G. E. KISILEVSKII

ON THE ORDEREDNESS OF CHARACTERISTIC MATRIX-FUNCTIONS OF DISSIPATIVE VOLTERRA OPERATORS

(Presented by Academician L. S. Pontryagin, 2 VI 1964)

Let us consider the class \(\Omega_2^+\) of all simple dissipative Volterra operators with two-dimensional imaginary component, acting in a separable Hilbert space \(\mathfrak H\), and let \(K_2^+\) be the set of the corresponding characteristic matrix-functions. By virtue of \((^1)\), the one-cell character of an operator \(A \in \Omega_2^+\) is equivalent to the orderedness of its characteristic matrix-function \(W(z) \in K_2^+\). In the present paper, conditions are established for the orderedness of matrix-functions from \(K_2^+\).

1. Every matrix-function \(W(z) \in K_2^+\) can be represented \((^2)\) in the form of a multiplicative Stieltjes integral

\[ W(z)=\int_0^l e^{2iz\,dH(t)} \quad \left( \dot H(t)=\int_0^t P(x)\,dx,\quad P(x)\geq 0,\quad \operatorname{sp}P(x)=1 \right), \tag{1} \]

where it may be assumed \((^3)\) that the rank of the matrix-function \(P(t)\) is equal to 1 almost everywhere. Representation (1) is unique if and only if the matrix-function \(W(z)\) is ordered. It follows from \((^3)\) that every matrix-function \(W(z)\in K_2^+\) is uniquely representable in canonical form

\[ W(z)=e^{im_0z I}\int_0^{l_0} e^{2iz\,dH_0(t)} = e^{im_0z I} W_0(z),\quad \text{where}\quad H_0(t)=\int_0^t P_0(x)\,dx; \tag{2} \]

\(P_0(x)\) is a Hermitian nonnegative matrix-function with summable elements, whose values are projection matrices of rank 1; \(m_0=2l-\sigma\), where \(2l\) is the weight and \(\sigma\) is the type of the matrix-function \(W(z)\). Here the matrix-function \(W_0(z)\) is equal to \(I\) for \(\sigma=l\) and is ordered if \(\sigma>l\). Since the scalar matrix-function \(e^{im_0z I}\) \((m_0>0)\) is not ordered \((^3)\), for the matrix-function \(W(z)\) to be ordered it is necessary and sufficient that it have no scalar divisors. We shall call an ordered divisor\(^*\) of the matrix-function \(W(z)\) maximal if its type is equal to the type \(\sigma\) of the matrix-function \(W(z)\). In \((^4)\) the existence of maximal ordered divisors was established and it was proved that every such divisor has the form \(W_3(z)W_0(z)\), where \(W_3(z)\) is some matrix-function from \(K_2^+\) of weight \(m_0\). Hence follows

Lemma 1. If two maximal ordered divisors of \(W(z)\) have no common divisors, then the matrix-function \(W(z)\) is scalar.

Lemma 2. If

\[ W_1(z)=\int_0^{l_1} e^{2iz\,dH_1(t)} \left( H_1(t)=\int_0^t P_1(x)\,dx \right) \]

is an ordered divisor of the matrix-function \(W(z)\) and \(2l_1>m_0\), then \(P_1(t)\equiv P_0(t)\) \((0\leq t\leq l_1-m_0/2)\).

Proof. We shall first prove that the matrix-functions \(W_1(z)\) and \(W_0(z)\) have a common divisor. Assuming the contrary, take a maximal ordered divisor of the matrix-function \(W(z)\) and consider its divisor \(W_2(z)\) of weight \(2l_1\). Then, as is easy to show, the matrix-functions \(W_1(z)\) and \(W_2(z)\) also have no common divisors. Let the matrix-function \(W(z)\) be characteristic for some operator \(A\in\Omega_2^+\). There exist \((^2)\) invariant subspaces of the operator \(A\) relative to the operator \(A\) corres—

\(^*\) According to \((^2)\), a divisor of \(W(z)\) is its right divisor.

