MATHEMATICS

M. S. BRODSKII

ON THE TRIANGULAR REPRESENTATION OF CERTAIN OPERATORS WITH COMPLETELY CONTINUOUS IMAGINARY PART

(Presented by Academician V. I. Smirnov on 16 IV 1960)

Let \(A\) be a bounded operator with purely real spectrum, acting in a Hilbert space \(H\). If \(P\) is some projector in \(H\), then by \(A_P\) we shall denote the operator \(PAP\), considered in the subspace \(PH\), and by \(\sigma(A_P)\) the spectrum of the operator \(A_P\). We shall say that the projector \(P\) cuts the spectrum of the operator \(A\) at the point \(t\) if the subspace \(PH\) is invariant with respect to \(A\) and if
\(\sigma(A_P) \subset (-\infty,t]\), \(\sigma(A_{E-P}) \subset [t,\infty)\).

We assign the operator \(A\) to the class \(\Omega\) if:

I. The imaginary part \(K=\dfrac{A-A^*}{2i}\) of the operator \(A\) is completely continuous.

II. The entire spectrum of the operator \(A\) lies on the real axis.

III. There exists a system of projectors \(P_t(-\infty<t<\infty)\) such that
\(P_{t_1}\le P_{t_2}\) for \(t_1<t_2\), and each projector \(P_t\) cuts the spectrum of the operator \(A\) at the point \(t\).

From results of Yu. I. Lyubich and V. I. Matsaev \((^1)\) it follows that every operator \(A\) satisfying conditions I and II, as well as the condition

\[ \int_0^\varepsilon \ln^+ \ln^+ M(\delta)\,d\delta<\infty \quad \left(M(\delta)=\sup_{|\operatorname{Im}\lambda|\ge \delta}\|(A-\lambda E)^{-1}\|\right) \tag{1} \]

belongs to the class \(\Omega\). In particular, as V. I. Matsaev showed, condition (1) will be fulfilled if the series
\(\sum_{n=1}^{\infty}\dfrac{|\omega_n|}{n}\) converges, where \(\omega_n\) are the eigenvalues of the operator \(K\), numbered with multiplicities taken into account in decreasing order of their absolute values.

In the present article the problem of reducing operators of the class \(\Omega\) to triangular form is solved.

1. Let the operator \(A\) belong to the class \(\Omega\). Then on some closed set of real numbers \(\mathfrak M\) \((\min \mathfrak M=0,\ \max \mathfrak M=1)\) there exists a function \(E(x)\) \((x\in\mathfrak M)\) having the following properties \((^2)\):
   1) the values of the function \(E(x)\) are projectors in \(H\);
   2) if \(x_1<x_2\), then \(E(x_1)<E(x_2)\);
   3) \(E(0)=0,\ E(1)=E\);
   4) the function \(E(x)\) is continuous on \(\mathfrak M\);
   5) each subspace \(E(x)H\) is invariant with respect to \(A\);
   6) the set of values of the function \(E(x)\) contains all projectors \(P_t\) \((-\infty<t<\infty)\);
   7) if \((a,b)\) is an interval complementary to the set \(\mathfrak M\), then there does not exist a subspace \(H_0\) invariant with respect to \(A\) and satisfying the condition
   \(E(a)H\subset H_0\subset E(b)H\).

Let us note \((^3)\) that if \(P\) and \(P'\) are projection operators onto subspaces invariant with respect to \(A\), and \(P<P'\), then, by virtue of condition 1),

\[ \sigma(A_{P'})=\sigma(A_P)+\sigma(A_{P'-P}),\qquad \sigma(A_{E-P})=\sigma(A_{E-P'})+\sigma(A_{P'-P}). \tag{2} \]

The first of these equalities shows that the function \(\alpha(x)=\max \sigma(A_{E(x)})\) is nondecreasing. Introduce the function \(\alpha(x+0)\), defining it at the point \(x\) in the usual way if every interval \((x,x+\varepsilon)\) contains points of the set \(\mathfrak M\), and putting \(\alpha(x+0)=\alpha(x)\) otherwise.

