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right half-lattice CMV operators and a similar result for associated Green’s functions) discussed in this paper, but we also refer to the monumental two-volume treatise by Simon [71] (see also [70] and [72]) and the exhaustive bibliography therein. For classical results on orthogonal polynomials on the unit circle we refer, for instance, to [6], [41]–[43], [53], [75]–[77], [80], [81]. More recent references relevant to the spectral theoretic content of this paper are [22], [38]–[40], [49], [50], [51], [66], [69], and [87]. The full-lattice CMV operators U on Z are closely related to an important, and only recently intensively studied, completely integrable nonabelian version of the defocusing nonlinear Schr¨odinger equation (continuous in time but discrete in space), a special case of the Ablowitz–Ladik system. Relevant references in this context are, for instance, [1]–[5], [37], [44]–[47], [57], [59]–[62], [68], [79], and the literature cited therein. We emphasize that the case of matrix-valued coefficients αk is considerably less studied than the case of scalar coefficients. We should also emphasize that while there is an extensive literature on orthogonal matrix-valued polynomials on the real line and on the unit circle, we refer, for instance, to [7], [9], [10, Ch. VII], [13], [16], [18], [23]–[34], [35], [36], [54], [55], [56], [58], [63]–[65], [67], [74], [83]–[86], and the large body of literature therein, the case of CMV operators with matrix-valued Verblunsky coefficients appears to be a much less explored frontier. The only references we are aware of in this context are Simon’s treatise [71, Part 1, Sect. 2.13] and the recent papers [8], [19], [21], and [73]. Finally, a brief description of the content of each section in this paper: In Section 2 we review, and in part, extend the basic Weyl–Titchmarsh theory for half-lattice CMV operators with scalar Verblunsky coefficients originally derived in [49], and recall its intimate connections with transfer matrices and orthogonal Laurent polynomials. The principal result of this section, Theorem 2.3, then provides a necessary and sufficient condition for the difference between the full-lattice CMV operator U and its “decoupling” into a direct sum of appropriate left and right half-lattice CMV operators to be of rank one. The same result is also derived for the resolvent differences of U and the resolvent of its decoupling into a direct sum of left and right half-lattice CMV operators. Theorem 2.3 is in sharp contrast to the familiar Jacobi case, since decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients results in a perturbation of rank two. While this difference compared to Jacobi operators was noticed first by Simon [71, Sect. 4.5], we explore it further here and provide a complete discussion of this decoupling phenomenon, including its extension to the matrix-valued case, which represents a new result. We conclude this section with a discussion of half-lattice Green’s functions in Lemma 2.7, extending a result in [49]. In Section 3 we develop all these results for CMV operators with m × m, m ∈ N, matrix-valued Verblunsky coefficients. In particular, in Theorem 3.6, the principal result of this section, we provide a necessary and sufficient condition for the difference between the full-lattice CMV operator U and its decoupling into a direct sum of appropriate left and right half-lattice CMV operators to be of minimal rank m. Finally, Appendix A summarizes basic facts on matrix-valued Caratheodory and Schur functions relevant to this paper.
2. CMV operators with scalar coefficients This section is devoted to a study of CMV operators associated with scalar Verblunsky coefficients. We derive a criterion under which a difference of a full-lattice CMV operator and a direct sum of two half-lattice CMV operators is of rank one. The same condition will also imply a similar result for the resolvents of these operators. At the end of the section we establish relations that hold between Weyl–Titchmarsh m-functions associated with the above operators and derive explicit expressions for half-lattice Green’s matrices. We start by introducing basic notations used throughout this paper. Let s(Z) be the space of complex-valued sequences and ℓ2(Z) ⊂ s(Z) be the usual Hilbert space of all square summable MINIMALRANKDECOUPLINGOFCMVOPERATORS 3
complex-valued sequences with scalar product (·, ·)ℓ2(Z) linear in the second argument. The standard basis in ℓ2(Z) is denoted by
⊤ {δk}k∈Z, δk = (..., 0,..., 0, 1 , 0,..., 0,... ) , k ∈ Z. (2.1) k |{z} N R m 2 Z 2 Z Cm For m ∈ and J ⊆ an interval, we will identify ⊕j=1ℓ (J ∩ ) and ℓ (J ∩ ) ⊗ and then use the simplified notation ℓ2(J ∩ Z)m. For simplicity, the identity operators on ℓ2(J ∩ Z) and ℓ2(J ∩ Z)m are abbreviated by I without separately indicating its dependence on m and J. The identity m × m matrix is denoted by Im. Throughout this section we make the following basic assumption:
Hypothesis 2.1. Let α = {αk}k∈Z ∈ s(Z) be a sequence of complex numbers such that
αk ∈ D, k ∈ Z. (2.2)
Given a sequence α satisfying (2.2), we define the following sequence of positive real numbers {ρk}k∈Z by
2 1/2 ρk = 1 − |αk| , k ∈ Z. (2.3) Following Simon [71], we call {αk}k∈Z the Verblunsky coefficients in honor of Verblunsky’s pio- neering work in the theory of orthogonal polynomials on the unit circle [80], [81]. Next, we also introduce a sequence of 2 × 2 unitary matrices Θk by
−α ρ Θ = k k , k ∈ Z, (2.4) k ρ α k k
and two unitary operators V and W on ℓ2(Z) by their matrix representations in the standard basis of ℓ2(Z) as follows,
. . .. 0 .. 0 Θ Θ V = 2k−2 , W = 2k−1 , (2.5) Θ Θ 2k 2k+1 . . 0 .. 0 .. where
V2k−1,2k−1 V2k−1,2k W2k,2k W2k,2k+1 =Θ2k, =Θ2k+1, k ∈ Z. (2.6) V − V W W 2k,2k 1 2k,2k 2k+1,2k 2k+1,2k+1
Moreover, we introduce the unitary operator U on ℓ2(Z) by
U = V W, (2.7) 4 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
or in matrix form, in the standard basis of ℓ2(Z), by ...... 0 0 −α0ρ−1 −α−1α0 −α1ρ0 ρ0ρ1 ρ ρ α ρ −α α α ρ 0 U = −1 0 −1 0 0 1 0 1 (2.8) 0 −α ρ −α α −α ρ ρ ρ 2 1 1 2 3 2 2 3 ρ1ρ2 α1ρ2 −α2α3 α2ρ3 0 0 . . . . . ...... − −− − + − + = ρ ρδeven S + (α ρδeven − α ρδodd)S − αα
+ ++ + + + ++ ++ + (αρ δeven − α ρ δodd)S + ρ ρ δodd S , (2.9) where δeven and δodd denote the characteristic functions of the even and odd integers,
δeven = χ2Z , δodd =1 − δeven = χ2Z+1 (2.10) and S±, S++, S−− denote the shift operators acting upon s(Z), that is, S±f(·)= f ±(·)= f(·± 1) for f ∈ s(Z), S++ = S+S+, and S−− = S−S−. Here the diagonal entries in the infinite matrix (2.8) are given by Uk,k = −αkαk+1, k ∈ Z. As explained in the introduction, in the recent literature on orthogonal polynomials on the unit circle, such operators U are frequently called CMV operators. Next we recall some of the principal results of [49] needed in this paper.
