GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 5, 1997, 439-442

ON FACTORIZATION AND PARTIAL INDICES OF UNITARY MATRIX-FUNCTIONS OF ONE CLASS

G. JANASHIA AND E. LAGVILAVA

Abstract. An effective factorization and partial indices are found for a class of unitary matrix functions.

Let R denote a normed ring of functions defined on the unit circle of a complex plane, say, a ring H^{α} of Hölder functions with a usual norm, 0 < α < 1, which can be decomposed into the direct sum of its subrings R = R^{+} + R_{0}^{−}, where the elements R^{+} are the boundary values of analytic functions defined within the unit circle, and the elements R_{0}^{−} are the boundary values of analytic functions defined outside the unit circle and vanishing at infinity. Also, let R^{−} = R_{0}^{−} + ℂ, where ℂ denotes the ring of complex numbers. M_{q}(R) will denote the ring of square q × q matrix-functions with entries from R. It is well known that the invertible matrix-function G(t) ∈ M_{q}(R) is factored as G(t) = G^{+}(t)D(t)G^{−}(t), where G^{±}(t) ∈ M_{q}(R^{±}), (G^{±}(t))^{−1} ∈ M_{q}(R^{±}), and D(t) = ∥d_{ij}(t)∥ is a diagonal matrix with entries d_{ii}(t) = t^{n_{i}}. According to Muskhelishvili, integers n_{1}, n_{2}, …, n_{q}, are called partial indices G(t) which can be used to determine the number of linearly independent solutions of the corresponding homogeneous singular integral equation [1].

By a polar decomposition of an arbitrary invertible matrix-function G(t)= S(t)U(t) into a positive definite factor S(t) and a unitary factor U(t) and by using the factorization type for positive definite matrix-functions (S(t))^{2} = S^{+}(t)(S^{+}(t))^{∗} = SU_{1}U_{1}^{∗}S [2], [3] we see, in particular, that partial indices for positive matrices are equal to zero. Thus, taking into account the equality G(t) = SU_{1}U_{1}^{−1}U = S^{+}U_{2}, where (S^{+})^{±1} ∈ M_{q}(R^{+}) and U_{2} is a unitary matrix, we can say (at least formally) that the general problem of finding partial indices is reducible to the problem of finding such indices for unitary matrix-functions.

439

1991 Mathematics Subject Classification. 47A68, 35Q15.

Key words and phrases. Factorization of matrix function, partial indexes.

1072-947X/97/0900-0439$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

Since partial indices are unstable in general [4], it is interesting to select classes of unitary matrix-functions with zero partial indices. In this connection we note the following

Proposition. Partial indices of a unitary matrix-function U(t) ∈ M_{q}(R) with det U(t) = 1 are equal to zero if and only if there exists a positive definite matrix-function S(t) such that S(t)U(t) ∈ M_{q}(R^{+}).

The sufficiency immediately follows from

U(t) = S^{−1}(t)S(t)U(t) = Y^{+}(t)^{∗}Y^{+}(t)(S(t)U(t) and U = (U^{∗})^{−1}.

The necessity follows from the fact that if

U(t) = U^{+}(t)U^{−}(t) = (U^{+}(t)^{∗})^{−1}(U^{−}(t)^{∗})^{−1} = U_{1}^{−}U_{1}^{+} = S_{1}O_{1}S_{2}O_{2},

where U_{1}^{−} = S_{1}O_{1} and U_{1}^{+} = S_{2}O_{2} is a polar decomposition, then S_{1}^{−2}U ∈ M_{q}(R_{+}).

The above proposition remains valid if there exists a factor S(t) on the right side or if R^{+} is replaced by R^{−}.