We shall need the following properties of the functions \(E(x)\) and \(\alpha(x)\).

a) The projector \(E(x)\) splits the spectrum of the operator \(A\) at the point \(\alpha(x)\).

b) The function \(\alpha(x)\) is continuous from the left.

c) If \(x<x'\) \((x,x'\in\mathfrak M)\), then the spectrum of the operator \(A_{E(x')-E(x)}\) lies in the segment \([\alpha(x+0),\alpha(x')]\).

Indeed, if \(\lambda\in\sigma(A_{E(x')-E(x)})\), then, by virtue of (2), \(\lambda\in[\alpha(x),\alpha(x')]\). Let there exist a strictly decreasing sequence \(\{x_n\}\subset\mathfrak M\) tending to \(x\). Then \(\alpha(x_n)\geqslant\alpha(x+0)\) and, consequently, for any \(\varepsilon>0\),
\[ E(x)=\lim_{n\to\infty} E(x_n)\geqslant P_{\alpha(x+0)-\varepsilon}. \]
Therefore
\[ \lambda\in\sigma(A_{E-E(x)})\subset\sigma(A_{E-P_{\alpha(x+0)-\varepsilon}})\subset[\alpha(x+0)-\varepsilon,\infty), \]
i.e. \(\lambda\in[\alpha(x+0),\infty)\).

d) If \((a,b)\) is a complementary interval of the set \(\mathfrak M\), then
\[ \dim\{E(b)H\ominus E(a)H\}=1, \tag{3} \]
and
\[ (E(b)-E(a))A(E(b)-E(a))=\alpha(b)(E(b)-E(a)). \tag{4} \]

In fact, \(\sigma(A_{E(b)-E(a)})\subset[\alpha(a),\alpha(b)]\). Let \(\lambda\) belong to the spectrum of the operator \(A\) and let \(\alpha(a)<\lambda<\lambda_1<\alpha(b)\). Since some projector \(E(x)\) splits the spectrum of the operator \(A\) at the point \(\lambda_1\), we have \(\lambda\leqslant\alpha(x)\leqslant\lambda_1\), which is impossible. Thus the set \(\sigma(A_{E(b)-E(a)})\) consists either of the two points \(\alpha(a)\) and \(\alpha(b)\), or of the single point \(\alpha(b)\). If the first case occurs, then, by a well-known theorem of Riesz, the operator \(A_{E(b)-E(a)}\) has a nontrivial invariant subspace \(H_0\). But then the subspace \(H_1=E(a)H\oplus H_0\) is invariant with respect to \(A\), and \(E(a)H\subset H_1\subset E(b)H\), which contradicts property 7) of the function \(E(x)\). Thus the operator \(A_{E(b)-E(a)}\) has only one spectral point \(\alpha(b)\). Since its imaginary part is completely continuous, it has the form \(\alpha(b)E+B\), where \(B\) is a completely continuous operator\({}^{(4)}\). If we suppose that the subspace \(E(b)H\ominus E(a)H\) is not one-dimensional, then the operator \(A_{E(b)-E(a)}\) will have a nontrivial invariant subspace by the theorem of Aronszajn and Smith\({}^{(5)}\), which again leads to a contradiction.

1. We shall say that the points \(0=x_0<x_1<\cdots<x_n=1\) \((x_k\in\mathfrak M)\) form a \(\delta\)-partition of the set \(\mathfrak M\), if \(x_{k-1}\) and \(x_k\) \((k=1,2,\ldots,n)\) are either endpoints of an interval complementary to \(\mathfrak M\), or satisfy the condition \(x_k-x_{k-1}<\delta\). It is easy to verify that the integral
   \[ \int_{\mathfrak M}\alpha(x)\,dE(x)=\lim_{\delta\to0}\sum_{k=1}^{n}\alpha(\xi_k)\Delta E_k \quad (x_{k-1}<\xi_k\leqslant x_k,\ \xi_k\in\mathfrak M,\ \Delta E_k=E(x_k)-E(x_{k-1})) \]
   converges in norm in the sense of S. O. Shatunovskii. We also note that, by virtue of (4),
   \[ \lim_{\delta\to0}\left\|\sum_{k=1}^{n}\Delta E_k\frac{A-A^*}{2i}\Delta E_k\right\|=0. \]