Lemma 2.2. Let z ∈ C\{0} and suppose {u(z, k)}k∈Z, {v(z, k)}k∈Z ∈ s(Z). Then the following items (i)–(iii) are equivalent: (i) (Uu(z, ·))(k)= zu(z, k), (Wu(z, ·))(k)= zv(z, k), k ∈ Z. (2.11) (ii) (Wu(z, ·))(k)= zv(z, k), (V v(z, ·))(k)= u(z, k), k ∈ Z. (2.12) u(z, k) u(z, k − 1) (iii) = T (z, k) , k ∈ Z, (2.13) v(z, k) v(z, k − 1) where the transfer matrices T (z, k), z ∈ C\{0}, k ∈ Z, are given by α z 1 k , k odd, ρk 1/z αk! T (z, k)= (2.14) αk 1 1 , k even. ρk 1 α k! Here U, V , and W are understood in the sense of difference expressions on s(Z) rather than difference operators on ℓ2(Z). it Z If one sets αk0 = e , t ∈ [0, 2π), for some reference point k0 ∈ , then the CMV operator (denoted in this case by U (t)) splits into a direct sum of two half-lattice operators U (t) and U (t) acting k0 −,k0−1 +,k0 2 2 on ℓ ((−∞, k0 − 1] ∩ Z) and on ℓ ([k0, ∞) ∩ Z), respectively. Explicitly, one obtains (t) (t) (t) 2 2 U = U ⊕ U on ℓ ((−∞, k0 − 1] ∩ Z) ⊕ ℓ ([k0, ∞) ∩ Z) k0 −,k0−1 +,k0 (2.15) it if αk0 = e , t ∈ [0, 2π). it Z (Strictly, speaking, setting αk0 = e , t ∈ [0, 2π), for some reference point k0 ∈ contradicts our basic Hypothesis 2.1. However, as long as the exception to Hypothesis 2.1 refers to only one site, MINIMALRANKDECOUPLINGOFCMVOPERATORS 5
we will safely ignore this inconsistency in favor of the notational simplicity it provides by avoiding (t) (t) the introduction of a properly modified hypothesis on {α } Z.) Similarly, one obtains V , W , k k∈ k0 k0 V (t) , and W (t) , so that ±,k0 ±,k0 V (t) = V (t) ⊕ V (t) , W (t) = W (t) ⊕ W (t) , k0 −,k0−1 +,k0 k0 −,k0−1 +,k0 U (t) = V (t)W (t),U (t) = V (t) W (t) . (2.16) k0 k0 k0 ±,k0 ±,k0 ±,k0 For simplicity we will abbreviate U = U (t=0) = V (t=0)W (t=0) = V W . (2.17) ±,k0 ±,k0 ±,k0 ±,k0 ±,k0 ±,k0
It is instructive to introduce one more sequence of Verblunsky coefficients β = {βk}k∈Z ∈ s(Z) by −it βk = e αk, k ∈ Z, (2.18)
it so that the sequence {ρk}k∈Z is unchanged and αk0 = e corresponds to βk0 = 1. Then the CMV operators Uβ and U±,k0;β associated with β are unitarily equivalent to the corresponding CMV operators U and U (t) associated with α. Indeed, one verifies that α ±,k0;α e−it/2 0 −α ρ e−it/2 0 −β ρ k k = k k , k ∈ Z, (2.19) 0 eit/2 ρ α 0 eit/2 ρ β k k k k and hence, setting A to be the following diagonal unitary operator on ℓ2(Z),
−it/2 it/2 A = e δodd + e δeven, (2.20) one obtains for the full-lattice and the direct sum of half-lattice CMV operators, ∗ ∗ ∗ AUαA = [AVαA][A WαA ]= VβWβ = Uβ, (2.21) AU (t) A∗ = AV (t) A A∗W (t) A∗ = V W = U . (2.22) k0;α k0;α k0;α k0;β k0;β k0;β We refer to [50, Sect. 3] for additional results on CMV operators with Verblunsky coefficients related via (2.18). Now we turn to our principal result of this section.
Theorem 2.3. Fix t ,t ∈ [0, 2π), k ∈ Z, z ∈ C\∂D, and let U (t1,t2) denote the following unitary 1 2 0 k0 operator on ℓ2(Z),
U (t1,t2) = U (t1) ⊕ U (t2) . (2.23) k0 −,k0−1 +,k0 −1 Then U − U (t1,t2) and (U − zI)−1 − U (t1,t2) − zI are of rank one if and only if the relation k0 k0 −it2/2 it2/2 t1 = 2arg i(αk0 e − e ) holds. Otherwise, these differences are of rank two. In particular, −1 U − U (t) and (U − zI)−1 − U (t) − zI are of rank two for any t ∈ [0, 2π). k0 k0 Proof. Similar to (2.23) we introduce unitary operators V (t1,t2) and W (t1,t2) by k0 k0 V (t1,t2) = V (t1) ⊕ V (t2) and W (t1,t2) = W (t1) ⊕ W (t2) on ℓ2(Z). (2.24) k0 −,k0−1 +,k0 k0 −,k0−1 +,k0 Then U (t1,t2) = V (t1,t2)W (t1,t2) by (2.16), and hence, it follows from (2.5) that k0 k0 k0
(t1,t2) (t ,t ) V W − W , k0 odd, U − U 1 2 = k0 (2.25) k0 (t1,t2) V − V W, k0 even. ( k0 6 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
For k odd, D = W − W (t1,t2) is block-diagonal with all its 2 × 2 blocks on the diagonal being zero 0 k0 except for one which has the following form
it1 Dk0−1,k0−1 Dk0−1,k0 −αk0 ρ0 −e 0 = − −it . (2.26) D − D ρ α 0 e 2 k0,k0 1 k0,k0 k0 k0
Thus, the difference U − U (t1,t2) in (2.25) is always of rank one or two and it is precisely of rank one k0 if and only if the 2 × 2 matrix in (2.26) is of rank 1. The latter case is equivalent to
Dk0−1,k0−1 Dk0−1,k0 it1 −it2 i(t1−t2) 0 = det = e αk0 + e αk0 − e − 1 D − D k0,k0 1 k0,k0 it2 −it2 it1 −it2 αk0 − e = (αk0 − e ) e + e α − e−it2 k0 −it2/2 it2/2 −it2 it1 iαk0 e − e = (αk0 − e ) e − , (2.27) −it2/2 it2/2 iαk0 e − e !
−it2/2 it2/2 which holds if and only if t1 = 2arg i(αk0 e − e ) . The case of even k0 follows similarly. Finally, the statement for the resolvents follows from the result for U − U (t1,t2) and the following k0 identity,
(U − zI)−1 − (U (t1,t2) − zI)−1 = −(U − zI)−1 U − U (t1,t2) (U (t1,t2) − zI)−1. (2.28) k0 k0 k0 h i
Next we present formulas that link various spectral theoretic objects associated with half-lattice CMV operators U (t) for different values of t ∈ [0, 2π). We start with an analog of Lemma 2.2 for ±,k0 difference expressions U (t) , V (t) , and W (t) . In the special case t = 0 it is proven in [49, Lem. ±,k0 ±,k0 ±,k0 2.3] and the general case below follows immediately from the special case and the observation of unitary equivalence in (2.22).
(t) (t) Lemma 2.4. Fix t ∈ [0, 2π), k0 ∈ Z, z ∈ C\{0}, and let pˆ (z,k,k0) , rˆ (z,k,k0) ∈ + k≥k0 + k≥k0 s([k , ∞) ∩ Z). Then the following items (i)–(iii) are equivalent: 0
(i) U (t) pˆ(t)(z, ·, k ) (k)= zpˆ(t)(z,k,k ), +,k0 + 0 + 0 W (t) pˆ(t)(z, ·, k ) (k)= zrˆ(t)(z,k,k ), k ≥ k . (2.29) +,k0 + 0 + 0 0 (ii) W (t) pˆ(t)(z, ·, k )(k)= zrˆ(t)(z,k,k ), +,k0 + 0 + 0 V (t) rˆ(t)(z, ·, k ) (k)=ˆp(t)(z,k,k ), k ≥ k . (2.30) +,k0 + 0 + 0 0 (t) (t) pˆ+ (z,k,k0) pˆ+ (z, k − 1, k0) (iii) (t) = T (z, k) (t) , k>k0, rˆ+ (z,k,k0) rˆ+ (z, k − 1, k0) zeitrˆ(t)(z, k , k ), k odd, pˆ(t)(z, k , k )= + 0 0 0 (2.31) + 0 0 −it (t) (e rˆ+ (z, k0, k0), k0 even. MINIMALRANKDECOUPLINGOFCMVOPERATORS 7
(t) (t) Similarly, let pˆ (z,k,k0) , rˆ (z,k,k0) ∈ s((−∞, k0] ∩ Z). Then the following items − k≤k0 − k≤k0 (iv)–(vi)are equivalent:
(iv) U (t) pˆ(t)(z, ·, k ) (k)= zpˆ(t)(z,k,k ), −,k0 − 0 − 0 W (t) pˆ(t)(z, ·, k ) (k)= zrˆ(t)(z,k,k ), k ≤ k . (2.32) −,k0 − 0 − 0 0 (v) W (t) pˆ(t)(z, ·, k )(k)= zrˆ(t)(z,k,k ), −,k0 − 0 − 0 V (t) rˆ(t)(z, ·, k ) (k)=ˆp(t)(z,k,k ), k ≤ k . (2.33) −,k0 − 0 − 0 0 (t) (t) pˆ− (z, k − 1, k0) −1 pˆ− (z,k,k0) (vi) (t) = T (z, k) (t) , k ≤ k0, rˆ− (z, k − 1, k0) rˆ− (z,k,k0) −eitrˆ(t)(z, k , k ), k odd, pˆ(t)(z, k , k )= − 0 0 0 (2.34) − 0 0 −it (t) (−ze rˆ− (z, k0, k0), k0 even.
