in the form of an integral operator and on the sign of the function \(g\).

Here we shall prove that this theorem is a special case of a general theorem for arbitrary normal operators in the space \(L_2(B)\). We consider operator equations of the type \(Ax = \lambda Nx\) and applications, as well as an analogous theorem in a semi-ordered space.

II. Let \(H\) be one of the following spaces: \(L_2(B)\)—the class of measurable and square-summable complex functions on a measurable set \(B \subset R_n\) of positive measure, or the Hilbert space of sequences \(l_2\) \((B=N)\), or the complex space \(R_n\) with the ordinary Euclidean metric \((B=\{1,2,\ldots,n\})\).

We shall always regard the elements of the space \(H\) as functions on \(B\). Let \(f,h \in H\) (the representatives are fixed so that \(f(s), h(s)\) are defined for all \(s \in B\)). We divide \(B\) into the measurable sets

\[ A_1=\{s\mid h(s)\ne 0\}, \qquad A_2=\{s\mid h(s)=0,\ f(s)=0\}, \]

\[ A_3=\{s\mid h(s)=0,\ f(s)\ne 0\} \]

and denote the characteristic function of the set \(A_i\) by \(\varphi_i\); then define a function \(g(s)\), with values in the Riemann sphere of complex numbers \(S\), by

\[ g(s)= \begin{cases} \dfrac{f(s)}{h(s)}, & \text{for } s\in A_1,\\[6pt] b, & \text{for } s\in A_2,\quad b\in S \text{ arbitrary, but fixed},\\ \infty, & \text{for } s\in A_3. \end{cases} \]

Changes of the values of \(g(s)\) on sets of measure zero are allowed; the class \(g\) will be denoted by \(\left(\dfrac{f}{h}\right)\) and called the ratio of \(f\) to \(h\).

Consider the following regions \(K\) on the Riemann sphere:

\[ K_1=\{z\mid |z-a|\le r\}, \qquad K_2=\{z\mid |z-a|\ge r\}, \qquad \{K_3=\{z\mid \operatorname{Re} e^{i\vartheta}(z-c)\ge 0\}. \]

Let \(A\) be a linear (closed) normal operator whose domain of definition \(D_A\) is dense in \(H\). Then the spectrum \(\sigma(A)\ne \varnothing\), and for \(z\in \rho(A)\) the resolvent \(R_z=(A-zE)^{-1}\) is a normal, bounded operator with dense domain of definition, and \(\|R_z\|=d_z^{-1}\), where

\[ d_z=\min_{\zeta\in\sigma(A)} |z-\zeta|. \]

Theorem 1. Using \(f_0\in D_A\), \(f_0\ne 0\), form \(f_1=Af_0\) and the ratio

\[ g(s)=\left(\frac{f_1}{f_0}\right). \]

Suppose that the set of values of \(g(s)\) is situated in the circle \(K\). Then the intersection \(K\cap\sigma(A)\ne \varnothing\).

Proof. We shall assume that \(K\cap\sigma(A)=\varnothing\) and arrive at a contradiction.

1. \(K=K_1\). Since \(A_3=\varnothing,\ b\ne\infty\), we have

\[ g(s)=f_1(s)[f_0(s)+\varphi_2(s)]^{-1}+b\varphi_2(s) \]

is a measurable and bounded function. For the bounded multiplication operator \(G\), defined by the formula \(Gf=g(s)f(s)\), we have \(\|G-aE\|\le r<d_a\). From \(Af_0=Gf_0\) it follows that \(f_0=R_a(G-aE)f_0\), and (since \(\|f_0\|\ne 0\))

\[ \|f_0\|\le \|R_a\|\cdot \|G-aE\|\cdot \|f_0\|<d_d^{-1}d_a\|f_0\|. \]

1. \(K=K_2\) and \(A\) is bounded. With the aid of the measurable, bounded function

\[ \hat g=f_0[f_1-af_0+\varphi_2]^{-1}+\varphi_2(b-a)^{-1} \]

we define the multiplication operator \(\hat G f=\hat g(s)f(s)\). From \(Af_0=f_1\) it follows that \(\hat G(A-aE)f_0=f_0\). Since \(\|f_0\|\ne0,\ \|\hat G\|\le \frac1r,\ \|A-aE\|<r\), we obtain a contradiction.

