partial indices of a class of second order matrix-functions

ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2007, Vol. 42, No. 3, pp. 139–145. c© Allerton Press, Inc., 2007.Original Russian Text c© A. G. Kamalyan, A. V. Sargsyan, 2007, published in Izvestiya NAN Armenii. Matematika, 2007, No. 3, pp. 33–43.

SPECTRAL THEORY

Partial Indices of a Class of Second Order Matrix-Functions*

Armen G. Kamalyan 1* and Anna V. Sargsyan1

1Yerevan State University, ArmeniaReceived November 18 2006

Abstract—The paper studies Winer–Hopf factorization problem for matrix-functions of the secondorder defined on the unit circle of the complex plane. The element G21 of these matrix-functionsis assumed to be representable by combinations of the other three elements and two functionsmeromorphic in the interior and exterior of the circle. Some formulas for partial indices are obtained.

MSC2000 numbers: 30E25, 45G05

DOI: 10.3103/S106836230703003X

Key words: Matrix-function; factorization; partial indices.

1. INTRODUCTION

By Γ we denote the unit disc in the complex plane C, by W the Wiener algebra of functions thatare continuous in Γ and representable as absolutely convergent Fourier series. Besides, by W n×m wedenote the set of all those matrix-functions (MF) of the order n × m, whose components belong to W .For brevity, we use W n and W instead of the common notation W n×1 and W 1×1. We define the followingprojections acting in W n×m:

P−i (f)(t) =

i−1∑

k=−∞〈f〉ktk, P+

i (f)(t) =∞∑

k=i

〈f〉ktk,

P0ij(f)(t) =

j−1∑

k=i

〈f〉ktk, t ∈ Γ, 〈f〉k =1

2πi

∫

Γf(z) z−k−1dz,

where i < j and i, j ∈ Z. Besides, we assume that P0ij(f)(t) = 0 for i ≥ j and introduce the notation

W n×m+ = P+

0 (W n×m), W n×m− = P−

1 (W n×m) and◦

Wn×m

− = P−0 (W n×m).

By factorization of a G ∈ W 2×2 we mean its representation in the form

G = G− ∧ G+,

where G−, G−1− ∈ W 2×2

− , G+, G−1+ ∈ W 2×2

+ , ∧(t) = diag [tκ1 , tκ2], t ∈ Γ, κ1, κ2 ∈ Z and κ1 ≤ κ2. Thenumbers κ1 and κ2 are called partial indices of G, and if κ1 = κ2 = 0, then the above factorization iscalled canonical.

We assume that G11, G12, G22, h1, h2 ∈ W and there exist a nonnegative, integer N and somepolynomials q− and q+, such that their roots lie correspondingly in Γ+ = {z ∈ C, |z| < 1} and Γ− =

C \ Γ+ and p− =q+ h1

τN ∈ W−, p+ = q− h2 ∈ W+, where τ(t) = t (t ∈ Γ). Without loss of generality we

assume that deg q+ ≤ N and note that any rational functions with poles lying outside Γ can be taken forh1, h2.

*E-mail: [email protected]

139

140 KAMALYAN, SARGSYAN

In the present paper, we consider the problem on finding the partial indices of those MF-s G ∈ W 2×2

that have the form

G =

⎛

⎝ G11 G12

h1G11 + h2G22 − h1h2G12 G22

⎞

⎠ . (1.1)

It is known that an MF-s G has a factorization in W (see Chapter II in [1] and Chapter V in [2]) ifand only if det G(t) �= 0 (t ∈ Γ), and the latter is equivalent to the conditions Vi(t) �= 0 (t ∈ Γ, i = 1, 2),where

V1 = (q−G11 − p+G12)τN

q+ q−and V2 =

(τ−N q+G22 − p−G12

) τN

q+ q−.

The possibility of effective factorizations (1.1) was noted in [3], so-called EF-algorithm for factorizationof (1.1) form MF when h1 = 0 or h2 = 0 was suggested in [4]—[6].

