UDC 513.88 + 517.948.35 + 517.948.5

MATHEMATICS

V. M. BRODSKII, M. S. BRODSKII

ON AN ABSTRACT TRIANGULAR REPRESENTATION OF LINEAR BOUNDED OPERATORS AND THE MULTIPLICATIVE DECOMPOSITION OF THE CORRESPONDING CHARACTERISTIC FUNCTIONS

(Presented by Academician S. L. Sobolev on 25 X 1967)

Let \(\mathfrak H\) be a separable Hilbert space and \(\mathfrak R=\mathfrak R_{\mathfrak H}\) the ring of linear bounded operators acting in \(\mathfrak H\). A completely continuous operator \(B\in\mathfrak R\) will be assigned to the class \(\mathfrak S_\omega\) \((^{1,2})\) if the eigenvalues \(s_j(B)\) of the nonnegative operator \((B^*B)^{1/2}\), numbered in decreasing order with multiplicities counted, satisfy the condition
\[ B_{\mathfrak S_\omega}=\sum_j \frac{s_j(B)}{2j-1}<\infty . \]
The introduction of the norm \(\|\cdot\|_{\mathfrak S_\omega}\) turns the class \(\mathfrak S_\omega\) into a symmetrically normed ideal* of the ring \(\mathfrak R\). In the present article, operators \(A\in\mathfrak R\), the imaginary components of which belong to \(\mathfrak S_\omega\), are reduced to an abstract triangular form, and the corresponding characteristic functions—to a multiplicative one. This generalizes the results of one of the authors \((^{4,5})\), pertaining to the case when \(A\) has a purely real spectrum. Another approach to the problem of triangular representation of operators of the class considered by us is contained in the works of L. de Branges \((^{6,7})\).

1. Let us specify a closed chain of orthoprojectors \(\mathfrak P=\{P\}\) in \(\mathfrak H\) and functions \(F_1(P)\), \(F_2(P)\), \(G(P)\) \((P\in\mathfrak P)\) with values in \(\mathfrak R\). We define the integral
   \[ \int_{\mathfrak P} F_2(P)\,dG(P)\,F_1(P) \]
   as the limit in Shatunovskii’s sense (in the sense of one topology or another) of integral sums of the form
   \[ \sum_{j=1}^{m} F_2(Q_j)\bigl(G(P_j)-G(P_{j-1})\bigr)F_1(Q_j) \tag{1} \]
   \[ (\min\mathfrak P=P_0<P_1<\cdots<P_m=\max\mathfrak P,\; P_{j-1}\le Q_j\le P_j;\; P_j,Q_j\in\mathfrak P). \]
   With the additional restriction \(P_{j-1}<Q_j\) or \(P_{j-1}=Q_j\), we shall write instead of \(\int_{\mathfrak P}\) respectively \(\int_{(\mathfrak P]}\) or \(\int_{[\mathfrak P}\).

If the chain \(\mathfrak P\) is maximal and is an increasing sequence \(\{P_j\}_0^\infty\), then, obviously: a) \(P_0=0\), \(P_\infty=I\); b) \(\dim\{P_j\mathfrak H\}=j\), c) \(P_j\to I\). Such a chain will be called discrete.

Let \(\mathfrak P=\{P_j\}_0^\infty\) be a maximal discrete chain and let \(\varphi(P)\) \((P\in\mathfrak P)\) be a bounded scalar function. It is easy to verify that then
\[ \int_{(\mathfrak P]} \varphi(P)\,dP = \sum_{j=1}^{\infty}\varphi(P_j)\Delta P_j \quad (\Delta P_j=P_j-P_{j-1}). \tag{2} \]
The integral and the series in (2) converge strongly.

* The authors adhere to the terminology adopted in the monographs of I. Ts. Gokhberg and M. G. Krein \((^{2,3})\).

Lemma 1. If the operator \(A\in\mathfrak R\) has a maximal discrete chain \(\mathfrak P=\{P_j\}_0^\infty\), then

\[ A=\int_{[\mathfrak P]}\alpha(P)\,dP+2i\int_{[\mathfrak P]} PA_I\,dP \left(A_I=\frac{A-A^*}{2i}\right), \tag{3} \]

where the values \(\alpha(P_j)\) of the scalar function \(\alpha(P)\) are determined from the relations
\(\alpha(P_j)\Delta P_j=\Delta P_j A\Delta P_j\).
The integrals in the equality (3) converge strongly. In the case when \(A_I\in\mathfrak S_\omega\), the integral
\[ \int_{[\mathfrak P]} PA_I\,dP \]
converges uniformly.

Proof. For given \(h\in\mathfrak H\) and \(\varepsilon>0\) choose a natural number \(N=N(h,\varepsilon)\) so that the inequality
\(\|(I-P_N)h\|<\varepsilon/2\|A\|\) is satisfied. Any continuation of the partition
\(P_0<P_1<\cdots<P_N<I\) has the form
\(Q_0<Q_1<\cdots<Q_n=I\), where \(Q_j=P_j\) \((j=1,2,\ldots,N)\). Since

\[ A=\sum_{j=1}^n \Delta Q_j A\Delta Q_j +2i\sum_{j=1}^n Q_{j-1}A_I\Delta Q_j \quad(\Delta Q_j=Q_j-Q_{j-1}) \]

and, consequently,

\[ \begin{aligned} \left\|(A-\hat A)h -2i\sum_{j=1}^n Q_{j-1}A_I\Delta Q_j h\right\|^2 &= \left\|\sum_{j=N+1}^n \Delta Q_j A\Delta Q_j h -\sum_{j=N+1}^n \Delta P_j A\Delta P_j h\right\|^2 \\ &\le (2\|A\|)^2\|(I-P_N)h\|^2<\varepsilon^2 \\ \left(\hat A=\sum_{j=1}^\infty \alpha(P_j)\Delta P_j =\sum_{j=1}^\infty \Delta P_j A\Delta P_j =\int_{[\mathfrak P]}\alpha(P)\,dP\right), \end{aligned} \]

the integral
\[ \int_{[\mathfrak P]} PA_I\,dP \]
exists in the sense of strong convergence. At the same time relation (3) has been proved.

If \(A_I\in\mathfrak S_\omega\), then \(A_I-\hat A_I\in\mathfrak S_\omega\), where
\[ \hat A_I=\sum_{j=1}^\infty \Delta P_j A_I\Delta P_j \]
(see (2), p. 74). Moreover, evidently,
\[ \Delta P_j(A_I-\hat A_I)\Delta P_j=0\quad(j=1,2,\ldots). \]
It follows from this (see (3), p. 132) that, in the sense of uniform convergence, the integral
\[ \int_{\mathfrak P} P(A_I-\hat A_I)\,dP \]
exists. In particular, for a given \(\varepsilon>0\) there is a partition of the chain \(\mathfrak P\) such that any continuation of it
\(0=P'_0<P'_1<\cdots<P'_k=I\) will satisfy the inequality

\[ \left\| \int_{\mathfrak P} P(A_I-\hat A_I)\,dP -\sum_{j=1}^k P'_{j-1}(A_I-\hat A_I)\Delta P'_j \right\|<\varepsilon. \]

Since
\[ P'_{j-1}\hat A_I\Delta P'_j=0\quad(j=1,2,\ldots,k), \]
the integral
\[ \int_{[\mathfrak P]} PA_I\,dP \]
converges uniformly.

The lemma is proved. Consider an operator \(A\in\mathfrak R\) with an imaginary component from \(\mathfrak S_\omega\). Denote by \(\mathfrak H_0\) the closure of the linear span of its root subspaces corresponding to nonreal points of the spectrum, and introduce the orthoprojectors \(P^{(0)}\) and \(P^{(1)}\) respectively onto \(\mathfrak H_0\) and \(\mathfrak H\ominus\mathfrak H_0\). Let \(A_0\) be the operator induced in \(\mathfrak H_0\), and \(A_1\) the operator in \(\mathfrak H_1\) defined by the formula
\[ A_1h=P^{(1)}Ah\quad(h\in\mathfrak H_1). \]
The operator \(A_0\) has a maximal discrete chain
\(\mathfrak P^{(0)}=\{Q_j\}_0^\infty\), and the operator \(A_1\)* has such a maximal chain—

* We consider, for definiteness, only the case when \(\dim\mathfrak H_0=\infty\).
** The whole spectrum of the operator \(A_1\) lies on the real axis \((^8)\).

for which \(\mathfrak P^{(1)}=\{Q\}\), such that for every real \(t\) there is an orthoprojector \(Q_t\in\mathfrak P^{(1)}\) cutting its spectrum at \(t\) \((^4)\). Define the functions \(\alpha_0(Q)\) \((Q\in\mathfrak P^{(0)})\) and \(\alpha_1(Q)\) \((Q\in\mathfrak P^{(1)})\) by setting

\[ \alpha_0(Q_j)\Delta Q_j=\Delta Q_jA\Delta Q_j,\qquad \alpha_1(Q)=\max\sigma[A_1/Q], \]

where \(\sigma[A_1/Q]\) is the spectrum of the restriction of the operator \(QA_1Q\) to the subspace \(Q\mathfrak H_1\).