1. \(K=K_2\), \(A\) is unbounded. A trivial case.

2. \(K=K_3\). From the assumption \(K\cap\sigma(A)=\varnothing\) it follows that \(C=(S-K)\cap\rho(A)\ne\varnothing\); on every straight line perpendicular to the boundary \(K'\) of the half-plane \(K\), there is an open interval belonging to \(C\) and tangent to \(K'\). Let \(\xi\in C\) and lie in such an interval. Define \(h_0=f_1-\xi f_0\ne0,\ h_1=f_0\ne0\). Then \(f_1=Af_0,\ f_0=R_\xi(f_1-\xi f_0),\ R_\xi h_0=h_1\). For the relation \(\tilde g(s)=\left(\frac{h_1}{h_0}\right)\) we have \(\tilde A_1=\tilde A_1\cup\tilde A_3,\ \tilde A_2=\tilde A_2,\ \tilde A_3=\varnothing\). Put \(\tilde g(s):=(b-\xi)^{-1}\); then \(\tilde g(s)=(g(s)-\xi)^{-1}\). Let the finite disk \(\widetilde K\) be the image of the domain \(K\) under the transformation \(w=(z-\xi)^{-1}\). For \(\widetilde K,\tilde g,R_z\) the theorem has already been proved in (1), consequently \(\widetilde K\cap\sigma(R_z)\ne\varnothing\); since \(K\cap\sigma(A)=\varnothing\), it follows that \(\widetilde K\cap\sigma(R_z)=\{0\}\). By a more refined argument one can show here that inside the disk \(\widetilde K\) there are points of the spectrum \(\sigma(R_z)\), so that \(K\cap\sigma(A)\ne\varnothing\).

III. The theorem can be generalized to the case when the operator equation has the form \(Ax=\lambda Nx\). Let the operator \(N\) be defined on the whole space \(H\), bounded, symmetric, and positive definite, \(0\notin\sigma(N)\). \(N\) has a root \(Q,\ QQ=N\), with the same properties. We call \(\chi(N)=\|N\|\cdot\|N^{-1}\|\) the condition number of the operator \(N\). (F. L. Bauer and A. S. Householder \((^2)\) introduced this concept for finite matrices.) Let \(A\) be a self-adjoint operator on \(D_A\subset H\) and \(\hat A\) the operator defined on \(D_{\hat A}=\{q\mid q=Qf,\ f\in D_A\}\) by \(\hat A=Q^{-1}AQ^{-1}\). In the natural way we define \(\sigma_N(A)=\sigma(\hat A)\).

Theorem 2. By means of \(f\in D_A,\ f\ne0\), form \(Af=f_1,\ Nf=f_0\) and the relation \(g\left(\frac{f_1}{f_2}\right)\). Let the set of values of the function \(g\) lie in the disk \(K\):

\[ K_1=\{z\mid |z-a|\le r\} \quad\text{or}\quad K_2=\{z\mid |z-a|\ge r\}. \]

Then \(\hat K\cap\sigma_N(A)\ne\varnothing\), where \(\hat K\) is defined as

\[ \hat K_1=\{z\mid |z-a|\le r\chi(N)^{1/2}\}, \quad\text{or}\quad \hat K_2=\{z\mid |z-a|\ge r\chi(N)^{-1/2}\}. \]