If a, b ∈ W and p, q, r, s are some polynomials satisfying the condition s2 + pq = r2, then takingin (1.1) G11 = a − bs, G22 = a + bs, G12 = bp, h+ = h− = (r-s)

/p, we come to a representation G =

aI + bR, where R is an arbitrary polynomial matrix which satisfies the condition trR = 0 and possessesa determinant which is the square of some other polynomial. The set of those MF-s G which arerepresentable in the form G = aI + bR, where trR = 0 and R is a polynomial matrix, is known asDaniel–Khrapkov class. The factorization problem of MF-s from the Daniel–Khrapkov class is arisingin some problems of mechanics (see eg. [7], [8]). Efficient factorization methods for MF-s from this classunder additional conditions were suggested by many authors (see eg. [4], [7]-[13] and their referencelists).

In the present paper, we obtain some explicit formulas for the partial indices of MF-s of the form (1.1).These formulas permit to find the partial indices of an MF-s G by means of the ranks of finite number ofblock matrices, the block elements of which are the Fourier coefficients of some MF-s. These MFs areexplicitly determined by G and the factors of the functions Vi (i = 1, 2).

2. A REPRESENTATION OF THE MATRIX–FUNCTION G

Henceforth, we assume that Vi(t) �= 0 (t ∈ Γ) and

χi = indVi =12π

var arg Vi(t)∣∣∣∣t∈Γ

, i = 1, 2, χ0 = max{χ1, χ2}.

A triangular MF-s

V =

⎛

⎜⎜⎝τ−χ1V1

τN−χ1

q+ q−G12

0 τ−χ2V2

⎞

⎟⎟⎠ , τ ∈ Γ

permits a canonical factorization V = V− V+, where the factors V± =

⎛

⎝ V ±1 V ±

12

0 V ±2

⎞

⎠ can be recovered by

the formulas

V ±i = expP±

0

(ln

(τ−χiVi

)), i = 1, 2,

V +12 = V +

2 P+0

(τN−χ1 G12

V −1 V +

2 q+ q−

), V −

12 = V −1 P−

0

(τN−χ1 G12

V −1 V +

2 q− q+

).

An MF-s G of the form (1.1) permits a decomposition G = τχ0G0, where G0 = AB and

A =

⎛

⎜⎝q+

τN 0

p− 1

⎞

⎟⎠

⎛

⎜⎝τχ1−χ0 0

0 τχ2−χ0

⎞

⎟⎠ V−, B = V+

⎛

⎜⎝1 0

p+ q−

⎞

⎟⎠

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42 No. 3 2007

PARTIAL INDICES OF A CLASS 141

Denoting ν− = N + |χ1 − χ2|, n− = deg q+, ν+ = deg q− and taking into account that q+τ−ν−A−1 ∈W 2×2

− and q−B−1 ∈ W 2×2+ , we obtain

n−∑

k=0

〈A−1〉m−k〈q+〉k = 0, m = ν− + 1, ν− + 2, . . . , (2.1)

ν+∑

k=0

〈B−1〉m−k〈q−〉k = 0, m = −1,−2, . . . (2.2)

Now, writing

U = τ−ν−P−ν−

(B−1

)P+

ν−

(A−1

)

one can see that

q−U =(q−B−1 − q−P+

ν−

(B−1

))τ−ν−P+

ν−

(A−1

).

Hence q−U ∈ W 2×2+ , and consequently

ν+∑

k=0

〈U〉m−k〈q−〉k = 0, m = −1,−2, . . . (2.3)

3. THE MAIN LEMMA

We need the following families of Hankel operators H−j :

◦W 2

− −→ W 2+, H+

j : W 2+ −→

◦W 2

− (j ∈ Z)and Toeplitz operators Tj : W 2

+ → W 2+ (j ∈ Z):

H−j ϕ = P+

0

(τ j

(A−1ϕ

)), H+

j ϕ = P−0

(τ j

(B−1ϕ

)),

Tjϕ = P+0

(τ j (G0ϕ)

),

where j = (j + |j|)/ 2, j = (j − |j|)/ 2, j ∈ Z. For j > 0, j ∈ Z, by Lj we denote the space of thevector-polynomials

∑j−1k=0 ϕkzk (ϕk ∈ C

n) and for j < 0, j ∈ Z, the space of vector-polynomials de-pending on z−1 of the form

∑−1k=j ϕkz

k. For j = 0, we assume that L0 = {0}. In addition, we considerthe following space of finite-dimensional operators:

Kj = H+j H−

j

∣∣∣L−(ν−+j)

, j ∈ Z.

Lemma 1. A vector-function ϕ belongs to Ker Tj if and only if there exists some ψ ∈ Ker Kj suchthat ϕ = τ jB−1H−

j (ψ). Besides, the following equalities are true:

dim Ker Tj = ν− + 2j − dim Im Kj , j ∈ Z. (3.1)

Proof: It is known (see [14]) that

Im H−j = Im

(H−

j

∣∣∣L−(ν−+j)

), j ∈ Z. (3.2)

Hence

H−j (Ker Kj) = Ker

(H+

j

∣∣∣Im H−

j

). (3.3)

Therefore, to prove the first assertion of our lemma it suffices to verify that ϕ ∈ Ker Tj if and only if

ϕ0 = τ−jBϕ ∈ Ker

(H+

j

∣∣∣Im H−

j

).



Let ϕ ∈ Ker Tj (j ∈ Z). Then ϕ ∈ W 2+, f = τ−jGϕ ∈

◦W 2

− and τ jA−1f = τ−jBϕ = ϕ0 ∈ W 2+. Conse-

quently, ϕ0 = H−j f and H+

j ϕ0 = 0, i.e. ϕ0 ∈ Ker

(H+

j

∣∣∣Im H−

j

). Now let ϕ0 ∈ Ker

(H+

j

∣∣∣Im H−

j

).

Then by (3.2) there exists some f ∈ L−(ν−+j) such that ϕ0 = H−j f . Therefore, ϕ = τ jB−1ψ ∈ W 2

+

since H+j ϕ0 = 0. Besides, the equality ϕ0 = τ jA−1f − P0

(τ jA−1f

)implies that τ−jAψ = f − g,

where g = τ−jAH+j ∈ W 0

−. Hence Tjϕ = P+0

(t−jAψ

)= P+

0 (f − g) = 0, i.e. ϕ ∈ Ker Tj .

To prove the equality (3.1), observe that by the just proved assertion and (3.3)

dim Ker Tj = dim Ker

(H+

j

∣∣∣Im H−

j

)

= dim Im H−j − dim Im

(H+

j

∣∣∣Im H−

j

)

= dim Im H−j − dim Im Kj .

It remains to observe that dim Im H−j = ν− + nj (see [14], Proposition 5.2).

4. DIMENSIONS OF THE KERNELS

We proceed to calculation of the numbers dim Ker Tj (j ∈ Z).

Proposition 1 If j ≤ −ν−, then dim Ker Tj = 0.

Proof: Let q ∈ Ker Kj , ϕ ∈ H−q. Then ϕ ∈ Ker H+j by the definition of the operator Kj , hence ϕ ∈ W 2

+

and ψ = τ jB−1ϕ ∈ W 2+. Besides,

〈ϕ〉0 = 〈ϕ〉1 = ... = 〈ϕ〉−j−1 = 0

by the equality ϕ = τ−jBψ, and ϕ =∑∞

m=0 zm∑ν−

k=1〈A−1〉m+k〈q〉−k since q ∈ L−ν− . In addition,∑ν−k=1〈A−1〉m+k〈q〉−k = 0 (m = 0, . . . , ν− − 1) since ν− − 1 ≤ −j − 1. Observing that 〈q+〉0 �= 0 and

n− ≤ ν− and using (2.1) we obtain

〈ϕ〉ν− =ν−∑

k=1

〈A−1〉ν−+k〈q〉−k

= −ν−∑

k=1

n−∑

i=1

〈q+〉i〈q+〉0

〈A−1〉ν−+k−i〈q〉−k

= −n−∑

i=1

〈q+〉i〈q+〉0

ν−∑

k=1

〈A−1〉ν−+k−i〈q〉−k = 0.