Using this proposition one can establish the following

Theorem. Partial indices of a unitary matrix-function U(t) = ∥u_{ij}(t)∥ with det U = 1 of the form

u_{ij}(t) = α_{ij}^{+}(t), u_{qj}(t) = \overline{α_{qj}^{+}(t)} for 1 ≤ i ≤ q − 1, 1 ≤ j ≤ q, (1)

where α_{ij}^{+}(t) are polynomials, are equal to zero if and only if the condition

∑_{j=0}^{q} |α_{qj}^{+}(0)|^{2} ≠ 0  (2)

is fulfilled.

To prove the sufficiency of condition (2), for given U(t) one should define a positive definite matrix-function S(t) = ∥s_{ij}(t)∥ and X^{−}(t) = ∥x_{ij}^{−}(t)∥ ∈ M_{q}(R^{−}) by the equation

S(t)U(t) = X^{−}(t).  (3)

Condition (1) implies that s_{ij}(t) = const, 1 ≤ i ≤ q − 1, 1 ≤ j ≤ q − 1, \overline{s}_{qi}(t) ∈ R^{+}, 1 ≤ i ≤ q − 1 are polynomials. We set s_{ij} = δ_{ij}, 1 ≤ i ≤ q − 1, 1 ≤ j ≤ q − 1, where δ_{ij} is Kronecker’s symbol and denote φ_{i}^{+} = \overline{s}_{qi}(t) = s_{iq}(t), 1 ≤ i ≤ q − 1, and φ_{q} = s_{qq}(t). Equation (3) can now be rewritten as

α_{jk}^{+} + φ_{j}^{+}\overline{α}_{qk}^{+} = x_{jk}^{−}, 1 ≤ j ≤ q − 1, 1 ≤ k ≤ q,  (4)

∑_{j=1}^{q−1} α_{jk}^{+}\overline{φ}_{j}^{+} + φ_{q}\overline{α}_{qk}^{+} = x_{qk}^{−}, 1 ≤ k ≤ q.  (5)

Condition (2) implies α_{qj}^{+}(0) ≠ 0 for some j = p. Let F_{p}^{+} be some part of the series (α_{qp}^{+})^{−1} such that F_{p}^{+}α_{qp}^{+} = 1 + a_{N}t^{N} + a_{N+1}t^{N+1} + ⋯ with sufficiently large N. When k = p, by (4) we obtain

φ_{j}^{+} = ℙ(\overline{F} _{p}^{+}α_{jp}^{+}), 1 ≤ j ≤ q − 1,  (6)

where ℙ is the projecting operator from R into R^{+}, ℙ(R_{0}) = 0. Setting

φ_{q} = 1 + ∑_{j=1}^{q−1} |φ_{j}^{+}|^{2},  (7)

we see that S(t) is a positive definite matrix-function. It remains for us to check whether (4) is fulfilled for k ≠ p and (5). After substituting α_{jp}^{+} from (4) into ∑_{j=1}^{q−1} \overline{α}_{jk}^{+}α_{jp}^{+} + α_{qk}^{+}\overline{α}_{qp}^{+} = δ_{kp}, 1 ≤ k ≤ q, we obtain

(∑_{j=1}^{q−1} \overline{α}_{jk}^{+}φ_{j}^{+} + α_{qk}^{+})\overline{α}_{qp}^{+} ∈ R^{−} = δ_{kp}, 1 ≤ k ≤ q. The multiplication by \overline{F} _{p}^{+} gives ∑_{j=1}^{q−1} \overline{α}_{jk}^{+}φ_{j}^{+} + α_{qk}^{+} = y_{qk}^{−} ∈ R^{−}, 1 ≤ k ≤ q. Diagonalizing these linear equations with respect to φ_{j}^{+} without taking k = i into account we find that (4) is fulfilled for k = i. Finally, ∑_{j=1}^{q−1} α_{jk}^{+}\overline{φ}_{j}^{+} + φ_{q}\overline{α}_{qk}^{+} = ∑_{j=1}^{q−1}(\overline{α}_{jk}^{+} + φ_{j}^{+}\overline{α}_{qk}^{+})\overline{φ}_{j}^{+}+\overline{α}_{qk}^{+} ∈ R^{−}. Thus (5) is also fulfilled, which completes the proof.