Theorem 1. If \(A\in\mathfrak R\) and \(A_I\in\mathfrak S_\omega\), then

\[ A=\int_{\mathfrak P}\alpha(P)\,dP+2i\int_{[\mathfrak P]} PA_IdP, \tag{4} \]

where \(\mathfrak P\) is a maximal chain in \(\mathfrak H\), which is the union of the chains \(\mathfrak P_0=\{QP^{(0)}\}\) \((Q\in\mathfrak P^{(0)})\) and \(\mathfrak P_1=\{P^{(0)}+QP^{(1)}\}\) \((Q\in\mathfrak P^{(1)})\),

\[ \alpha(P)= \begin{cases} \alpha_0(Q), & P=QP^{(0)} \qquad (Q\in\mathfrak P^{(0)}),\\ \alpha_1(Q), & P=P^{(0)}+QP^{(1)} \qquad (Q\in\mathfrak P^{(1)}). \end{cases} \]

The first integral in (4) converges strongly, and the second uniformly.

Proof. By Lemma 1 and the main theorem of \((^4)\),

\[ P^{(0)}AP^{(0)}=A_0P^{(0)}=\int_{\mathfrak P_0}\alpha(P)\,dP+2i\int_{[\mathfrak P_0]} PA_IdP, \tag{5} \]

\[ P^{(1)}AP^{(1)}=A_1P^{(1)}=\int_{\mathfrak P_1}\alpha(P)\,dP+2i\int_{\mathfrak P_1}P^{(1)}PA_IdP. \tag{6} \]

The integrals in (6) converge uniformly. From the relations \(P^{(0)}AP^{(1)}=2iP^{(0)}A_IP^{(1)}\), \((P^{(0)}AP^{(1)})^2=0\) it follows that the operator \(P^{(0)}AP^{(1)}\) is Volterra. Since it possesses the chain \(\mathfrak P\), in the sense of uniform convergence \((^3,^9)\),

\[ P^{(0)}AP^{(1)}=\int_{\mathfrak P}PP^{(0)}AP^{(1)}\,dP =2i\int_{\mathfrak P}PP^{(0)}P^{(1)}\,dP =2i\int_{\mathfrak P_1}P^{(0)}PA_IdP. \tag{7} \]

The assertion of the theorem follows from (5), (6), (7) and the equality

\[ A=P^{(0)}AP^{(1)}+P^{(1)}AP^{(1)}+P^{(0)}AP^{(1)}. \]

Formula (4) shows that an operator \(A\in\mathfrak R\) with imaginary component from \(\mathfrak S_\omega\) can be represented as the sum of a normal operator and a Volterra operator possessing one and the same maximal chain.

2. Let us specify an operator node \(\theta=\begin{pmatrix} A&K&J\\ \mathfrak H&\mathfrak G \end{pmatrix}\) and consider its characteristic operator-function \((^{10},{}^{11})\)

\[ W_\theta(\lambda)=I-2iK^*(A-\lambda E)^{-1}KJ. \]

Lemma 2. Let \(\mathfrak S\) be an arbitrary symmetrically normed ideal of the ring \(\mathfrak R_{\mathfrak G}\). If \(K^*K\in\mathfrak S\) and \(T\in\mathfrak R_{\mathfrak H}\), then \(K^*TK\in\mathfrak S\) and
\[ \|K^*TK\|_{\mathfrak S}\le 2\|T\|\,\|K^*K\|_{\mathfrak S}. \]

Lemma 3. If \(A\) possesses a discrete maximal chain \(\mathfrak P=\{P_j\}_0^\infty\) and the operator \(K^*K\) belongs simultaneously to the symmetrically normed ideals \(\mathfrak S_\omega\) and \(\mathfrak S\), then

\[ W_\theta(\lambda)=\prod_{j=1}^{\infty}\left(I+2i\,\frac{K^*\Delta P_jKJ}{\lambda-\lambda_j}\right), \qquad (\Delta P_jA\Delta P_j=\lambda_j\Delta P_j), \tag{8} \]

where the infinite product converges in the norm of the ideal \(\mathfrak S\).