By repetitions of similar argument, we get 〈ϕ〉ν−+1 = 〈ϕ〉ν−+2 = ... = 0, i.e. ϕ = 0. It remains to observethat by Lemma 1 Ker Tj = {0}.

Proposition 2 If j ≥ ν+, then

dim Ker Tj = ν− + 2j − ν+. (4.1)

Proof: Observe that if j > 0 and q ∈ Lj , then ϕ = τ−jAq ∈◦

W 2− and H−

j ϕ = q. Hence

Lj ⊂ Im H−j . (4.2)



Further, we can represent the vector-function y ∈ W 2+ in the form y = q + q− y0, where y0 ∈ W 2

+ andq is a vector-polynomial, whose degree does not exceed ν+ − 1. Besides, q−B−1y0 ∈ W 2

+ and thereforeH+

0 y = H+0 q by the equality B−1y = B−1q + q−B−1y0, i.e.

Im H+0 = Im

(H+

0

∣∣Lν+

). (4.3)

Defining an operator T ′ :◦

W 2− →

◦W 2

− by the formula T ′y = P−0 (By), one can prove that Ker T ′ =

Im H+0 . Indeed, if ϕ ∈ Im H+

0 and y ∈ W 2+ is such that ϕ = H+

0 y, then Bϕ = y − BP+0

(B−1y

)∈ W 2

+,i.e. ϕ ∈ Ker T ′. Conversely, if ϕ ∈ Ker T ′, then ψ = Bϕ ∈ W 2

+ and ϕ = H+0 ψ.

Observe now, that B has a factorization in W , since det B(t) �= 0 (t ∈ Γ). It is known that dim Ker T ′

coincides with the sum of positive partial indices of B, and all partial indices of B are nonnegative byanalyticity inside the circle (see eg. [14]). Hence, dim Ker T ′ coincides with the total index of B. On theother hand, the total index of B is equal to the sum of multiplicities of zeros of the function det B, insidethe circle, and by the definition of the MF-s B this number equals ν+:

dim Im H+0 = ν+. (4.4)

Using (3.2) and (4.2) we obtain that for j ≥ 0

Im H+0 ⊃ Im Kj = Im

(H+

0

∣∣H−

j

(L−(ν−+j)

))

= Im(

H+0

∣∣ImH−

j

)⊃ Im

(H+

0

∣∣Lj

).

Besides, if j ≥ ν+, then Im H+0 ⊃ Im Kj ⊃ Im H+

0 by (4.3). Consequently, if j ≥ ν+, then dim Im Kj =dim Im H+

0 = ν+ by (4.4). Hence (4.1) follows by (3.1), and the proof is complete.

Now, assuming that ν− + ν+ ≥ 2 and j ∈ Z satisfies the conditions −ν− + 1 ≤ j ≤ ν+ − 1 and ν− +j > 1, we introduce the following matrices Bj and Aj of the orders 2

(ν+ − j

)× 2

(j + ν− − 1

)and

2(ν− + j − 1

)× 2

(ν− + j

)respectively:

Bj =

⎛

⎜⎜⎜⎜⎜⎜⎝

〈B−1〉−1−j 〈B−1〉−2−j ... 〈B−1〉−ν−−j+1

〈B−1〉−2−j 〈B−1〉−3−j ... 〈B−1〉−ν−−j

... ... ... ...

〈B−1〉−ν+ 〈B−1〉−ν+−1 ... 〈B−1〉−(ν++ν−)+2−j

⎞

⎟⎟⎟⎟⎟⎟⎠,

Aj =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 〈A−1〉ν−−1 〈A−1〉ν−−2 ... 〈A−1〉1−j

0 0 〈A−1〉ν−−1 ... 〈A−1〉2−j

... ... ... ... ...

0 0 0 ... 〈A−1〉ν−−1

⎞

⎟⎟⎟⎟⎟⎟⎠.