Lemma. For every \(\varepsilon>0\) there exists a partition \(0=x_0<x_1<\cdots<x_n=1\) of the set \(\mathfrak M\) such that
\[ \left\|\sum_{k=1}^{n}\Delta E_k A\Delta E_k-\int_{\mathfrak M}\alpha(x)\,dE(x)\right\|<\varepsilon \quad (\Delta E_k=E(x_k)-E(x_{k-1})). \]

Proof. Let \(\varepsilon>0\). There exists a partition
\(0=t_0<t_1<\cdots<t_m=1\) of the set \(\mathfrak M\) such that \(t_{k-1}\) and \(t_k\) \((k=1,2,\ldots,m)\) are either endpoints of an interval complementary to \(\mathfrak M\), or satisfy the condition
\(\alpha(t_k)-\alpha(t_{k-1}+0)<\varepsilon/4\). Moreover, there exists a constant \(M\) such that

\[ \left\|\left(A_{E(t_k)-E(t_{k-1})}-\lambda E\right)^{-1}\right\|<M \]

\[ (k=1,2,\ldots,m;\ \operatorname{Im}\lambda=0,\quad \lambda\bar\in[\alpha(t_{k-1}+0)-\varepsilon/4,\ \alpha(t_k)+\varepsilon/4]). \]

Let \(0=x_0<x_1<\cdots<x_n=1\) be a refinement of the above partition satisfying the conditions

\[ \left\|\sum_{k=1}^{n}\Delta E_k\,\frac{A-A^*}{2}\,\Delta E\right\| < \min\left\{\frac1M,\frac{\varepsilon}{4}\right\} \quad (\Delta E_k=E(x_k)-E(x_{k-1})), \tag{5} \]

\[ \left\|\sum_{k=1}^{n}\alpha(\xi_k)\Delta E_k-\int_{\mathfrak M}\alpha(x)\,dE(x)\right\| < \frac{\varepsilon}{4} \quad (x_{k-1}<\xi_k\le x_k). \tag{6} \]

Then

\[ \left\|\sum_{k=1}^{n}\Delta E_k A\Delta E_k - \sum_{k=1}^{n}\Delta E_k\,\frac{A+A^*}{2}\,\Delta E_k \right\| < \frac{\varepsilon}{4}. \tag{7} \]

Each segment \([x_{p-1},x_p]\) belongs to some segment \([t_{p'-1},t_{p'}]\). Therefore, for
\(\lambda\bar\in[\alpha(t_{p'-1}+0)-\varepsilon/4,\alpha(t_{p'})+\varepsilon/4]\),
\(\operatorname{Im}\lambda=0\),

\[ \left\|\left(A_{E(x_p)-E(x_{p-1})}-\lambda E\right)^{-1}\right\| \le \left\|\left(A_{E(t_{p'})-E(t_{p'-1})}-\lambda E\right)^{-1}\right\| <M, \]

\[ \left\| \left(A_{E(x_p)-E(x_{p-1})}-\lambda E\right) - \left(\left(\frac{A+A^*}{2}\right)_{E(x_p)-E(x_{p-1})}-\lambda E\right) \right\| < \frac1M < \left\|\left(A_{E(x_p)-E(x_{p-1})}-\lambda E\right)^{-1}\right\|^{-1}. \]

This proves that the entire spectrum of the operator
\(\left(\dfrac{A+A^*}{2}\right)_{E(x_p)-E(x_{p-1})}\) lies in the segment
\([\alpha(t_{p'-1}+0)-\varepsilon/4,\alpha(t_{p'})+\varepsilon/4]\).
Suppose the points \(x_{p-1}\) and \(x_p\) are not endpoints of an interval complementary to \(\mathfrak M\). Then the points \(t_{p'-1}\) and \(t_{p'}\) also have this property, and, consequently,

\[ \left\| \Delta E_p\,\frac{A+A^*}{2}\,\Delta E_p-\alpha(\xi_p)\Delta E_p \right\| = \left\| \left(\frac{A+A^*}{2}\right)_{\Delta E_p}-\alpha(\xi_p)E \right\| \le \frac{\varepsilon}{2}. \tag{8} \]

If, however, the points \(x_{p-1}\) and \(x_p\) are endpoints of a complementary interval, then by virtue of (4),
\(\Delta E_p\dfrac{A+A^*}{2}\Delta E_p-\alpha(\xi_p)\Delta E_p=0\).
Thus inequality (8) holds for all \(p=1,2,\ldots,n\), and consequently,

\[ \left\| \sum_{k=1}^{n}\Delta E_k\,\frac{A+A^*}{2}\,\Delta E_k - \sum_{k=1}^{n}\alpha(\xi_k)\Delta E_k \right\| \le \frac{\varepsilon}{2}. \tag{9} \]

The assertion of the lemma follows from inequalities (7), (9), and (6).

1. Let \(K\) be some bounded operator. To each \(\delta\)-partition
   \(0=x_0<x_1<\cdots<x_n=1\) of the set \(\mathfrak M\) we assign the integral sum

\[ \sum_{k=1}^{n}E(\xi_k)K\Delta E_k \quad (x_{k-1}\le \xi_k\le x_k,\quad \Delta E=E(x_k)-E(x_{k-1})). \tag{10} \]

If, as \(\delta \to 0\), the sums (10) converge in norm to some operator \(B\), then we shall write
\[ B=\int_{\mathfrak M} E(x)K\,dE(x). \]

A completely continuous operator will be called Volterra if zero is the only point of its spectrum.

Theorem\(^*\). Let \(A\) be an operator of class \(\Omega\); let \(\alpha(x)\) and \(E(x)\) be the functions defined above. Then
\[ A=\int_{\mathfrak M}\alpha(x)\,dE(x)+2i\int_{\mathfrak M}E(x)K\,dE(x) \qquad \left(K=\frac{A-A^*}{2i}\right). \tag{11} \]

Proof. If
\[ 0=x_0<x_1<\cdots<x_n=1 \]
is an arbitrary partition of the set \(\mathfrak M\), then
\[ A:=\sum_{s,t=1}^n \Delta E_s A\Delta E_t =\sum_{k=1}^n \Delta E_k A\Delta E_k +2i\sum_{s<t}\Delta E_s K\Delta E_t \qquad \left(K=\frac{A-A^*}{2i}\right). \tag{12} \]
By virtue of the lemma proved above, there exists such a sequence of partitions of the set \(\mathfrak M\) for which the first summand in the sum (12) converges in norm to
\[ \int_{\mathfrak M}\alpha(x)\,dE(x). \]
At the same time the second summand will converge in norm to some operator \(B\). Since the operator
\[ 2i\sum_{s<t}\Delta E_s K\Delta E_t \]
is completely continuous and some natural power of it is equal to zero, it is Volterra. Consequently, the operator \(B\) is also Volterra. Moreover,
\[ 2i\sum_{s<t}\Delta E_s K\Delta E_t -\left(2i\sum_{s<t}\Delta E_s K\Delta E_t\right)^* = 2iK-2i\sum_{s=1}^n \Delta E_s K\Delta E_s, \]
and therefore
\[ \frac{B-B^*}{2i}=K. \]
Let us also note that each subspace \(E(x)H\) is invariant for the operator
\[ 2i\sum_{s<t}\Delta E_s K\Delta E_t, \]
and hence also for the operator \(B\). Finally, taking into account that, by (4),
\[ (E(b)-E(a))K(E(b)-E(a))=0 \]
for every interval \((a,b)\) complementary to \(\mathfrak M\), we obtain
\[ B=2i\int_{\mathfrak M}E(x)K\,dE(x). \]

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
14 IV 1960

REFERENCES

1. Yu. I. Lyubich, V. I. Macaev, DAN, 131, No. 1 (1960).
2. L. A. Sakhnovich, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 1 (8), 180 (1959).
3. M. S. Brodskii, M. S. Livshits, UMN, 13, issue (79), 3 (1958).
4. I. Ts. Gohberg, M. G. Krein, UMN, 12, issue (74), 43 (1957).
5. N. Aronszajn, R. J. Smith, Ann. Math., 60, 316 (1954).

\(^*\) For the particular case
\[ \sum_{\alpha=1}^{\infty}|\omega_\alpha|^p<\infty \quad (p>1), \]
where \(\omega_\alpha\) are the eigenvalues of the operator
\[ \frac{A-A^*}{2i}, \]
a triangular representation of the form (11) was obtained by I. Ts. Gohberg and M. G. Krein.