(t) (t) p± (z,k,k0) q± (z,k,k0) C In the following, we denote by (t) and (t) , z ∈ \{0}, four linearly r± (z,k,k0) k∈Z s± (z,k,k0) k∈Z independent solutions of (2.13) with the initial conditions:
(t) zγ (t) zγ p+ (z, k0, k0) , k0 odd, q+ (z, k0, k0) , k0 odd, = γ = −γ (2.35) r(t)(z, k , k ) γ s(t)(z, k , k ) −γ + 0 0 ( γ , k0 even, + 0 0 ( γ , k0 even. (t) γ (t) γ p− (z, k0, k0) , k0 odd, q− (z, k0, k0) , k0 odd, = −γ = γ (2.36) r(t)(z, k , k ) −zγ s(t)(z, k , k ) zγ − 0 0 ( γ , k0 even, − 0 0 ( γ , k0 even,
−it/2 (t) (t) (t) (t) where γ = e . Then it follows that p± (z,k,k0), q± (z,k,k0), r± (z,k,k0), and s± (z,k,k0), k k, k0 ∈ Z, are Laurent polynomials in z, that is, finite linear combinations of terms z , k∈ Z, with complex-valued coefficients. Since all of the above sequences satisfy the same recursion relation (2.13) which can have at most two linearly independent solutions, these sequences satisfy various identities. Some of them we state in the following lemma. 8 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
−itj /2 Lemma 2.5. Let t1,t2 ∈ [0, 2π) and γj = e , j =1, 2. Then
(t2) (t1) (t1) q± (z, ·, k0) q± (z, ·, k0) p± (z, ·, k0) = Re(γ γ ) + iIm(γ γ ) , (2.37) (t2) 1 2 (t1) 1 2 (t1) s± (z, ·, k0) s± (z, ·, k0) r± (z, ·, k0) (t2) (t1) (t1) p± (z, ·, k0) q± (z, ·, k0) p± (z, ·, k0) = iIm(γ γ ) + Re(γ γ ) , (2.38) (t2) 1 2 (t1) 1 2 (t1) r± (z, ·, k0) s± (z, ·, k0) r± (z, ·, k0) (t2) (t1) (t1) q− (z, ·, k0) γ γ − γ γ z q+ (z, ·, k0) γ γ + γ γ z p+ (z, ·, k0) = 1 2 1 2 + 1 2 1 2 , (2.39) (t2) 2zk0 (mod 2) (t1) 2zk0 (mod 2) (t1 ) s− (z, ·, k0) s+ (z, ·, k0) r+ (z, ·, k0) (t2) (t1) (t1) p− (z, ·, k0) γ1γ2 + γ1γ2z q+ (z, ·, k0) γ1γ2 − γ1γ2z p+ (z, ·, k0) (t ) = (t ) + (t ) , (2.40) 2 2zk0 (mod 2) 1 2zk0 (mod 2) 1 r− (z, ·, k0) s+ (z, ·, k0) r+ (z, ·, k0) (t2) (t1) q− (z, ·, k0 − 1) iIm(γ γ + α γ γ ) q+ (z, ·, k0) = 1 2 k0 1 2 (t2) ρ (t1) s− (z, ·, k0 − 1) k0 s+ (z, ·, k0) (t1) Re(γ γ + α γ γ ) p+ (z, ·, k0) + 1 2 k0 1 2 , (2.41) ρ (t1) k0 r+ (z, ·, k0) (t2) (t1) p− (z, ·, k0 − 1) Re(γ γ − α γ γ ) q+ (z, ·, k0) = 1 2 k0 1 2 (t2) ρ (t1) r− (z, ·, k0 − 1) k0 s+ (z, ·, k0) (t1) iIm(γ γ − α γ γ ) p (z, ·, k0) + 1 2 k0 1 2 + . (2.42) ρ (t1) k0 r+ (z, ·, k0)
−it2/2 it2/2 In particular, whenever t1 = 2arg i(αk0 e − e ) identity (2.42) simplifies to
(t ) (t ) 2 it2 1 p− (z, ·, k0 − 1) i e − αk p+ (z, ·, k0) = 0 . (2.43) (t2) ρ (t1 ) r− (z, ·, k0 − 1) k0 r+ (z, ·, k0) Proof. Since both sides of (2.37)–(2.42) satisfy the same recursion relation (2.13) it suffices to check these equalities only at one point, say at point k = k0. Substituting (2.35) and (2.36) into (2.37)– (2.40) one verifies the first four identities. Using (2.35) and (2.36) once again and applying transfer matrix T (z, k0) to the left hand-sides of (2.41)–(2.43) one verifies the last three identities.
Next, following [49] we introduce half-lattice Weyl–Titchmarsh m-functions associated with the CMV operators U (t) by ±,k0
(t) (t) (t) −1 m (z, k )= ±(δ , (U + zI)(U − zI) δ ) 2 Z ± 0 k0 ±,k0 ±,k0 k0 ℓ ( ∩[k0,±∞)) (2.44) (t) ζ + z C D = ± dµ± (ζ, k0) , z ∈ \∂ , D ζ − z I∂ where (t) (t) 2 Z D dµ± (ζ, k0)= d(δk0 , EU (ζ)δk0 )ℓ ( ∩[k0,±∞)), ζ ∈ ∂ , (2.45) ±,k0 (t) and dE (t) (·) denote the operator-valued spectral measures of the operators U , U ±,k0 ±,k0
(t) U = dE (t) (ζ) ζ. (2.46) ±,k0 U D ±,k0 I∂ MINIMALRANKDECOUPLINGOFCMVOPERATORS 9
Then following the steps of [49, Cor. 2.14] one verifies that
q(t)(z, ·, k ) p(t)(z, ·, k ) ± 0 (t) ± 0 2 Z 2 C D (t) + m± (z, k0) (t) ∈ ℓ ([k0, ±∞) ∩ ) , z ∈ \(∂ ∪{0}). (2.47) s± (z, ·, k0) r± (z, ·, k0) The special case t = 0 of the next result is proven in [49, Cor. 2.16 and Thm. 2.18]. The general case of t ∈ [0, 2π) stated below follows along the same lines and hence we omit the details for brevity.
Theorem 2.6. Let t ∈ [0, 2π) and k0 ∈ Z. Then there exist unique Caratheodory (resp. anti- (t) Caratheodory) functions M± (·, k0) such that
u(t)(z, ·, k ) q(t)(z, ·, k ) p(t)(z, ·, k ) ± 0 + 0 (t) + 0 2 Z 2 (t) = (t) + M± (z, k0) (t) ∈ ℓ ([k0, ±∞) ∩ ) , v± (z, ·, k0) s+ (z, ·, k0) r+ (z, ·, k0) z ∈ C\(∂D ∪{0}). (2.48)
(t) u± (z,k,k0) In addition, sequence (t) satisfies (2.13) and is unique (up to constant scalar multiples) v± (z,k,k0) k∈Z among all sequence that satisfy (2.13) and are square summable near ±∞.
(t) (t) We will call u± (z, ·, k0) and v± (z, ·, k0) Weyl–Titchmarsh solutions of U. Similarly, we will (t) (t) call m± (z, k0) as well as M± (z, k0) the half-lattice Weyl–Titchmarsh m-functions associated with U (t) . (See also [69] for a comparison of various alternative notions of Weyl–Titchmarsh m-functions ±,k0 for U+,k0 .) Using (2.39)–(2.42), (2.47), and Theorem 2.6 one also verifies that (t) (t) C D M+ (z, k0)= m+ (z, k0), z ∈ \∂ , (2.49) (t) M+ (0, k0)=1, (2.50) (t) (t) (t) Re(ak0 )+ iIm(bk0 )m− (z, k0 − 1) (1 − z)m− (z, k0)+(1+ z) M− (z, k0)= (t) = (t) iIm(ak0 ) + Re(bk0 )m− (z, k0 − 1) (1 + z)m− (z, k0)+(1 − z) m(t)(z, k )+1 − z m(t)(z, k ) − 1 − 0 − 0 C D = (t) (t) , z ∈ \∂ , (2.51) m− (z, k0)+1 + z m− (z, k0) − 1 it (t) αk0 + e M− (0, k0)= it , (2.52) αk0 − e (t) (t) Re(ak0+1) − iIm(ak0+1)M− (z, k0 + 1) m− (z, k0)= (t) Re(bk0+1)M− (z, k0 + 1) − iIm(bk0+1) z M (t)(z, k )+1 − M (t)(z, k ) − 1 − 0 − 0 C D = (t) (t) , z ∈ \∂ , (2.53) z M− (z, k0)+1 + M− (z, k0) − 1
−it −it Z (t) where ak =1+ e αk and bk =1 − e αk, k ∈ . In particular, one infers that M± are analytic at z = 0. (t) Z Next, we introduce the Schur (resp. anti-Schur) functions Φ± (·, k), k ∈ , by
(t) (t) M± (z, k) − 1 C D Φ± (z, k)= (t) , z ∈ \∂ . (2.54) M± (z, k)+1 10 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Then by (2.53) and (2.54),
(t) (t) (t) 1+Φ± (z, k) (t) z − Φ− (z, k) C D M± (z, k)= (t) , m− (z, k)= (t) , z ∈ \∂ . (2.55) 1 − Φ± (z, k) z +Φ− (z, k)
Moreover, it follows from (2.35), (2.54), and Theorem 2.6 that
(t) it v± (z,k,k0) ze (t) , k odd, (t) u± (z,k,k0) Z C D Φ± (z, k)= (t) k ∈ , z ∈ \∂ , (2.56) it u± (z,k,k0) e (t) , k even, v± (z,k,k0)
(t) (t) where u± (·,k,k0) and v± (·,k,k0) are the sequences defined in (2.48). Since the Weyl–Titchmarsh (t) u± (z,k,k0) solution (t) is unique up to a multiplicative constant, we conclude from (2.56) that v± (z,k,k0) k∈Z −it (t) e Φ± (·, k) is actually t-independent. Thus, fixing t1,t2 ∈ [0, 2π), one computes
(t2) i(t2−t1) (t1) Φ± (·, k)= e Φ± (·, k), (2.57) (t ) iIm(ei(t2−t1)/2) + Re(ei(t2−t1)/2)M 1 (·, k) (t2) ± Z M± (·, k)= (t ) , k ∈ . (2.58) i(t2−t1)/2 i(t2−t1)/2 1 Re(e )+ iIm(e )M± (·, k)
Finally, following [49], we obtain the following identities,
(t) k0 (mod 2) (t) r+ (z,k,k0)= z p+ (1/z,k,k0), (2.59)
(t) k0 (mod 2) (t) s+ (z,k,k0)= −z q+ (1/z,k,k0), (2.60)
(t) (k0+1) (mod2) (t) r− (z,k,k0)= −z p− (1/z,k,k0), (2.61)
(t) (k0+1) (mod 2) (t) s− (z,k,k0)= z q− (1/z,k,k0), (2.62)
and provide formulas for the resolvents of the half-lattice CMV operators U (t) and the full-lattice ±,k0 CMV operator U. In the special case t = 0 these formulas were obtained in (2.63)–(2.66), (2.171), (2.172), and (3.7) of [49]. MINIMALRANKDECOUPLINGOFCMVOPERATORS 11
−1 Lemma 2.7. Let t ∈ [0, 2π), k ∈ Z, and z ∈ C\(∂D ∪{0}). Then the resolvent U (t) − zI is 0 ±,k0 given in terms of its matrix representation in the standard basis of ℓ2([k , ±∞) ∩ Z) by 0 −k (mod 2) −1 z 0 U (t) − zI (k, k′)= +,k0 2z (t) (t) ′ ′ ′ p+ (z,k,k0)v+ (z, k , k0), k
−k0 (mod 2) −1 ′ −z (U − zI) (k, k )= (t) (t) 2z[M+ (z, k0) − M− (z, k0)] (t) (t) ′ ′ ′ u− (z,k,k0)v+ (z, k , k0), k
(t) (t) ′ ′ ′ u− (z,k,k0)u+ (1/z, k , k0), k
3. CMV operators with matrix-valued coefficients In the remainder of this paper, Cm×m denotes the space of m × m matrices with complex-valued Cm entries endowed with the operator norm k·kCm×m (we use the standard Euclidean norm in ). The m×m ∗ m adjoint of an element γ ∈ C is denoted by γ , Im denotes the identity matrix in C , and the real and imaginary parts of γ are defined as usual by Re(γ) = (γ + γ∗)/2 and Im(γ) = (γ − γ∗)/(2i). 12 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Remark 3.1. For simplicity of exposition, we find it convenient to use the following conventions: We denote by s(Z) the vector space of all C-valued sequences, and by s(Z)m = s(Z) ⊗ Cm the vector space of all Cm-valued sequences; that is, . . (φ(k))1 φ(−1) (φ(k))2 m m φ = {φ(k)}k∈Z = φ(0) ∈ s(Z) , φ(k)= . ∈ C , k ∈ Z. (3.1) . φ(1) . . (φ(k))m . m×n m n m×n Moreover, we introduce s(Z) = s(Z) ⊗ C , m,n ∈ N, that is, Φ = (φ1,...,φn) ∈ s(Z) , m where φj ∈ s(Z) for all j =1,...,n. We also note that s(Z)m×n = s(Z)⊗Cm×n, m,n ∈ N; which is to say that the elements of s(Z)m×n can be identified with the Cm×n-valued sequences, . . Φ(−1) (Φ(k))1,1 ... (Φ(k))1,n . . m×n Φ= {Φ(k)} Z = Φ(0) , Φ(k)= . . ∈ C , k ∈ Z, (3.2) k∈ . . Φ(1) (Φ(k))m,1 ... (Φ(k))m,n . . by setting Φ = (φ1,...,φn), where . . (Φ(k)) φj (−1) 1,j φ = φ (0) ∈ s(Z)m, φ (k)= . ∈ Cm, j =1,...,n, k ∈ Z. (3.3) j j j . φj (1) (Φ(k))m,j . . For the elements of s(Z)m×n we define the right-multiplication by n × n matrices with complex- valued entries by
c1,1 ... c1,n n n ΦC = (φ ,...,φ ) . . = φ c ,..., φ c ∈ s(Z)m×n (3.4) 1 n . . j j,1 j j,n j=1 j=1 cn,1 ... cn,n X X for all Φ ∈ s(Z)m×n and C ∈Cn×n. In addition, for any linear transformation A : s(Z)m → s(Z)m, m×n we define AΦ for all Φ = (φ1,...,φn) ∈ s(Z) by m×n AΦ = (Aφ1,..., Aφn) ∈ s(Z) . (3.5) Given the above conventions, we note the subspace containment: ℓ2(Z)m = ℓ2(Z) ⊗ Cm ⊂ s(Z)m and ℓ2(Z)m×n = ℓ2(Z) ⊗ Cm×n ⊂ s(Z)m×n. We also note that ℓ2(Z)m represents a Hilbert space with scalar product given by ∞ m 2 m (φ, ψ)ℓ2 (Z)m = (φ(k))j (ψ(k))j , φ, ψ ∈ ℓ (Z) . (3.6) −∞ j=1 k=X X Finally, we note that a straightforward modification of the above definitions also yields the Hilbert space ℓ2(J)m as well as the sets ℓ2(J)m×n, s(J)m, and s(J)m×n for any J ⊂ Z. We start by introducing our basic assumption: MINIMALRANKDECOUPLINGOFCMVOPERATORS 13
Hypothesis 3.2. Let m ∈ N and assume α = {αk}k∈Z is a sequence of m×m matrices with complex entries1 and such that Z kαkkCm×m < 1, k ∈ . (3.7)
Given a sequence α satisfying (3.7), we define two sequences of positive self-adjoint m×m matrices {ρk}k∈Z and {ρk}k∈Z by