Let an operator \(J\in\mathfrak R_{\mathfrak G}\) satisfy the conditions \(J=J^*\), \(J^2=I\). An operator-function of the complex variable \(W(\lambda)\), whose values belong to \(\mathfrak R_{\mathfrak G}\), will be assigned to the class \(\Omega_J\) if: 1) \(W(\lambda)\) is holomorphic in the domain \(G_W\) obtained by deleting from the extended complex plane some bounded set having no nonreal limit points; 2) \(W(\infty)=I\); 3) \(W^*(\lambda)JW(\lambda)-J\ge0\) \((\operatorname{Im}\lambda>0,\ \lambda\in G_W)\); 4) \(W^*(\lambda)JW(\lambda)-J=0\) \((\operatorname{Im}\lambda=0,\ \lambda\in \overline{G_W})\);

5) all operators \(W(\lambda)-I\) \((\lambda \in G_W)\) are completely continuous. From 3) and 4) it follows that in a neighborhood of the infinitely distant point the function \(W(\lambda)\) expands into a series of the form

\[ W(\lambda)=I+\frac{2i}{\lambda}H_WJ+\ldots \qquad (H_W \geqslant 0). \]

Theorem 2*. Let \(W(\lambda)\in \Omega_J\). If the operator \(H_W\) belongs simultaneously to the symmetrically normed ideals \(\mathfrak{S}_\omega\) and \(\mathfrak{S}\), then

\[ W(\lambda)=\prod_{j=1}^{n}\left(I+\frac{2i}{\lambda-\lambda_j}\,F_jJ\right) \int_{0}^{1}\left(I+\frac{2i}{\lambda-\alpha(x)}\,dF(x)J\right) \qquad (n\leqslant \infty), \tag{9} \]

where \(\lambda_j\) are nonreal numbers; \(\alpha(x)\) is a scalar function, continuous from the left and nondecreasing; \(F_j\) are one-dimensional positive operators satisfying the condition \(F_jJF_j=\operatorname{Im}\lambda_jF_j\); \(F(x)\) \((0\leqslant x\leqslant 1)\) is a positive, strictly increasing, absolutely continuous operator-function whose values belong to the ideal \(\mathfrak{S}\). The integral products

\[ \prod_{k=1}^{m}\left(I+\frac{2i}{\lambda-\alpha(\xi_k)}\,\Delta F_kJ\right) \]

\[ (0=x_0<x_1<\ldots<x_m=1,\; x_{k-1}<\xi_k\leqslant x_k,\quad \Delta F_k=F(x_k)-F(x_{k-1})) \]

converge, in the sense of Shatunovskii, in the norm of the ideal \(\mathfrak{S}\). In the sense of the same norm there converges the product of the factors \(I+\dfrac{2i}{\lambda-\lambda_j}\,F_jJ\) when \(n=\infty\).

Proof. The function \(W(\lambda)\) is the characteristic function for some simple node

\[ \theta=\begin{pmatrix} AKJ \\ \mathfrak{H}\ \mathfrak{G} \end{pmatrix} \]

with completely continuous operator \(K\) \((^5,{}^{10})\). Moreover,
\(W_\theta(\lambda)=W_0(\lambda)W_1(\lambda)\), where \(W_0(\lambda)\) and \(W_1(\lambda)\) are the characteristic operator-functions of the projections of the node \(\theta\) onto the subspaces \(\mathfrak{H}_0\) and \(\mathfrak{H}_1\), introduced in the first section of the article. Applying to the functions \(W_0(\lambda)\) and \(W_1(\lambda)\), respectively, Lemma 3 of the present article and Theorem 1 of article \((^5)\), we obtain formula (9). The convergence of the multiplicative integral in the sense of the norm of the ideal \(\mathfrak{S}\) follows from Lemma 2.

Odessa Institute of National Economy
Odessa Pedagogical Institute
named after K. D. Ushinskii

Received
20 X 1967

REFERENCES

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* Multiplicative representations of functions of the class \(\Omega_J\) \((H_W\in\mathfrak{S}_1)\), where \(\mathfrak{S}_1\) is the ideal of all nuclear operators, were studied in detail by Yu. P. Ginzburg by purely analytic methods. In particular, for this case he established Theorem 2 as well \((^{12,13})\).