spaces \(\mathfrak H_1\) and \(\mathfrak H_2\), so that \(W_k(z)=\operatorname{Pr}_{\mathfrak H_k} W(z)\) \((k=1,2)\). Denote by \(\mathfrak G_0\) the smallest invariant subspace of the operator \(A\) containing \(\mathfrak H_1\) and \(\mathfrak H_2\), and let \(V_0(z)=\operatorname{Pr}_{\mathfrak G_0} W(z)\). We shall show that the type \(\sigma_0\) of the matrix function \(V_0(z)\) is equal to \(2l_1\). Indeed, since the matrix functions \(W_k(z)\) \((k=1,2)\) are ordered and are divisors of \(V_0(z)\), it follows that \(\sigma_0 \geqslant 2l_1\). On the other hand, in \((^2)\) it was proved that \(\sigma_0 \leqslant 2l_1\). Thus, \(W_1(z)\) and \(W_2(z)\) are maximal ordered divisors of the matrix function \(V_0(z)\). Therefore, by Lemma 1, the matrix function \(V_0(z)\) must be scalar, that is, \(V_0(z)=e^{2il_1 z}I\), which is impossible, since \(V_0(z)\) is a divisor of \(W(z)\) and \(2l_1>m_0\). In view of the orderedness of the matrix functions \(W_1(z)\) and \(W_0(z)\), \(P_1(t)\equiv P_0(t)\) on some interval of the form \([0,a]\) \((a>0)\). Let \([0,a_0]\) be the largest interval on which the matrix functions \(P_1(t)\) and \(P_0(t)\) coincide. Then it is not difficult to prove that \(a_0 \geqslant l_1-m_0/2\). We note that an assertion analogous to Lemma 2 also holds for left divisors.

Lemma 3. Let the matrix functions
\[ W_k(z)=\int_0^a e^{2iz\,dH_k(t)} \quad \left(H_k(t)=\int_0^t P_k(x)\,dx,\quad k=1,2\right) \]
be ordered. If
\[ W_2(z)W_1(z)=e^{2iaz}I, \]
then
\[ P_2(t)P_1(a-t)\equiv 0. \]

Theorem 1. If the matrix functions
\[ W_k(z)=\int_0^{l_k} e^{2iz\,dH_k(t)} \quad \left(H_k(t)=\int_0^t P_k(x)\,dx,\quad k=1,2,\ldots,n\right) \]
are ordered, then, for the orderedness of the matrix function
\[ W(z)=W_n(z)\cdots W_1(z), \]
it is necessary and sufficient that, for each \(k=1,\ldots,n-1\), the measure of the set of all \(t\) from the interval \((0,\delta)\) for which
\[ P_{k+1}(t)P_k(l_k-t)=0 \]
be positive for every \(\delta>0\).

Proof. It is sufficient to prove the theorem for \(n=2\). Suppose that the matrix function \(W(z)\) is not ordered. Representing it in canonical form and using Lemma 2, we obtain
\[ P_1(t)\equiv P_0(t)\quad (0\leqslant t\leqslant l_1-m_0/2); \]
\[ P_2(l_2-t)\equiv P_0(l_0-t)\quad (0\leqslant t\leqslant l_2-m_0/2), \]
and since
\[ (l_1-m_0/2)+(l_2-m_0/2)=l_1+l_2-m_0=l_0, \]
it follows that
\[ \int_0^{m_0/2} e^{2iz\,dH_2(t)} \times \int_{l_1-m_0/2}^{l_1} e^{2iz\,dH_1(t)} = e^{im_0z}I, \]
whence, by Lemma 3,
\[ P_2(t)P_1(l_1-t)\equiv 0 \quad (0\leqslant t\leqslant m_0/2). \]
If
\[ P_2(t)P_1(l_1-t)\equiv 0 \quad (0\leqslant t\leqslant \delta_0), \]
then, as is easy to show, the matrix function \(W(z)\) has a scalar divisor and, consequently, is not ordered.

2. Lemma 4. Let
\[ W(z)=\int_{x_0}^{x} e^{2iz\,dH(t)} \quad \left(H(t)=\int_{x_0}^{t} P(\xi)\,d\xi\right). \]
If
\[ \sup_{t',t''\in[x_0,x]}\|P(t')-P(t'')\|=\omega, \]
then
\[ \|W(iy)\|\geqslant e^{-2y(1-\omega)\Delta x} \quad (\Delta x=x-x_0,\ y<0). \]

Lemma 5. Let
\[ W_k(z)=\int_0^l e^{2iz\,dH_k(t)} \quad \left(H_k(t)=\int_0^t P_k(x)\,dx,\quad k=1,2\right). \]
If
\[ mE(P_1\ne P_2)\leqslant \varepsilon, \]
then
\[ \|W_1(z)-W_2(z)\|\leqslant 2\varepsilon |z|e^{2l|z|}. \]

Lemma 6. Let
\[ W(z)=\int_{0}^{l} e^{2iz\,dH(t)},\qquad W_k(z)=\int_{t_{k-1}}^{t_k} e^{2iz\,dH(t)} \quad (0=t_0<t_1<\cdots<t_n=l), \]
\(\sigma\) be the type of the matrix-function \(W(z)\), and \(\sigma_k\) the type of the matrix-function \(W_k(z)\). If for no \(x_1,x_2\in[0,l]\) is the matrix-function
\[ \int_{x_1}^{x_2} e^{2iz\,dH(t)} \]
scalar, then
\[ \sigma=\sum_{k=1}^{n}\sigma_k . \]

Theorem 2. If for no \(x_1,x_2\in[0,l]\) is the matrix-function
\[ \int_{x_1}^{x_2} e^{2iz\,dH(t)} \]
scalar, then the matrix-function
\[ W(z)=\int_{0}^{l} e^{2iz\,dH(t)} \]
is ordered.