∗ 1/2 Z e ρk = [Im − αkαk] , k ∈ , (3.8) ∗ 1/2 Z ρk = [Im − αkαk] , k ∈ , (3.9)
Cm×m Cm×m and two sequences of m×m matricese with positive real parts, {ak}k∈Z ⊂ and {bk}k∈Z ⊂ by
ak = Im + αk, k ∈ Z, (3.10)
bk = Im − αk, k ∈ Z. (3.11)
Then (3.7) implies that ρk and ρk are invertible matrices for all k ∈ Z, and using elementary power series expansions one verifies the following identities for all k ∈ Z,
±1 ±1 e ∗ ±1 ±1 ∗ ∗ −2 −2 ∗ ρk αk = αkρk , αkρk = ρk αk, akρk ak = akρk ak, (3.12) ∗ −2 −2 ∗ ∗ −2 −2 ∗ ∗ −2 −2 ∗ bkρk bk = bkρk bk, akρk bk + akρk bk = bkρk ak + bkρk ak =2Im. (3.13) e e e According to Simon [71], we call α the Verblunsky coefficients in honor of Verblunsky’s pioneering e k e e work in the theory of orthogonal polynomials on the unit circle [80], [81]. Next, we introduce a sequence of 2 × 2 block unitary matrices Θk with m × m matrix coefficients by −α ρ Θ = k k , k ∈ Z, (3.14) k ρ α∗ k k and two unitary operators V and W on ℓ2(Z)m bye their matrix representations in the standard basis of ℓ2(Z)m by
. . .. 0 .. 0 Θ2k−2 Θ2k−1 V = , W = , (3.15) Θ Θ 2k 2k+1 . . 0 .. 0 .. where
V2k−1,2k−1 V2k−1,2k W2k,2k W2k,2k+1 =Θ2k, =Θ2k+1, k ∈ Z. (3.16) V − V W W 2k,2k 1 2k,2k 2k+1,2k 2k+1,2k+1 Moreover, we introduce the unitary operator U on ℓ2(Z)m as the product of the unitary operators V and W by U = VW, (3.17)
1 m×m We emphasize that αk ∈ C , k ∈ Z, are general (not necessarily normal) matrices. 14 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
or in matrix form in the standard basis of ℓ2(Z)m, by ...... ∗ 0 0 −α0ρ−1 −α0α−1 −ρ0α1 ρ0ρ1 ∗ ∗ ∗ ρ0ρ−1 ρ0α−1 −α0α1 α0ρ1 0 U = ∗ . (3.18) 0 −αe2ρ1 −eα2eα1 −ρ2α3 ρ2ρ3 ∗ ∗ ∗ ρ2ρ1 ρ2α −α α3 α ρ3 0 e1 2 2 0 . . . . . .. .. e .. e e .. .. e ∗ ∗ Z U Here terms of the form −α2kα2k−1 and −α2kα2k+1, k ∈ , represent the diagonal entries 2k−1,2k−1 and U2k,2k of the infinite matrix U in (3.18), respectively. Then, with δeven and δodd defined in (2.10), and by analogy with (2.9), we see that as an operator acting upon ℓ2(Z)m, U can be represented by − −− − ∗ ∗ − ∗ + + ∗ U = ρρ δevenS + [ρ(α ) δeven − α ρ δodd]S − α α δeven − α α δodd ∗ + + ++ + + ++ ++ + [α ρ δeven − ρ α δodd]S + ρ ρ δoddS . (3.19) We continue to call the operator U on ℓ2(Z)m the CMV operator since (3.14)–(3.18) in the context of the scalar-valued semi-infinitee (i.e.,e half-lattice) case weree e obtained by Cantero, Moral, and Vel´azquez in [17] in 2003. Then, in analogy with Lemma 2.2, the following result is proven in [19]: m×m Lemma 3.3. Let z ∈ C\{0} and {U(z, k)}k∈Z, {V (z, k)}k∈Z be two C -valued sequences. Then the following items (i)–(iii) are equivalent: (i) (UU(z, ·))(k)= zU(z, k), (WU(z, ·))(k)= zV (z, k), k ∈ Z. (3.20) (ii) (WU(z, ·))(k)= zV (z, k), (VV (z, ·))(k)= U(z, k), k ∈ Z. (3.21) U(z, k) U(z, k − 1) (iii) = T(z, k) , k ∈ Z. (3.22) V (z, k) V (z, k − 1) Here U, V, and W are understood in the sense of difference expressions on s(Z)m×m rather than difference operators on ℓ2(Z)m (cf. Remark 3.1) and the transfer matrices T(z, k), z ∈ C\{0}, k ∈ Z, are defined by ρ−1α zρ−1 k k k odd, −1 −1 −1 ∗ , k z ρk ρk αk! T (z, k)= e e (3.23) −1 ∗ −1 ρ α ρ k k k , k even. ρ−1 ρ−1α k k k! Definition 3.4. If for some reference point, k ∈ Z, we modify Hypothesis 3.2 by allowing α = γ, 0 k0 where γ ∈ Cm×m is unitary, then the resultinge operatorse defined in (3.15)–(3.17) will be denoted by V(γ), W(γ) and U(γ). k0 k0 k0 Z Remark 3.5. Strictly, speaking, allowing αk0 to be unitary for some reference point k0 ∈ contradicts our basic Hypothesis 3.2. However, as long as the exception to Hypothesis 3.2 refers to only one site, we will safely ignore this inconsistency in favor of the notational simplicity it provides by avoiding (γ) the introduction of a properly modified hypothesis on α = {α } Z and will refer to U as a CMV k k∈ k0 operator. The operator U(γ) splits into a direct sum of two half-lattice operators U(γ) and U(γ) acting k0 −,k0−1 −,k0 2 m 2 m on ℓ ((−∞, k0 − 1] ∩ Z) and on ℓ ([k0, ∞) ∩ Z) , respectively. Explicitly, one obtains U(γ) = U(γ) ⊕ U(γ) in ℓ2((−∞, k − 1] ∩ Z)m ⊕ ℓ2([k , ∞) ∩ Z)m. (3.24) k0 −,k0−1 +,k0 0 0 MINIMALRANKDECOUPLINGOFCMVOPERATORS 15
Similarly, one obtains W(γ) , V(γ) and W(γ) , V(γ) such that −,k0−1 −,k0−1 +,k0 +,k0
V(γ) = V(γ) ⊕ V(γ) , W(γ) = W(γ) ⊕ W(γ) , (3.25) k0 −,k0−1 +,k0 k0 −,k0−1 +,k0 and hence U(γ) = V(γ) W(γ) , U(γ) = V(γ)W(γ). (3.26) ±,k0 ±,k0 ±,k0 k0 k0 k0
For the special case when γ = Im, we simplify our notation by writing U = U(γ=Im) = V(γ=Im)W(γ=Im) = V W (3.27) k0 k0 k0 k0 k0 k0 U = U(γ=Im) = V(γ=Im)W(γ=Im) = V W (3.28) ±,k0 ±,k0 ±,k0 ±,k0 ±,k0 ±,k0 In analogy with the scalar case, when emphasizing dependence of these operators on the sequence (γ) α = {α } Z, we write, for example, U and U . Also in analogy to the scalar case, let σ, τ ∈ k k∈ α α;k0 Cm×m be unitary, and let A and A be the unitary operators defined on ℓ2(Z)m by A A = σδe odd + τδeven, = τδodd + σδeven, (3.29) Then, for the full-lattice and direct sum of half-lattice CMV operators, we obtain the following e analogs of (2.21) and (2.22):
∗ ∗ ∗ AUβA = AVβA AWβA = VαWα = Uα, (3.30) ∗ ∗ ∗ AU(γ) A∗ = AV(γ) A∗ AW(γ) A∗ = V(σγτ )W(σγτ ) = U(σγτ ). (3.31) β;k0 eβ;k0 e β;k0 α;k0 α;k0 α;k0 ∗ −1/2 1/2 where β = {βk}k∈Z, and αk = σβkτ .e In particular,e when σ = γ , τ = γ , we note, by (3.27) and (3.31), that AU(γ) A∗ = V W = U . (3.32) β;k0 α;k0 α;k0 α;k0 The unitary transformations cited in (3.29)–(3.31) are relevant to our next result because of the following observation about n × n complex matrices: Let α ∈ Cm×m, where we are not assuming that α is unitary. Then, α has a (not necessarily unique) polar decomposition, α = U|α|, where U, |α| ∈ Cm×m, U is unitary, and |α| = (α∗α)1/2 ≥ 0 is nonnegative (cf., e.g., [52, Theorem 3.1.9 ∗ Cm×m (c)]). The nonnegative matrix |α| can then be diagonalized: |α| = U0 DU0, where U0,D ∈ , m×m U0 is unitary, and D is diagonal. Thus, each α ∈ C has a (not necessarily unique) factorization of the form α = σβτ, (3.33) where, σ, β, τ ∈ Cm×m, where σ and τ are unitary, and where β is diagonal. We now present our principal result on rank m perturbations: Theorem 3.6. Given U , fix Z and let ∗ be a factorization for , as described α k0 ∈ αk0 = σk0 βk0 τk0 αk0 Cm×m Cm×m in (3.33), where σk0 , τk0 ∈ are unitary, and where βk0 ∈ is the diagonal matrix β = diag[β ,...,β ]. Let γ = σ θ τ ∗ , j = 1, 2, where θ = diag[eit1 ,...,eitm ], and θ = k0 k0,1 k0,m j k0 j k0 1 2 (γ ,γ ) diag[eis1 ,...,eism ]. Let U 1 2 denote the unitary operator acting on ℓ2(Z)m and defined by α;k0
U(γ1,γ2) = U(γ1) ⊕ U(γ2) . (3.34) α;k0 α;−,k0−1 α;+,k0
Then, for an arbitrary unitary matrix γ ∈ Cm×m, the difference U − U(γ) has rank greater than α α;k0 −1 m, while the differences U − U(γ1,γ2) and (U − zI)−1 − U(γ1,γ2) − zI are of rank m if and only α α;k0 α α;k0 if t = 2 arg[i(β e−isj /2 − eisj /2)], j =1,...,m, and otherwise possess rank greater than m. j k0,j 16 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Proof. U(γ) = V(γ) W(γ) by (3.26), and hence, it follows from (3.15) that α;k0 α;k0 α;k0 (γ) (γ) Vα Wα − W , k0 odd, U − U = α;k0 (3.35) α α;k0 (γ) Vα − V Wα, k0 even. ( α;k0 For k odd, D = W − W(γ) is block-diagonal with all of its 2m × 2m blocks on the diagonal being 0 α α;k0 zero except for one which has the following form
Dk0−1,k0−1 Dk0−1,k0 −αk0 + γ ρk0 A = = ∗ ∗ (3.36) Dk ,k −1 Dk ,k ρk α − γ 0 0 0 0 0 k0 e Note that the following subspaces have only trivial intersection with ker(A) since ρk0 and ρk0 are invertible matrices, ξ 0 e S = ∈ C2m | ξ ∈ Cm , S = ∈ C2m | η ∈ Cm . (3.37) 1 0 2 η As a consequence, rank(A) > m. Further note, for any c ∈ C, that ξ ∈ ker(A) (3.38) cξ only when ξ =0 ∈ Cm. To see this, assume that (3.38) holds for some c ∈ C and some ξ 6=0 ∈ Cm. It follows that ∗ 0 cξ ξ −cξ¯ ∗(α − γ)ξ + |c|2ξ∗ρ ξ = A = k0 k0 (3.39) 0 ξ cξ ξ∗ρξ + cξ∗(α − γ)∗ξ k0 and hence by conjugation in the first line of (3.39) that e ∗ ∗ 2 ∗ 0= −cξ (αk0 − γ) ξ + |c| ξ ρk0 ξ, (3.40) ∗ ∗ ∗ 0= ξ ρξ + cξ (αk0 − γ) ξ. (3.41) e ∗ 2 Summing these equations, we see that ξ (ρk0 + |c| ρk0 )ξ = 0. However, strict positivity of the 2 Cm self-adjoint matrix ρk0 + |c| ρk0 implies that ξ =0 ∈ ; a contradiction. Noting again that rank(A) > m, assume that rank(eA)= m. Given that 2em C = S1 ⊕ S2, ker(A) ∩ Sj = {0}, j =1, 2, (3.42) then there exists a matrix M ∈ Cm×m such that ξ ker(A)= | ξ ∈ Cm . (3.43) Mξ However, this implies the existence of some ξ 6= 0 ∈ Cm and some c ∈ C such that (3.38) holds; thus, again a contradiction. Hence, the rank of A, and as a consequence the rank of U − U(γ) , are α α;k0 greater than m. The proof when k0 ∈ Z is even is similar to that just given. Now consider the rank of the difference U − U(γ1,γ2), first noting by (3.25) and (3.26) that α α;k0 U(γ1,γ2) = V(γ1,γ2)W(γ1,γ2) and hence, by (3.15) that α;k0 α;k0 α;k0
(γ1,γ2) (γ ,γ ) Vα Wα − W , k0 odd, U − U 1 2 = α;k0 (3.44) α α;k0 (γ1,γ2) Vα − V Wα, k0 even. ( α;k0 We again proceed by assuming that k ∈ Z is odd, and letting D = W − W(γ1,γ2) be the block- 0 α α;k0 diagonal matrix all of whose 2m × 2m blocks on the diagonal are zero except for one which has the MINIMALRANKDECOUPLINGOFCMVOPERATORS 17
following form:
∗ Dk0−1,k0−1 Dk0−1,k0 −αk0 ρk0 −σk0 θ1τk0 0 = ∗ − ∗ ∗ (3.45) Dk ,k −1 Dk ,k ρk α 0 τk θ σ 0 0 0 0 0 k0 0 2 k0 e τ ∗ 0 σk0 0 −βk0 + θ1 κk0 k0 = ∗ ∗ ∗ , (3.46) 0 τk κk β − θ 0 σ 0 0 k0 2 k0
where κ = (I − β β∗ )1/2 = (I − β∗ β )1/2 = diag[κ ,...,κ ]. Then, the rank of the k0 m k0 k0 m k0 k0 k0,1 k0,m difference D = W − W(γ1,γ2) equals the rank of the matrix α α;k0
−βk0 + θ1 κk0 B = ∗ ∗ . (3.47) κk β − θ 0 k0 2
Since all m × m matrices on the right-hand side of (3.47) are diagonal, performing an equivalent set of permutations on the rows and on the columns of the matrix B in (3.47) results in a block diagonal matrix with 2 × 2 blocks arrayed along the diagonal, each of the form,
itj −βk0,j + e κk0,j Bj = −isj , j =1, . . . , m. (3.48) κk ,j β − e 0 k0,j
It follows that rank(B) = m precisely when rank(Bj ) = 1 for all j = 1,...,m, and otherwise, that rank(B) > m. We now observe that each Bj has the form given in (2.26). Hence, by the derivation following from (2.27) we see that det(Bj ) = 0, and hence that rank(Bj ) = 1, precisely −isj /2 isj /2 when tj = 2 arg[i(βk0,j e − e )]. As in the scalar case discussed in Theorem 2.3, the statement in this theorem for the resolvents follows from the result just proven for the difference U − U(γ1,γ2) and from the identity, α α;k0
−1 −1 (U − zI)−1 − U(γ1,γ2) − zI = −(U − zI)−1 U − U(γ1,γ2) U(γ1,γ2) − zI . (3.49) α α;k0 α α α;k0 α;k0
Remark 3.7. In particular, we note that the difference U − U(γ) has rank greater than m when α α;k0 γ = I . m Next we present formulas for different unitary γ ∈ Cm×m, linking various spectral theoretic objects associated with half-lattice CMV operators U(γ) . For the special case when γ = I , these ±,k0 m objects, and the relationships described below, have proven exceptionally useful (see, e.g., [19], [20], [49], [50], and [87]). We begin with an analog of Lemma 3.3 for difference expressions U(γ) , V(γ) , and W(γ) . For ±,k0 ±,k0 ±,k0 the special case γ = Im, the result below is proven in [19, Lemma 2.4]. The general case below follows immediately from the special case and the observation of unitary equivalence in (3.32). 18 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
m×m (γ) Lemma 3.8. Let z ∈ C\{0}, k0 ∈ Z, and let γ ∈ C be unitary. Let P (z,k,k0) , + k≥k0 (γ) m×m R (z,k,k0) be two C -valued sequences. Then the following items (i)–(iii) are equiva- + k≥k0 lent: b b (i) U(γ) P (γ)(z, ·, k ) (k)= zP (γ)(z,k,k ), +,k0 + 0 + 0 (γ) (γ) (γ) W P (z, ·, k0) (k)= zR (z,k,k0), k ≥ k0. (3.50) +,k0b + b + (γ) (γ) (γ) (ii) W P (z, ·, k0)(k)= zR (z,k,k0), +,k0 b+ b+ (γ) (γ) (γ) V R (z, ·, k0) (k)= P (z,k,k0), k ≥ k0. (3.51) +,k0 b+ +b P (γ)(z,k,k ) P (γ)(z, k − 1, k ) + b 0 T b + 0 (iii) (γ) = (z, k) (γ) , k>k0, R (z,k,k0) R (z, k − 1, k0) b+ b+ (γ) (γ) zγR+ (z, k0, k0), k0 odd, P+b (z, k0, k0)= ∗ (γ) b (3.52) (γ R+ (z, k0, k0), k0 even. b b (γ) (γ) b m×m Similarly, let P (z,k,k0) , R (z,k,k0) be two C -valued sequences. Then the − k≤k0 − k≤k0 following items (iv)–(vi)are equivalent: b b (iv) U(γ) P (γ)(z, ·, k ) (k)= zP (γ)(z,k,k ), −,k0 − 0 − 0 (γ) (γ) (γ) W P (z, ·, k0) (k)= zR (z,k,k0), k ≤ k0. (3.53) −,k0b − b − (γ) (γ) (γ) (v) W P (z, ·, k0)(k)= zR (z,k,k0), −,k0 b− b− (γ) (γ) (γ) V R (z, ·, k0) (k)= P (z,k,k0), k ≤ k0. (3.54) −,k0 b− −b P (γ)(z, k − 1, k ) P (γ)(z,k,k ) − b 0 T b −1 − 0 (vi) (γ) = (z, k) (γ) , k ≤ k0, R− (z, k − 1, k0) R− (z,k,k0) b b (γ) (γ) −γR− (z, k0, k0), k0 odd, P−b (z, k0, k0)= ∗ (γ) b (3.55) (−zγ R− (z, k0, k0), k0 even. b b (γ) (γ) P± (z,k,k0) Qb± (z,k,k0) C Next, we denote by (γ) and (γ) , z ∈ \{0}, four linearly independent R± (z,k,k0) k∈Z S± (z,k,k0) k∈Z solutions of (3.21) satisfying the following initial conditions:
(γ) zγ1/2 (γ) zγ1/2 P+ (z, k0, k0) γ−1/2 , k0 odd, Q+ (z, k0, k0) −γ−1/2 , k0 odd, = − = − (3.56) (γ) γ 1/2 (γ) −γ 1/2 R+ (z, k0, k0) 1/2 , k even, S+ (z, k0, k0) 1/2 , k even. γ 0 γ 0 (γ) γ1/2 (γ) γ1/2 P− (z, k0, k0) −1/2 , k0 odd, Q− (z, k0, k0) −1/2 , k0 odd, = −γ = γ (3.57) (γ) −zγ−1/2 (γ) zγ−1/2 R− (z, k0, k0) 1/2 , k even, S− (z, k0, k0) 1/2 , k even. γ 0 γ 0 (γ) Z Cm×m Then, P± (z,k,k0), Q(γ)±(z,k,k0), R(γ)±(z,k,k0), and S(γ)±(z,k,k0), k, k0 ∈ , are - valued Laurent polynomials in z. MINIMALRANKDECOUPLINGOFCMVOPERATORS 19
m×m Lemma 3.9. Let z ∈ C\{0}, k0 ∈ Z, and let γj ∈ C , j = 1, 2, be unitary. With C1 = 1 −1/2 1/2 1/2 −1/2 1 −1/2 1/2 1/2 −1/2 2 (γ1 γ2 + γ1 γ2 ) and D1 = 2 (γ1 γ2 − γ1 γ2 ), then,
(γ2) (γ1) (γ1) Q± (z, ·, k0) Q± (z, ·, k0) P± (z, ·, k0) = C + D (3.58) (γ2) (γ1) 1 (γ1) 1 S± (z, ·, k0) S± (z, ·, k0) R± (z, ·, k0) (γ2) (γ1) (γ1) P± (z, ·, k0) Q± (z, ·, k0) P± (z, ·, k0) = D + C . (3.59) (γ2) (γ1) 1 (γ1) 1 R± (z, ·, k0) S± (z, ·, k0) R± (z, ·, k0)
k0 (mod 2) −1 −1/2 1/2 1/2 −1/2 k0 (mod 2) −1 −1/2 1/2 1/2 −1/2 With C2 = (2z ) (γ1 γ2 − zγ1 γ2 ), D2 = (2z ) (γ1 γ2 + zγ1 γ2 ), then
(γ2) (γ1) (γ1) Q− (z, ·, k0) Q+ (z, ·, k0) P+ (z, ·, k0) = C + D , (3.60) (γ2) (γ1) 2 (γ1) 2 S− (z, ·, k0) S+ (z, ·, k0) R+ (z, ·, k0) (γ2) (γ1) (γ1) P− (z, ·, k0) Q+ (z, ·, k0) P+ (z, ·, k0) = D + C . (3.61) (γ2) (γ1) 2 (γ1) 2 R− (z, ·, k0) S+ (z, ·, k0) R+ (z, ·, k0) With C = 1 (γ−1/2ρ−1α γ−1/2 − γ1/2ρ−1α∗ γ1/2)+ 1 (γ−1/2ρ−1γ1/2 − γ1/2ρ−1γ−1/2) and D = 3 2 1 k0 k0 2 1 k0 k0 2 2 1 k0 2 1 k0 2 3 1 (γ−1/2ρ−1α γ−1/2 + γ1/2ρ−1α∗ γ1/2)+ 1 (γ−1/2ρ−1γ1/2 + γ1/2ρ−1γ−1/2), then 2 1 k0 k0 2 1 k0 k0 2 2 1 k0 2 1 k0 2 e e (γ2) (γ1) (γ1) Q− (z, ·, k0 − 1) Q (z, ·, k0) P (z, ·, k0) e = + e C + + D . (3.62) (γ2) (γ1) 3 (γ1) 3 S− (z, ·, k0 − 1) S+ (z, ·, k0) R+ (z, ·, k0) With C = − 1 (γ−1/2ρ−1α γ−1/2 + γ1/2ρ−1α∗ γ1/2)+ 1 (γ−1/2ρ−1γ1/2 + γ1/2ρ−1γ−1/2) and D = 4 2 1 k0 k0 2 1 k0 k0 2 2 1 k0 2 1 k0 2 4 − 1 (γ−1/2ρ−1α γ−1/2 − γ1/2ρ−1α∗ γ1/2)+ 1 (γ−1/2ρ−1γ1/2 − γ1/2ρ−1γ−1/2), 2 1 k0 k0 2 1 k0 k0 2 2 1 k0 2 1 k0 2 e e (γ2) (γ1) (γ1) P− (z, ·, k0 − 1) Q+ (z, ·, k0) P+ (z, ·, k0) e = e C + D . (3.63) (γ2) (γ1) 4 (γ1) 4 R− (z, ·, k0 − 1) S+ (z, ·, k0) R+ (z, ·, k0) Proof. Since each sequence in equations (3.58)–(3.63) satisfies the recurrence relation (3.22), it is necessary only to check equality at k = k0. Substituting (3.56), (3.57) into (3.58)–(3.61) suffices to verify the identities. In addition to use of (3.56), (3.57), application of the transfer matrix T(z, k0) in (3.23) to the left sides of (3.62), (3.63) suffices in the verification of the later two identities. Next, we introduce half-lattice Weyl-Titchmarsh m-functions associated with the half-lattice CMV operators, U(γ) , described in (3.26). ±,k0 m×m Let ∆k = {∆k(ℓ)}ℓ∈Z ∈ s(Z) , k ∈ Z, denote the sequences of m × m matrices defined by
Im, ℓ = k, (∆k)(ℓ)= k,ℓ ∈ Z. (3.64) (0, ℓ 6= k, Then using right-multiplication by m × m matrices on s(Z)m×m defined in Remark 3.1, we get the identity
X, ℓ = k, m×m (∆kX)(ℓ)= X ∈ C , (3.65) (0, ℓ 6= k, m×m m×m and hence consider ∆k as a map ∆k : C → s(Z) . In addition, we introduce the map ∗ Z m×m Cm×m Z ∆k : s( ) → , k ∈ , defined by ∗ Z m×m ∆kΦ=Φ(k), where Φ = {Φ(k)}k∈Z ∈ s( ) . (3.66) 20 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Similarly, one introduces the corresponding maps with Z replaced by [k0, ±∞) ∩ Z, k0 ∈ Z, which, ∗ for notational brevity, we will also denote by ∆k and ∆k, respectively. Z Cm×m (γ) Cm×m For k0 ∈ , and for a unitary γ ∈ , let dΩ± (·, k0) denote the -valued measures on ∂D defined by (γ) ∗ (γ) D dΩ± (ζ, k0)= d(∆k0 EU (ζ)∆k0 ), ζ ∈ ∂ , (3.67) ±,k0 (γ) where E (γ) (·) denotes the family of spectral projections for the half-lattice unitary operators U , U ±,k0 ±,k0
(γ) U = dE (γ) (ζ) ζ. (3.68) ±,k0 U D ±,k0 I∂ (γ) Then, the half-lattice Weyl-Titchmarsh m-functions, m± (z, k0), are defined by
m(γ)(z, k )= ±∆∗ (U(γ) + zI)(U(γ) − zI)−1∆ (3.69) ± 0 k0 ±,k0 ±,k0 k0 (γ) ζ + z C D = ± dΩ± (ζ, k0) , z ∈ \∂ , (3.70) D ζ − z I∂ with (γ) dΩ± (ζ, k0)= Im. (3.71) D I∂ (γ) As defined, we note that m± (z, k) are matrix-valued Caratheodory functions: see Appendix A for definition and further properties. Proof of the next result in the special case when γ = Im can be found in [19, Lemma 2.13, Theorem 2.17]. The proof in the case for general unitary γ ∈ Cm×m follows the same lines and is omitted here for the sake of brevity.
