MATHEMATICS

Yu. P. GINZBURG

ON THE FACTORIZATION OF ANALYTIC MATRIX-FUNCTIONS

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

1. We shall consider matrix-functions \(X(\xi)\), regular for \(|\xi|<1\), of finite order \(n\), with determinant not identically equal to zero. We shall say that such a matrix-function belongs to the class \(A^{(n)}\) if

\[ \sup_{0\leq \rho<1}\int_{0}^{2\pi}\ln^{+}\|X(\rho e^{i\vartheta})\|\,d\vartheta<\infty, \]

to the class \(D^{(n)}\), if the family of functions

\[ I_\rho(\varphi)=\int_{0}^{\varphi}\ln^{+}\|X(\rho e^{i\vartheta})\|\,d\vartheta \qquad (0\leq \rho<1,\quad 0\leq \varphi\leq 2\pi) \]

is uniformly absolutely continuous, to the class \(H_\delta^{(n)}\) \((0<\delta<\infty)\), if

\[ \sup_{0\leq \rho<1}\int_{0}^{2\pi}\|X(\rho e^{i\vartheta})\|^\delta\,d\vartheta<\infty, \]

to the class \(B^{(n)}\), if \(\sup_{|\xi|<1}\|X(\xi)\|<\infty\), and, finally, to the class \(\widetilde{B}^{(n)}\), if \(\|X(\xi)\|\leq 1\) for \(|\xi|<1\).

It is not hard to see that

\[ A^{(n)}\supset D^{(n)}\supset H_{\delta'}^{(n)}\supset H_{\delta''}^{(n)}\supset B^{(n)}\supset \widetilde{B}^{(n)} \qquad (\delta'<\delta''), \]

and that the matrix-function \(X(\xi)\) belongs to the class \(A^{(n)}\), \(D^{(n)}\), \(H_\delta^{(n)}\) \((\delta>0)\), \(B^{(n)}\) if and only if its elements belong to the corresponding scalar class \((^1)\).

In the present note we give “parametric representations” of the matrix classes \(A^{(n)}\), \(D^{(n)}\), \(H_\delta^{(n)}\), \(B^{(n)}\), \(\widetilde{B}^{(n)}\), analogous to those known \((^{1,2})\) for the scalar case. In proving them we make essential use of methods developed by V. P. Potapov in the works \((^{3,4})\), and in particular rely on his fundamental result concerning matrix-functions of the class \(\widetilde{B}^{(n)}\) (a particular case of the theorem concerning matrix-functions bounded in the indefinite metric \((^4)\)):

A matrix-function \(X(\xi)\) belongs to the class \(\widetilde{B}^{(n)}\) if and only if

\[ X(\xi)=\Pi(\xi)\int_{0}^{l}\exp\{k[\xi,\vartheta(t)]\}\,dE(t)\,U, \tag{1} \]

where \(\Pi(\xi)\) is a product of Blaschke type, computed by the formula

\[ \Pi(\xi)=\prod_{j=1}^{r}\left[\frac{\xi_j-\xi}{1-\bar{\xi}_j\xi}\,\frac{|\xi_j|}{\xi_j}\,P_j+(I-P_j)\right] \tag{2} \]

\[ (r\leq\infty,\quad |\xi_j|<1,\quad P_j\text{ are orthoprojections}); \]
\(\vartheta(t)\) is a nondecreasing scalar function \((0\leq\vartheta(t)\leq 2\pi)\), \(E(t)\) is a Hermitian increasing matrix-function \((\operatorname{sp}E(t)=t)\), \(U\) is a constant unitary matrix,

\[ k(\xi,\vartheta)=\frac{\xi+e^{i\vartheta}}{\xi-e^{i\vartheta}}. \]

2. The following theorem gives “parametric representations” of the matrix classes \(A^{(n)}\), \(D^{(n)}\), \(H_\delta^{(n)}\), \(B^{(n)}\), \(\widetilde{B}^{(n)}\).