Proof. Let \(\varepsilon\) be an arbitrary positive number \((\varepsilon<1/2,\ l/2)\). Using the well-known theorem of N. N. Luzin, construct a projection matrix-function \(P_1(t)\), continuous on the segment \([0,l]\), for which
\[ mE(P\ne P_1)\le \varepsilon^2, \]
and let
\[ W_1(z)=\int_{0}^{l} e^{2iz\,dH_1(t)} \]
\[ \left(H_1(t)=\int_{0}^{t} P_1(x)\,dx\right). \]
Choose numbers \(\delta>0\) and \(N>l/\delta\) so that the inequalities
\[ \|P_1(t')-P_1(t'')\|<\varepsilon \quad (|t'-t''|<\delta,\ t',t''\in[0,l]); \tag{3} \]
\[ \varepsilon l/n<e^{-2\varepsilon l/n}-e^{-4\varepsilon l/n} \quad (n>N), \tag{4} \]
hold, and consider the partition of the segment \([0,l]\) into \(n\) equal parts \((n>N)\) by the points
\[ t_k=kl/n\quad (k=0,1,\ldots,n). \]
Introducing the notation
\[ W_k(z)=\int_{t_{k-1}}^{t_k} e^{2iz\,dH(t)},\qquad W_k^{(1)}(z)=\int_{t_{k-1}}^{t_k} e^{2iz\,dH_1(t)},\qquad E_k=E\cap[t_{k-1},t_k],\quad \varepsilon_k=mE_k, \]
and using Lemmas 4 and 5, we obtain the estimates
\[ \|W_k^{(1)}(-i)\|\ge e^{(1-\varepsilon)2l/n},\qquad \|W_k(-i)-W_k^{(1)}(-i)\|\le 2\varepsilon_k e^{2l/n}. \tag{5} \]
Let \(\varepsilon_{k_j}<\varepsilon l/2n\) for \(j=1,2,\ldots,s\), and \(\varepsilon_{k_j}\ge \varepsilon l/2n\) for \(j=s+1,\ldots,n\). Since
\[ \varepsilon^2\ge \sum_{j=s+1}^{n}\varepsilon_{k_j} \ge \frac{\varepsilon l}{2n}(n-s), \]
it follows that
\[ s\ge \left(1-\frac{2\varepsilon}{l}\right)n. \]
Putting \(j=1,2,\ldots,s\) and using inequalities (4) and (5), we obtain
\[ \|W_{k_j}(-i)\|\ge \|W_{k_j}^{(1)}(-i)\| -\|W_{k_j}^{(1)}(-i)-W_{k_j}(-i)\| \ge e^{(1-\varepsilon)2l/n}-2\varepsilon_{k_j}e^{2l/n} \]
\[ >e^{(1-2\varepsilon)2l/n} \left[e^{2\varepsilon l/n}-\frac{\varepsilon l}{n}e^{4\varepsilon l/n}\right] >e^{(1-2\varepsilon)2l/n}, \]
whence it follows* that
\[ \sigma_{k_j}\ge (1-2\varepsilon)\frac{2l}{n} \quad (j=1,2,\ldots,s), \]
where \(\sigma_{k_j}\) is the type of the matrix-function \(W_{k_j}(z)\). By Lemma 6,
\[ \sigma=\sum_{j=1}^{n}\sigma_{k_j} \ge \sum_{j=1}^{s}\sigma_{k_j} \ge (1-2\varepsilon)\frac{2l}{n} \left(1-\frac{2\varepsilon}{l}\right)n =2l(1-2\varepsilon)\left(1-\frac{2\varepsilon}{l}\right), \]
and since, by (2), \(\sigma\le 2l\), and \(\varepsilon>0\) is arbitrary, \(\sigma=2l\), and the orderedness of the matrix-function \(W(z)\) follows from M. S. Brodskii’s criterion (2).

1. Consider the canonical representation (2) of the unordered matrix-function
   \[ W(z)=\int_{0}^{l} e^{2iz\,dH(t)}. \]
   Using Theorem 2, one can construct

* Since the matrix-function \(W_{k_j}(z)\) is unitary on the real axis, it follows from (3) that for all \(y<0\)
\[ \|W_{k_j}(x+iy)\|\le e^{-\sigma_{k_j}y}. \]

(a finite or infinite) system of pairwise nonintersecting intervals \(\Delta_k=(a_k,b_k)\), possessing the following properties:

\[ 1)\ \sum_k (b_k-a_k)=m_0;\qquad 2)\ \int_{a_k}^{b_k} e^{2izt}\,dH(t)=e^{i(b_k-a_k)z}I . \]

Let \(F\) be a set of positive measure belonging to the segment \([0,l]\), \(\mu(t)=m(F\cap[0,t])\). Introduce the notation \(P_F(t)=P(\nu(t))\), where \(\nu(t)\) is the function inverse to \(\mu(t)\).

Theorem 3. If \(F_0\) is the set obtained from the segment \([0,l]\) by discarding all the intervals \(\Delta_k\), then \(mF_0=l_0\) and \(P_{F_0}(t)\equiv P_0(t)\).

We shall say that the matrix-function \(P(t)\) has an orthogonally symmetric structure on the segment \([a,b]\) if there exist pairwise nonintersecting intervals \(\Delta_{mk}=(a_{mk},b_{mk})\subset [a,b]\) \((k=1,2,\ldots,n,\ n\ge 1;\ m=1,2)\), satisfying the conditions: 1) \(b_{1k}-a_{1k}=b_{2k}-a_{2k}=d_k\)

\[ (k=1,2,\ldots,n),\qquad \sum_{k=1}^n 2d_k=b-a; \]

2) between \(\Delta_{1k}\) and \(\Delta_{2k}\) there are no intervals \(\Delta_{ms}\) \((s>k)\); 3) \(P(a_{1k}+t)P(b_{2k}-t)\equiv 0\) \((0\le t\le d_k,\ k=1,2,\ldots,n)\).

Theorem 4. In order that the matrix-function

\[ W(z)=\int_a^b e^{2iz\,dH(t)} \qquad \left( H(t)=\int_0^t P(x)\,dx,\ 0\le a<b\le l \right) \]

be scalar, it is necessary and sufficient that there exist a sequence of sets \(F_1\subseteq F_2\subseteq\cdots\subseteq[a,b]\) such that \(\lim_{k\to\infty} mF_k=b-a\), and each of the matrix-functions \(P_{F_k}(t)\) has an orthogonally symmetric structure on the segment \([0,mF_k]\).

1. In the Hilbert space \(\mathcal L^2(0,l)\) consider the integral operator

\[ Af=2i\int_x^l f(t)\xi(t)\,dt\,\xi^*(x) \quad \bigl(\xi(x)=\|\varphi_1(x),\varphi_2(x)\|,\ \varphi_k\in\mathcal L^2(0,l), \]

\[ \xi(x)\xi^*(x)\equiv 1\bigr). \]

The operator \(A\) belongs to the class \(\Omega_2^{+}\), and the matrix-function

\[ W(z)=\int_0^l e^{2iz\,dH(t)} \qquad \left( H(t)=\int_0^t P(x)\,dx,\ P(x)=\xi^*(x)\xi(x) \right) \]

is characteristic for it. Let us note criteria for the unicellularity of the operator \(A\) that follow from the results obtained above.

Theorem 5. If the functions \(\varphi_k(x)\) \((k=1,2)\) are continuous on the segment \([0,l]\), then the operator \(A\) is unicellular.

Theorem 6. If the functions \(\varphi_k(x)\) \((k=1,2)\) are piecewise continuous on the segment \([0,l]\), then for the unicellularity of the operator \(A\) it is necessary and sufficient that, for each point of discontinuity \(t_{ki}\) of the function \(\varphi_k(x)\), the measure of the set of all \(t\) from the interval \((0,\delta)\) for which \(\xi(t_{ki}-t)\xi^*(t_{ki}+t)\ne 0\) be positive for every \(\delta>0\).

Theorem 7. If in some neighborhood of each point \(t\in(0,l)\) the vector-function \(\xi(t)\) has no mutually orthogonal values, then the operator \(A\) is unicellular.

Zhitomir Pedagogical Institute
named after I. Franko

Received
28 V 1964

REFERENCES

1. M. S. Brodskii, DAN, 138, No. 3, 512 (1961).
2. M. S. Brodskii, M. S. Livshits, UMN, 13, issue 1 (79), 3 (1958).
3. G. E. Kiselevskii, DAN, 159, No. 3 (1964).
4. G. E. Kiselevskii, Proceedings of the Republican Scientific Conference of Young Researchers, Kiev, 1964.