For finite matrices H. Baruch \((^1)\) had already obtained this result, but it can be sharpened. Let \(\mathfrak D=\mathfrak D(B)\) be the class of all measurable functions \(\delta(s)\) on \(B\) having the property

\[ 0<\delta_1\le \delta(s)\le \delta_2<\infty,\quad \text{for all } s\in B, \]

where \(\delta_1,\delta_2\) depend on the function. The multiplication operators \(\Delta h=\delta(s)h(s)\) and \(\Delta^{-1}\) are bounded. In addition to the operator \(A-\lambda N\), we shall consider \(\Delta A\Delta-\lambda\Delta N\Delta\)

on \(D_\Delta=\{q\mid q=\Delta^{-1}f,\ f\in D_A\}\). Since
\(\sigma_{\Delta N\Delta}(\Delta A\Delta)=\sigma_N(A)\) and
\(g(s)=\dfrac{Af}{f}=\dfrac{\Delta A\Delta\cdot \Delta^{-1}f}{\Delta N\Delta\cdot \Delta^{-1}f}\), in Theorem 2 one may replace \(\chi(N)\) by the quantity

\[ \overline{\chi}(N)=\inf_{\delta\in\mathscr D}\chi(\Delta N\Delta). \]

Obviously, \(\overline{\chi}(N)\geqslant 1\). For diagonal matrices \(\overline{\chi}(N)=1\). For real matrices of simple type

\[ N=\begin{pmatrix} N_1& &0\\ &\ddots&\\ 0& &N_0 \end{pmatrix},\qquad N_\nu=\begin{pmatrix} a_\nu&b_\nu\\ b_\nu&c_\nu \end{pmatrix}\quad(\nu=1,\ldots,m), \]

\(N_0\) is a diagonal matrix,

\[ \overline{\chi}(N)= \frac{\max_\nu\bigl(\sqrt{a_\nu c_\nu}+b_\nu\bigr)} {\min_\nu\bigl(\sqrt{a_\nu c_\nu}-c_\nu\bigr)}. \]

IV. Applications and examples. The definition, due to von Neumann, of a normal operator \((^6)\) makes it possible to construct examples of integral and differential operators. A closed operator is called normal if \(D_A=D_{A^*}\) and \(AA^*=A^*A\). A differential expression \(l\) of order \(n\) in a domain \(B\) will be called formally normal if \(ll^*y\) and \(l^*ly\) exist for every \(y\in C^{2n}(B)\), are continuous, and \(ll^*y=l^*ly\). For example, the ordinary differential expression of second order with real coefficients

\[ ly=py''+qy'+ry \]

is formally normal if and only if

\[ 2p(p''-q')+p'(q-p')=0,\qquad p(p^{\mathrm{IV}}-q''')+q(p'''-q'')+2r'(q-p')=0 \]

(in the self-adjoint case \(p'=q\)). Every differential expression with constant coefficients is formally normal.

Example 1.

\[ ly=y^{\mathrm{IV}}+a(y'''+y')=\lambda y, \]

\[ y(0)=y'(0),\qquad y(\pi)=y'(\pi),\qquad y''(0)=y''(\pi)=0, \]

is a normal and (except for \(a=0\)) non-self-adjoint operator. The trial function \(f_0=1+\sin x\) gives \(g=\sin x/(1+\sin x)\); consequently, there exists an eigenvalue \(\lambda\) with \(|\lambda-1/4|\leqslant 1/4\) for every \(a\).

Example 2. The operator

\[ Lu=-u_{xx}-u_{yy}+u_{xy}-2u_x=\lambda u \]

with boundary condition \(u=0\) on \(\Gamma\) in the triangle (see Fig. 1). The operator is normal. With the aid of the trial function
\(f_0(x,y)=y\{\sqrt3(1+x)-y\}\{\sqrt3(1-x)-y\}\) we obtain

\[ g(x,y)=\frac{6x(y-1)+4\sqrt3}{y^3-2\sqrt3y^2-3x^2y+3y}, \]

where \(g\) is positive inside \(B\) and unbounded on \(\Gamma\). In each half-plane
\(\operatorname{Re} e^{i\vartheta}(\lambda-8.84)\geqslant 0\),
\(-\pi/2\leqslant \lambda\leqslant \pi/2\), there lies an eigenvalue. (Applying the Ritz–Galerkin method, we obtain that the eigenvalue \(\lambda_1\) of smallest modulus satisfies \(|\lambda_1|\leqslant 14\).)

Fig. 1

Example 3. \(ly=-y''+x^4y\) in the space \(L_2(-\infty,\infty)\). Let \(L_0'\) be the operator on the set of finite twice differentiable functions; the closure \(L\) of this operator is already self-adjoint with discrete spectrum \((^7)\). The trial function \(f_0=e^{-\alpha x^2}\), \(\alpha>0\), gives \(g=2\alpha-4\alpha^2x^2+x^4\). Since

\[ \max_{\alpha>0}\min_x g(x)=0.75, \]

there exists an eigenvalue \(\lambda\geqslant 0.75\).

Example 4. The integral equation of the type \(Ay=\lambda [E+(N-E)]y\), where \(A\) and \(N-E\) are symmetric, completely continuous operators and \(\sigma(N-E)>-1\). Theorem 2 can be applied. Suppose we seek a number \(k\) for which the equation

\[ y(x)=\frac12\int_2^1[\sin(xs)+k\cos(xs)]y(s)\,ds \]

has a nontrivial solution. Since

\[ \alpha^2=\int_0^1\int_0^1 |\sin(xs)|^2\,dx\,ds =\frac{1}{2}(1-\operatorname{Si}(2))=0.0985, \]

there is no solution for \(k=0\). Rewrite the preceding equation in the form

\[ \frac12\int_0^1 \cos(xs)y(s)\,ds = \frac12\left[y(s)-\frac12\int_0^1 \sin(xs)y(s)\,ds\right]. \]

The spectrum \(\sigma(N-E)\) is symmetric with respect to zero and lies on the interval \(|z|\leq \tfrac12\alpha\), i.e. \(\chi(N)\leq \frac{2+\alpha}{2-\alpha}\) and \(\chi(N)^{1/2}\leq 1.1716\). With the trial function \(f\equiv 1\) we obtain \(g(x)=\sin x/(2x-1+\cos x)\), so that finally such a \(k=k_0\) exists in the interval

\[ 1.791\leq k_0<2.020. \]

V. Analogous properties hold for positive operators in partially ordered spaces. Let \(H\) be the real space \(C[0,1]\) of continuous functions with the cone \(K\) of nonnegative functions. Let \(A\) be a bounded linear operator which maps the cone \(K\) into itself. Then the following theorem is true.

Theorem 3. Let \(f_0\) be an interior point of the cone. Form \(f_1=Af_0\in K\) and the ratio \(g(s)=f_1/f_0\). Let \(0<m\leq g(s)\leq M\) for all \(s\in[0,1]\). Then the spectral radius \(r\) of the operator \(A\) lies in the interval

\[ m\leq r\leq M. \]

For finite matrices the theorem has long been known \((^4)\). Passage to the complex space \(C[0,1]\) gives no more general result. The proof uses certain properties of positive operators found by H. Schaefer \((^8)\).

Received
31 I 1964

REFERENCES

\(^1\) H. Bartsch, Arch. Math., 4, 133 (1953).
\(^2\) F. L. Bauer, A. S. Householder, Num. Math., 2, 308 (1960).
\(^3\) L. Collatz, Math. Zs., 47, 395 (1942).
\(^4\) L. Collatz, Eigenwertaufgaben mit technischen Anwendungen, 2 Aufl., Leipzig, 1963.
\(^5\) H. Ehrmann, Math. Zs., 82 H. 5, 403 (1963).
\(^6\) J. v. Neumann, Ann. Math., 33, 294 (1932).
\(^7\) M. A. Neumark, Lineare Differentialoperatoren, Berlin, 1960.
\(^8\) H. Schaefer, Pacific J., 10, 1009 (1960).