Theorem 1. The matrix-function \(X(\zeta)\) belongs to the class \(A^{(n)}\) if and only if the representation

\[ X(\zeta)=\Pi(\zeta)\cdot \prod_{j=1}^{s} \left[ \int_{0}^{l_j}\exp\{k(\zeta,\vartheta_j)\,dE_j(t)\} \right]\times \]

\[ {}\times \int_{0}^{2\pi}\exp\{k(\zeta,\vartheta)\,dS(\vartheta)\}\, U \int_{0}^{2\pi}\exp\{k(\zeta,\vartheta)\,M(\vartheta)\,d\vartheta\}, \tag{3} \]

holds, where \(\Pi(\zeta)\) is a product of the Blaschke type (2), \(s\leqslant\infty\), \(0\leqslant\vartheta_j\leqslant 2\pi\), \(E_j(t)\) and \(S(\vartheta)\) are Hermitian matrix-functions of bounded variation, with \(S(\vartheta)\) a continuous singular function, and \(M(\vartheta)\) a Hermitian summable matrix-function.

The function \(X(\zeta)\) belongs: 1) to the class \(D^{(n)}\), 2) to the class \(H_\delta^{(n)}\), 3) to the class \(B^{(n)}\) if and only if \(X(\zeta)\) admits representation (3) with parameters satisfying, in addition to those listed above, the following conditions:

1) \(E_j(t)\) and \(S(\vartheta)\) are Hermitian increasing matrix-functions;

2) \(E_j(t)\) and \(S(\vartheta)\) are Hermitian increasing and

\[ \int_{0}^{2\pi} e^{-2\pi\delta\lambda(\vartheta)}\,d\vartheta<\infty, \]

where \(\lambda(\vartheta)\) is the smallest eigenvalue of the matrix \(M(\vartheta)\);

3) \(E_j(t)\) and \(S(\vartheta)\) are Hermitian increasing, and \(\lambda(\vartheta)\) is a function bounded below (in particular, \(M(\vartheta)\geqslant 0\) for the class \(\widetilde B^{(n)}\)).

We indicate the main stages of the proof of this theorem. Using the fact that every function \(X(\zeta)\in\widetilde B^{(n)}\) can, inside the unit disk, be uniformly approximated by finite products of Blaschke type (see \((^4)\), p. 133), and also using representation (1), one can prove the validity of the following proposition:

Lemma. If \(X_j(\zeta)\in\widetilde B^{(n)}\) \((j=1,2)\), then there exist such \(Y_j(\zeta)\in\widetilde B^{(n)}\) that

\[ X_1(\zeta)X_2(\zeta)=Y_2(\zeta)Y_1(\zeta) \]

and

\[ \det X_1(\zeta)=\det Y_1(\zeta),\qquad \det X_2(\zeta)=\det Y_2(\zeta). \]

Relying on the lemma, as well as on the equality

\[ \det\int_{0}^{l}\exp\{f(t)\,dG(t)\} = \exp\left\{\int_{0}^{l} f(t)\,d[\operatorname{sp}G(t)]\right\}, \tag{4} \]

one can show that

\[ I(\zeta)\equiv \int_{0}^{l}\exp\{k[\zeta,\vartheta(t)]\,dE(t)\}= \]

\[ = \prod_{j=1}^{s} \left[ \int_{0}^{l_j}\exp\{k(\zeta,\vartheta_j)\,dE_j(t)\} \right]\cdot \int_{0}^{l'}\exp\{k[\zeta,\widetilde\vartheta(t)]\,d\widetilde E(t)\}U. \tag{5} \]

Here \(U\) is a constant unitary matrix; \(s(\leqslant\infty)\) is the number of intervals on which the function \(\vartheta(t)\) is constant; \(\vartheta_j\) is the value of \(\vartheta(t)\) on the \(j\)-th such interval; \(l_j\) is the length of this interval; \(E_j(t)\) and \(\widetilde E(t)\) are Hermitian increasing matrix-functions; \(\widetilde\vartheta(t)\) is a strictly increasing scalar function. The latter circumstance makes it possible, in the multiplicative integral occurring on the right in (5), to make a change of variable and write it in the form

\[ \int_{0}^{2\pi}\exp\{k(\zeta,\vartheta)\,d\Sigma(\vartheta)\}, \]

where \(\Sigma(\vartheta)\) is a continuous, nondecreasing Hermitian matrix function on \([0,2\pi]\).* Representing \(\Sigma(\vartheta)\) as the sum of a singular and an absolutely continuous function, and using the lemma, equality (4), and also certain known estimates (see (4), p. 229) for the multiplicative integral, we obtain a proof of the theorem for the class \(\widetilde B^{(n)}\), and hence also for the class \(B^{(n)}\). In order now to obtain a proof of representation (3) for the class \(A^{(n)}\), it is enough to note that \(X(\zeta)\in A^{(n)}\) if and only if \(X(\zeta)=y^{-1}(\zeta)Y(\zeta)\), where \(y(\zeta)\) is a scalar function of class \(\widetilde B\), not vanishing for \(|\zeta|<1\), and \(Y(\zeta)\in \widetilde B^{(n)}\).

From the theorem on the parametric representation of the scalar class \(D\) (1) it follows that, for \(X(\zeta)\) to belong to the class \(D^{(n)}\), it is necessary and sufficient that
\[ X(\zeta)=y^{-1}(\zeta)Y(\zeta),\quad \text{where } Y(\zeta)\in \widetilde B^{(n)},\quad y(\zeta)=\exp\left\{\int_0^{2\pi} k(\zeta,\vartheta)m(\vartheta)\,d\vartheta\right\} \]
(\(m(\vartheta)\) is a scalar function summable on \([0,2\pi]\)). Hence the validity of Theorem 1 for the class \(D^{(n)}\) follows.

As for the proof of the theorem in the part relating to the classes \(H_\delta^{(n)}\), it is carried out using the equality
\[ \|X(e^{i\vartheta})\|=\exp\{-2\pi\lambda(\vartheta)\}, \tag{6} \]
valid almost everywhere on \([0,2\pi]\) for functions \(X(\zeta)\in A^{(n)}\) represented by formula (3). Equality (6), which is the source of criteria for functions from \(D^{(n)}\) to belong to one or another narrower class, can be established with the help of Theorem 3 below and theorem (1) on the recovery of a maximal scalar function of class \(D\) from the moduli of its boundary values.**

1. A matrix function of class \(D^{(n)}\) is called inner if its boundary values on the unit circle are unitary almost everywhere. A function \(X(\zeta)\in D^{(n)}\) is called outer if, for every function \(Y(\zeta)\in D^{(n)}\) satisfying almost everywhere on \([0,2\pi]\) the relation
   \[ Y^*(e^{i\vartheta})Y(e^{i\vartheta})=X^*(e^{i\vartheta})X(e^{i\vartheta}), \tag{7} \]
   the inequality \(Y^*(\zeta)Y(\zeta)\le X^*(\zeta)X(\zeta)\) holds for \(|\zeta|<1\). It is obvious that if (7) holds for two outer functions \(X(\zeta)\) and \(Y(\zeta)\) from \(D^{(n)}\), then \(Y(\zeta)=UX(\zeta)\) (\(U\) is a constant unitary matrix).

Theorem 2. A matrix function \(X(\zeta)\in D^{(n)}\) is inner (outer) if and only if, in its representation (3), \(M(\vartheta)\equiv 0\) (respectively \(\Pi(\zeta)\equiv 1,\ E_j(t)=\mathrm{const},\ S(\vartheta)=\mathrm{const}\)).

The following proposition is used in one of its parts (together with the fact that boundedness of a function from \(D\) on the unit circle implies its boundedness for \(|\zeta|<1\)) for the proof of Theorem 2, while in another part it is obtained as its consequence:

In order that \(X(\zeta)\,(\in D^{(n)})\) be inner (outer), it is necessary and sufficient that its determinant \(\det X(\zeta)\) be an inner (outer) scalar function.

The theorem below on the uniqueness of the multiplicative representation of an outer function generalizes and refines the corresponding result of L. A. Sakhnovich (8).

* We note that P. Masani (5,6) made an error in changing variables in the integral \(I(\zeta)\) with a function \(\vartheta(t)\) which, generally speaking, has intervals of constancy, and as a consequence arrived at an incorrect result.

** In the case of boundedness of \(M(\vartheta)\) on \([0,2\pi]\), formula (6) follows directly from L. A. Sakhnovich’s theorem (7) on the boundary values of the multiplicative integral.

Theorem 3. If

\[ X_j(\xi)=\int_0^{2\pi}\exp\{k(\xi,\vartheta)M_j(\vartheta)\,d\vartheta\} \qquad (j=1,2;\ |\xi|<1), \]

where \(M_j(\vartheta)\) are Hermitian matrix functions summable on \([0,2\pi]\), and

\[ X_1^*(e^{i\vartheta})X_1(e^{i\vartheta}) = X_2^*(e^{i\vartheta})X_2(e^{i\vartheta}) \]

almost everywhere on \([0,2\pi]\), then \(M_1(\vartheta)=M_2(\vartheta)\) almost everywhere on \([0,2\pi]\).

For the proof, consider the matrix functions

\[ X_j(\xi;t)=\int_t^{2\pi}\exp\{k(\xi,\vartheta)M_j(\vartheta)\,d\vartheta\} \qquad (j=1,2;\ t\in[0,2\pi]). \]

It is not difficult to see that

\[ X_1^*(e^{i\vartheta};t)X_1(e^{i\vartheta};t) = X_2^*(e^{i\vartheta};t)X_2(e^{i\vartheta};t) \]

for each \(t\) and almost all \(\vartheta\) in \([0,2\pi]\). Since, on the basis of Theorem 2, the functions \(X_j(\xi;t)\in D^{(n)}\) are outer, it follows that

\[ X_2(\xi;t)=U(t)X_1(\xi;t) \tag{8} \]

(\(U(t)\) is a unitary matrix for each \(t\in[0,2\pi]\)). We now note that

\[ X_j(0;t)=\int_t^{2\pi}\exp\{-M_j(\vartheta)\,d\vartheta\}, \qquad (M_j^*(\vartheta)=M_j(\vartheta)), \]

and, as V. P. Potapov showed (see \((^4)\), p. 173), in this case the family of matrices \(M_j(t)\)* is uniquely recovered from the family of moduli

\[ R_j(t)=\sqrt{X_j^*(0;t)X_j(0;t)}. \]

Since it follows from (8) that \(R_1(t)=R_2(t)\) (\(t\in[0,2\pi]\)), the theorem is proved.

1. The exposition above, although formally not relying on the theorem on optimal factorization in the class \(H_2^{(n)}\) of summable positive matrix functions, first proved by M. G. Krein and presented in the works of Yu. A. Rozanov \((^{9-11})\), has points of contact with it. If this theorem is used, it is not difficult to show that:

A Hermitian positive matrix \(F(\vartheta)\) almost everywhere on \([0,2\pi]\) is the modulus of boundary values of functions of the classes \(A^{(n)}\) and \(D^{(n)}\) if and only if

\[ \left|\int_0^{2\pi}\ln\|F(\vartheta)\|\,d\vartheta\right|<\infty, \tag{9} \]

\[ \left|\int_0^{2\pi}\ln\|F^{-1}(\vartheta)\|\,d\vartheta\right|<\infty. \tag{10} \]

Analogous assertions may be formulated for the classes \(H_\delta^{(n)}\) \((0<\delta<\infty)\) and \(B^{(n)}\), replacing condition (9), respectively, by the conditions

\[ \int_0^{2\pi}\|F(\vartheta)\|^\delta\,d\vartheta<\infty, \qquad \sup_{0\le \vartheta\le 2\pi}\|F(\vartheta)\|<\infty. \]

In conclusion, I express my gratitude to D. Z. Arov, M. G. Krein, and V. P. Potapov for useful discussions of the results of this note.

Odessa Pedagogical Institute
named after K. D. Ushinsky

Received
28 V 1964

REFERENCES

1. I. I. Privalov, Boundary properties of analytic functions, 1950.
2. K. Hoffman, Banach spaces of analytic functions, 1963.
3. V. P. Potapov, DAN, 72, No. 5 (1950).
4. V. P. Potapov, Tr. Moscow Math. Soc., 4, 125 (1955).
5. R. Masani, C. R., 249, 906 (1959).
6. R. Masani, C. R., 251, 318 (1960).
7. L. A. Sakhnovich, UMN, 12, issue 3 (75), 205 (1957).
8. L. A. Sakhnovich, Tr. III All-Union Math. Congress, 1, 1956, p. 120.
9. Yu. A. Rozanov, Tr. V All-Union Conf. on Functional Analysis, Baku, 1961, p. 122.
10. Yu. A. Rozanov, UMN, 13, issue 2 (80), 93 (1958).
11. Yu. A. Rozanov, Stationary random processes, 1963.

* In \((^4)\) the arguments are carried out for the case \(JM_j(\vartheta)\ge 0\), but in the part of interest to us they rely essentially only on the \(J\)-Hermiticity of \(M_j(\vartheta)\) \((JM_j(\vartheta)=M_j^*(\vartheta)J)\).