Next, for j ∈ Z satisfying the conditions −ν− + 1 ≤ j ≤ ν+ − 1 we consider the following matrices Uj

and Kj of the order 2(ν+ − j

)× 2

(ν− + j

):

Uj =

⎛

⎜⎜⎜⎜⎜⎜⎝

〈U〉−1−j 〈U〉−2−j s 〈U〉−ν−−j

〈U〉−2−j 〈U〉−3−j s 〈U〉−ν−−j−1

s s s s

〈U〉−ν+ 〈U〉−ν+−1 s 〈U〉−(ν++ν−)+1−j

⎞

⎟⎟⎟⎟⎟⎟⎠



Kj = Uj + BjAj if ν− + j > 1 and Kj = Uj if ν− + j = 1.

Proposition 3 If j ∈ Z, ν− + ν+ ≥ 2 and −ν− + 1 ≤ j ≤ ν+ − 1, then

dim Ker Tj = ν− + 2j − rj , (4.5)

where rj = rangKj .

Proof: For q ∈ L−(ν−+j) we have the inclusions

τ jP−−j+1

(A−1

)q ∈

◦W

2−, τ jP+

ν−

(A−1

)q ∈ W 2

+

and the equality A−1 = P−−j+1

(A−1

)+ P0

−j+1,ν−

(A−1

)+ P+

ν−

(A−1

). So we obtain that H−

j q =

τ jP+ν−

(A−1

)q + g, where

g(t) = P+0

(τ jP0

−j+1,ν−

(A−1

)q)

(t) =ν−+j−2∑

k=0

〈g〉ktk, р

〈g〉k =−1∑

m=k−(ν−+j)+1

〈A−1〉k−m−j〈q〉m. (4.6)

Hence, by τ jP+ν−

(B−1

)P+

ν−

(A−1

)q ∈ W 2

+ we get Kjq = P−0

(τ jUh

)+ H+

j g, where h = τν−+jq.

Consequently, Kjq = 0 is equivalent to the infinite system of equalities

ν−+j−1∑

k=0

〈U〉m−k−j〈h〉k +ν−+j−2∑

k=0

〈B−1〉m−k−j〈g〉k = 0, m = −1,−2,−3, . . .

By (2.2) and (2.3) the above equalities hold for m = −1,−2, . . . if and only if they hold for m =−1, . . . ,−(ν+ − j). We denote

hj =(〈h〉T0 , . . . , 〈h〉T

ν−+j−1

)T, gj =

(〈g〉T0 , . . . , 〈g〉T

ν−+j−2

)T,

qj =(〈q〉T−ν−−j

, . . . , 〈q〉T−1

)T.

By (4.6), hj = qj and gj = Aj qj . So the condition q ∈ Ker Kj is equivalent to the equality Ujhj +Bj gj = 0. Rewriting this in the form Kj qj = 0, we finally obtain that Kjq = 0 only when Kj qj = 0.Consequently, dim Ker Kj = dim KerKj . Besides, by (3.1)

dim Ker Tj = ν− + 2j − dim Im Kj

= ν− + 2j −(2(ν− + j

)− dim KerKj

)= ν− + 2j − rj ,

and the proof is complete.

5. FORMULAS FOR PARTIAL INDICES

Assuming that r−ν− = ν−, rν+ = ν+, θj = 1 for j > 0 and θj = 0 for j ≤ 0 (j ∈ Z), we admits themain result of this paper.

Theorem 1. If Vi(t) �= 0 (i = 1, 2, t ∈ Γ), then the MF-s G defined by (1.1) admits factorization. Forthe partial indices the following equalities are true: κ1 = −ν− + ν0 + χ0, κ2 = ν+ − ν0 + χ0, whereν0 = 0 if ν− + ν+ ≤ 1 and ν0 = card {j ; rj = 2θj + rj−1 ; j = −ν− + 1, . . . , ν+} if ν− + ν+ ≥ 2.



Proof: It is known that dim Ker Tj is equal to -1 times the sum of negative partial indices of theMF-s τ−jG0. Besides, the partial indices of τ−jG0 are equal to κ1 − χ0 − j and κ2 − χ0 − j, hencedim Ker Tj > 0 for j ≥ κ1 − χ0. Consequently, −ν− ≤ κ1 − χ0 by Proposition 1. Because

κ1 + κ2 − 2χ0 = ind det G0 = ind det A + ind det B

= ind q+ − ind τN − |χ1 − χ2| + ind q− = ν+ − ν−,

we have κ2 − χ0 ≤ ν+. Thus, −ν− ≤ κ1 − χ0 ≤ κ2 − χ0 ≤ ν+ and κ1 + κ2 − 2χ0 = ν+ − ν−. Hencethe desired statement follows for the case ν− + ν+ ≤ 1.

By the method applied in [14], Proposition 2.1, one can verify that

κ1 − χ0 = η− + card {j|dim Ker Tj − dim Ker Tj−1 < 1; j = η− + 1, . . . , η+} , (5.1)

where η− and η+ are arbitrary integers satisfying the conditions η− ≤ κ1 − χ0 ≤ κ2 −χ0 ≤ η+. Besides,dim Ker Tj = ν− + 2j − rj (j = −ν−, . . . , ν+) by the definition of the numbers rj (j = −ν−, . . . , ν+)and Propositions 1-3 Consequently

dim Ker Tj − dim Ker Tj−1 = 2θj + rj−1 − rj . (5.2)

Taking η− = ν−, η+ = ν+ as in (5.1) and (5.2), we come to the desired statement for the case ν− + ν+ ≥2.

REFERENCES1. K. Clancey and I. Gohberg, “Factorization of Matrix Functions and Singular Integral Operators”, in

Operator Theory: Advances and Appl. 3 (Birkhauser, Basel 1981).2. G. S. Litvinchuk and I. M. Spitkovskii, Factorization of Matrix-Functions (Akademie-Verlag, Berlin

1987).3. V. N. Gavdzinskii, I. M. Spitkovskii, “On a Way of Effective Construction of Factorization”, Ukr. Mat. J.

[Ukrainian Mathematical Journal] 34 (1), 15-19 (1982).4. I. Feldman, I. Gohberg, N. Krupnik, “A Method of Explicit Factorization of Matrix Functions and Applica-

tions”, IEOT 18, 277-302 (1994).5. I. Feldman, I. Gohberg, N. Krupnik, “On Explicit Factorization and Applications”, IEOT 21, 430-459 (1995).6. I. Feldman, I. Gohberg, N. Krupnik, “An Explicit Factorization Algorithm”, IEOT, 49, 149-164 (2004).7. A. A. Khrapkov, “Certain Cases of Elastic Equilibrium of an Infinite Wedge With a Non-Symmetric Notch

at the Vertex, Subjected to the Concentrated Force”, Prikl. Mat. Mekch. [Applicable Mathematics andMechanics] 35, 625-637 (1971).

8. V. T. Daniele, “On the Solution of Two Coupled Wiener-Hopf Equations”, SIAM J. Appl. Math. 44 (4),667-680, (1984).

9. E. Meister, F.-O. Speck, “Wiener-Hopf Factorization of Certain Non-Rational Matrix Functions in Mathe-matical Physics”, in The Gohberg Anniversary Collection 11 (Birkhauser, Basel 1989).

10. S. Prosdorf, F. O. Speck, “A Factorization Procedure for Two by Two Matrix Functions on the Circle withTwo Rationally Independent Entries”, Proceedings of the Royal Society of Edinburgh 115A, 119-138 (1990).

11. A. B. Lebre, “Factorization in the Wiener Algebra of a Class of 2 × 2 Matrix Functions”, Integral Equationsand Operator Theory 12, 408-423 (1989).

12. A. G. Kamalyan, V. A. Ohanyan, “On Factorization of f-Circuit Matrix-Functions” Izv. NAN Armenii.Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 28 (5),44-62 (1993).

13. A. G. Kamalyan, V. A. Ohanyan, “A Constructive Method for Factorization Construction for a Classof Matrix-Functions” Izv. NAN Armenii. Matematika [Journal of Contemporary Mathematical Analysis(Armenian Academy of Sciences)] 28 (3), 44-62 (1993).

14. A. G. Kamalyan, “An Explicit Generalized Factorization of Bounded Holomorphic Functions” Izv. NANArmenii. Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)]32 (2), 19-38 (1997).


partial indices of a class of second order matrix-functions

Documents