The necessity follows from the fact that if ∑_{j=1}^{q} |α_{qj}^{+}(0)|^{2} = 0 and U = U^{+}U^{−} with invertible U^{±}, then U^{+} = U(U^{−})^{−1} and the elements of the last row of U^{+} belong both to R^{+} and R_{0}^{−} and therefore are equal to zero, which contradicts the invertibility of U^{+}.

Since the found positive matrix-function S(t) is effectively factored, the factorization of U(t) can be obtained effectively too.

Corollary 1. When condition (2) is fulfilled, the factorization of a unitary matrix-function of form (1) can be found as follows:

U(t) = (Y^{−}(t))^{∗}Y^{−}(t)S(t)U(t),

where Y^{−}(t) = ∥y_{ij}^{−}(t)∥, y_{ij}^{−}(t) = δ_{ij} for 1 ≤ i ≤ q − 1, 1 ≤ i ≤ q, y_{qj}^{−}(t) = −\overline{φ}_{j}^{+} with φ_{j}^{+} defined by (6), 1 ≤ j ≤ q − 1, y_{qq}(t) = 1, and (S(t))^{−1} = (Y^{−}(t))^{∗}Y^{−}(t).

The proof is obvious, since S(t)U(t) ∈ M_{q}(R^{−}), Y^{−}(t)^{∗} ∈ M_{q}(R^{∗}), and (Y^{−}(t)^{∗})^{−1} ∈ M_{q}(R^{∗}).

More can be said for the case with q = 2. For a unitary matrix-function U(t) of form (1) all functions α_{ij}^{∗}(t), 1 ≤ j ≤ q, may vanish simultaneously only for t = 0. Let α_{ij}^{+}(t) = t^{n_{ij}}a_{ij}^{+}(t) with polynomials a_{ij}^{+}(t), a_{ij}^{+}(0) ≠ 0 (for α_{ij}^{+}(t) = 0, n_{ij} = +∞) and let

n_{1} = min_{1≤j≤2} n_{1j}, n_{2} = − min_{1≤j≤2} n_{2j} = −n_{1}.  (8)

Then U(t) can be represented as U(t) = D(t)U_{1}(t), where U_{1}(t) is a unitary matrix of form (1) for which condition (2) holds, while D(t) is a diagonal matrix with d_{ii}(t) = t^{n_{i}}, i = 1, 2. Let U_{1}(t) = (Y_{1}^{−}(t))^{∗}Y_{1}^{−}(t)X_{1}^{−}(t) be the factorization of U_{1}(t) with lower triangular Y_{1}^{−}(t). Now we can formulate

Corollary 2. Partial indices of a second-order unitary matrix-function of form (1) are equal to n_{1}, −n_{1}, where n_{1} is defined by (8).

A factorization of U(t) can be found as follows:

U(t) = (D(t)y_{1}^{−}(t)^{∗}D(t)^{−1})D(t)Y_{1}^{−}(t)X_{1}^{−}(t).

The proof easily follows from the observation that D(t)Y_{1}^{−}(t)^{∗}D(t)^{−1} ∈ M_{2}(R^{+}) together with its inverse.

References

1. N. I. Muskhelishvili, Singular integral equations. (Translated from the Russian) P. Noordhoff, Groningen, 1953.

2. N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes. II Acta Math. 99(1958), 93–137.

3. I. Cokhberg and M. G. Krein, Systems of integral equations on a halfline with kernels depending on difference arguments. (Russian) Uspekhi Mat. Nauk 13(1958), No. 2, 3–72.

4. B. V. Bojarski, On stability of the Hilbert problem for homotopic vectors. (Russian) Bull. Acad. Sci. Georgian SSR (Soobshch. Akad. Nauk Gruz. SSR) 21(1958), No. 4, 391–398.

(Received 14.09.1995)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia