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Reports of the Academy of Sciences of the USSR
- Volume 158, No. 5
MATHEMATICS
KARL-PETER HADELER
ON THE SPECTRUM OF NORMAL OPERATORS AND THEIR PERTURBATIONS
(Presented by Academician L. S. Pontryagin on 29 IV 1964)
The present paper generalizes the results of the author’s preceding note \((^1)\) and some lemmas of I. M. Glazman \((^2)\). We consider normal operators in a Hilbert space.
Theorem 1. Let \(A\) and \(B\) be normal operators and let the spectrum \(\sigma(B)\) of the bounded operator \(B\) have the property
\[ \sigma(B)\subset \{z:\ |z-a|\leqslant r\}. \]
a) If there is a manifold \(F\subset D_A\) of dimension \(n\), \(1\leqslant n\leqslant \infty\), such that
\[ \|(A-B)f\|\leqslant \alpha \|f\| \quad \text{for all } f\in F, \]
then \(\{z:\ |z-a|\leqslant r+\alpha\}\) contains at least \(n\) points of the spectrum \(\sigma(A)\).
b) If in \(\{z:\ |z-a|\leqslant r+\alpha\}\) there are \(n\) points of the spectrum \(\sigma(A)\), \(1\leqslant n\leqslant \infty\), then there is a manifold \(F\) of dimension \(n\) such that
\[ \|(A-B)f\|\leqslant (2r+\alpha)\|f\| \quad \text{for all } f\in F. \]
c) If there is a subspace \(G\) with defect number \(n\), \(1\leqslant n<\infty\), such that
\[ \|(A-B)g\|>\alpha\|g\| \quad \text{for all } g\in G,\quad \alpha>r, \]
then the number of points of the spectrum \(\sigma(A)\) in \(\{z:\ |z-a|\leqslant \alpha-r\}\) is less than or equal to \(n\).
d) If the number of points of the spectrum \(\sigma(A)\) in \(\{z:\ |z-a|\leqslant \alpha+r\}\) is less than or equal to \(n<\infty\), then there is a subspace \(G\) with defect number not greater than \(n\), for which
\[ \|(A-B)g\|>\alpha\|g\| \quad \text{for all } g\in G. \]
The theorem remains true if in a), b), c), d) we replace all \(\leqslant\) by \(<\) and conversely (with the exception of the inequalities for \(n\)).
In the case \(r=0\), i.e. \(B=aE\), the theorem gives necessary and sufficient conditions \((^2)\).
Similar theorems hold not only for bounded operators \(B\). We shall give two special lemmas.
Lemma 1. Let \(A\) and \(B\) be normal operators,
\[ \sigma(B)\subset \{z:\ |z-a|\geqslant r\}. \]
If there is \(F\in D_A\cap D_B\), \(\|f\|=1\), \(\|Af-Bf\|\leqslant \alpha<r\), then
\[ \sigma(A)\cap \{z:\ |z-a|\geqslant r-\alpha\}\ne \varnothing. \]
Lemma 2. Let \(A\) and \(B\) be normal operators,
\[ \sigma(B)\subset \{z:\ \operatorname{Re} e^{i\varphi}(z-a)\geqslant 0\}=K,\qquad 0\leqslant \varphi<2\pi. \]
If there is \(f\in D_A\cap D_B\), \(f\ne 0\), such that \(Af=Bf\), then \(\sigma(A)\cap K\ne \varnothing\).
The proofs of Theorem 1 and of these lemmas are analogous to the proof in \((^1)\). The first part of Theorem 1 in \((^1)\) is a consequence of Theorem 1a), since the multiplication operator is a normal operator.
Theorem 2. Let \(A\) be a normal operator and, for \(f_0\in D_A\), \(f_0\ne 0\),
\[ Af_0=f_1. \]
Let \(K_c\) be the circle with center
\[ a=\frac{a_1}{a_0}+\frac{x\delta-1}{2ca_0} \]
and radius \(r=\)
\[ = \frac{c\delta+1}{2|c|a_0}, \qquad \text{where } a_0=(f_0,f_0), \quad a_1=(f_1,f_0), \quad a_2=(f_1,f_1), \quad \delta=a_2a_0-a_1\overline{a_1}, \]
\(c\ne0\) is an arbitrary complex number. Then \(K_c\cap\sigma(A)\ne\varnothing\).
Corollary. For fixed \(f_0\), with \(c=\delta^{-1/2}\), we obtain the minimal radius, namely, \(\hat a=a_1/a_0,\ \hat r^2=\delta/a_0^2\).
These circles are known as Krylov–Bogolyubov circles.
Theorem 3. Let \(A\) be a normal operator in \(L_2(D)\), \(D\subset R^n\) a measurable set of positive measure, and let \(q(x)\) be a measurable function on \(D\). Let \(Q\) be the closed convex hull of the spectrum of the function \(q(x)\), and let \(K\) be the smallest (closed) circle containing \(Q\). Let
\[ \widetilde A f=Af+qf \]
for \(f\in D_{\widetilde A}=D_A\) be the perturbed operator, and let \(\widetilde\sigma(\widetilde A)\) be the subset of all points of \(\sigma(\widetilde A)\) that do not belong to the pure residual spectrum. Then
\[ \widetilde\sigma(\widetilde A)\subset M=\{z:\ z=z_1+z_2,\ z_1\in\sigma(A),\ z_2\in K\}. \]
Let \(M'\) be the set of all \(z\) for which one can find a circle \(K_z\), \(\{z-Q\}\subset K_z,\ K_z\cap\sigma(A)=\varnothing\); then \(M'\subset M\) and even \(\widetilde\sigma(\widetilde A)\subset M'\).
Corollary 1. Suppose there is a sequence \(\{z_n\}\), \(z_n\in\sigma(A)\), such that
\[ \lim_{n\to\infty} d(z_n)=\infty, \qquad \text{where } d(z_n)=\inf_{\substack{z\in\sigma(A)\\ z\ne z_n}} |z_n-z|. \]
Then the components of the set \(M'\) corresponding to the points \(z_n\) asymptotically have the form \(\{z_n+Q\}\). These components, generally speaking, are not convex.
Corollary 2. Let \(A\) be a self-adjoint regular differential operator (with discrete spectrum) and \(q(x)\) a real, bounded, continuous function, \(|q(x)|\le b\), and let
\[ \widetilde A f=Af+iqf \]
for \(f\in D_{\widetilde A}=D_A\) be the perturbed operator. Then \(M'\) contains \(\sigma(\widetilde A)\) and is described explicitly. For \(z_n\in\sigma(A)\) define \(K_n=K_n^l\cup K_n^r\). If \(a_n=|z_n-z_{n-1}|\le 2b\), then \(K_n^l=K\cap\{z:\operatorname{Re}z\le0\}\); if \(a_n>2b\), then \(K_n^l\) is the closed domain bounded by the imaginary axis and the right branch of the hyperbola
\[ \left(x+\frac{a_n}{2}\right)^2-y^2=\frac14(a_n^2-4b^2). \]
Analogously, \(K_n^r\) is either \(K\cap\{z:\operatorname{Re}z\ge0\}\), or the domain bounded by the imaginary axis and the left branch of the hyperbola
\[ \left(x-\frac{a_{n+1}}{2}\right)^2-y^2=\frac14(a_{n+1}^2-4b^2). \]
Then
\[ M'=\bigcup_n \{z_n+K_n\}. \]
Institute of Applied Mathematics
of the University of Hamburg
Hamburg, FRG
Received
27 IV 1964
References
- K. P. Hadeler, DAN, 157, No. 2 (1964).
- I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Moscow, 1964.