Theorem 3.10. Let k ∈ Z. Then, for each unitary γ ∈ Cm×m, for P (γ) , Q(γ) , R(γ) , S(γ) 0 ±,k0 ±,k0 ±,k0 ±,k0 (γ) defined in (3.56) and (3.57), and for m± (z, k0) defined in (3.69), the following relations hold for z ∈ C\(∂D ∪{0}): (γ) (γ) (γ) (γ) 2 Z m×m U± (z, ·, k0)= Q± (z, ·, k0)+ P± (z, ·, k0)m± (z, k0) ∈ ℓ ([k0, ±∞) ∩ ) , (3.72) V (γ)(z, ·, k )= S(γ)(z, ·, k )+ R(γ)(z, ·, k )m(γ)(z, k ) ∈ ℓ2([k , ±∞) ∩ Z)m×m. (3.73) b± 0 ± 0 ± 0 ± 0 0 Cm×m (γ) C D Moreover,b there exist unique -valued functions M± (·, k0) such that for all z ∈ \(∂ ∪{0}), (γ) (γ) (γ) (γ) 2 Z m×m U± (z, ·, k0)= Q+ (z, ·, k0)+ P+ (z, ·, k0)M± (z, k0) ∈ ℓ ([k0, ±∞) ∩ ) , (3.74) (γ) (γ) (γ) (γ) 2 Z m×m V± (z, ·, k0)= S+ (z, ·, k0)+ R+ (z, ·, k0)M± (z, k0) ∈ ℓ ([k0, ±∞) ∩ ) . (3.75) Remark 3.11. Within the proof of Theorem 3.10, one observes that the sequences,
(γ) U± (z,k,k0) , (3.76) V (γ)(z,k,k ) ± 0 k∈Z defined by (3.74) and (3.75), are unique up to right-multiplication by constant m × m matrices. (γ) U (γ) Hence, we shall call U± (z, ·, k0) the Weyl–Titchmarsh solutions of . Note also that m± (z, k0), as well as M (γ)(z, k ), are said to be half-lattice Weyl–Titchmarsh m-functions associated with U(γ) . ± 0 ±,k0 (See also [69] for a comparison of various alternative notions of Weyl–Titchmarsh m-functions for U(γ) with scalar-valued Verblunsky coefficients.) +,k0 MINIMALRANKDECOUPLINGOFCMVOPERATORS 21
m×m For fixed k0 ∈ Z, z ∈ C\(∂D ∪{0}), and unitary γ ∈ C , using (3.70) and Theorem 3.10, we obtain, (γ) (γ) (γ) M+ (z, k0)= m+ (z, k0),M+ (0, k0)= Im. (3.77) In particular, by (3.77) and the uniqueness up to right multiplication by constant m × m matrices noted in Remark 3.11, note for (3.72)–(3.75) that (γ) (γ) (γ) (γ) U+ (z,k,k0)= U+ (z,k,k0), V+ (z,k,k0)= V+ (z,k,k0). (3.78)
Following the line of reasoning presented for the special case when γ = Im found in [19], one uses the equations in Lemmab 3.9, together with Theoremb 3.10, to obtain the following equations where C3,D3, and C4,D4 are defined in (3.62) and (3.63) respectively, with γ = γ1 = γ2. (γ) (γ) (γ) −1 M− (z, k0)= D3 + D4m− (z, k0 − 1) C3 + C4m− (z, k0 − 1) (3.79) (γ) (γ) = m− (z, k0)+ Im + z(m− (z, k0) − Im) (3.80) (γ) (γ) −1 C D × m− (z, k0)+ Im − z(m− (z, k0) − Im) , z ∈ \(∂ ∪{0}), M (γ)(0, k ) = [γ−1/2ρ−1γ−1/2 + γ1/2ρ−1γ1/2][γ−1/2ρ−1γ−1/2 − γ1/2ρ−1γ1/2]−1, (3.81) − 0 k0 k0 k0 k0 (γ) (γ) (γ) m− (z, k0)= z(M− (z, k0)+ Im) − (M− (z, k0) − Im) (3.82) e e (γ) (γ) −1 C D × z(M− (z, k0)+ Im) + (M− (z, k0) − Im) , z ∈ \(∂ ∪{0}). (γ) By their relations to the Caratheodory functions m± (z, k) given in (3.77) and (3.79), we see that (γ) M± (z, k) are also matrix-valued Caratheodory functions. Cm×m (γ) Z Next, we introduce the -valued Schur functions Φ± (·, k), k ∈ , by (γ) (γ) (γ) −1 C D Φ± (z, k)= M± (z, k) − Im M± (z, k)+ Im , z ∈ \∂ . (3.83) Then, by (3.82) and (3.83), one verifies that (γ) (γ) −1 (γ) C D M± (z, k)= Im − Φ± (z, k) Im +Φ± (z, k) , z ∈ \∂ , (3.84) (γ) (γ) −1 (γ) C D m− (z, k)= zIm +Φ− (z, k) zIm − Φ− (z, k) , z ∈ \∂ . (3.85) Moreover, by (3.56), (3.83), and Theorem 3.10, it follows, as in [19, Lemma 2.18], that for k ∈ Z, z ∈ C\∂D, zγ1/2V (γ)(z,k,k )U (γ)(z,k,k )−1γ1/2, k odd, Φ(γ)(z, k)= ± 0 ± 0 (3.86) ± 1/2 (γ) (γ) −1 1/2 (γ U± (z,k,k0)V± (z,k,k0) γ , k even, (γ) (γ) where U± (z,k,k0) and V± (z,k,k0) are sequences defined in (3.74) and (3.75). Since the Weyl- Titchmarsh solutions defined in (3.76) are unique up to right multiplication by a constant complex −1/2 (γ) −1/2 m × m matirx, (3.86) implies that γ Φ± (·, k)γ is γ-independent, and hence for unitary m×m γ1,γ2 ∈ C , and k ∈ Z, that
(γ2) 1/2 −1/2 (γ1) −1/2 1/2 Φ± (·, k)= γ2 γ1 Φ± (·, k)γ1 γ2 , (3.87)
(γ2) −1/2 1/2 1/2 −1/2 (γ1) −1/2 1/2 1/2 −1/2 M± (·, k)= (γ2 γ1 + γ2 γ1 )M± (·, k) + (γ2 γ1 − γ2 γ1 )
−1/2 1/2 1/2 −1/2 (γ1) −1/2 1/2 1/2 −1/2 −1 × (γ2 γ1 − γ2 γ1 )M± (·, k) + (γ2 γ1 + γ2 γ1 ) . (3.88) Full and half-lattice resolvent operators lie at the heart of our analysis of the Weyl-Titchmarsh theory for full and half-lattice CMV operators; in particular, as a tool in obtaining our Borg- Marchenko-type uniqueness results in [19] for CMV operators with matrix-valued coefficients. Hence, we conclude with a discussion of resolvent operators for a general unitary γ ∈ Cm×m. 22 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
First, we note the utility of the identities contained in the following lemma. This lemma was proven in [19, Lemma 3.2] for matrix-Laurent polynomial solutions of (3.22) defined by (3.56) in the special case when γ = Im. The identities listed below were central to the derivation of the full-lattice resolvent operator in [19, Lemma 3.3].
m×m Lemma 3.12. Let k, k0 ∈ Z and z ∈ C\{0}. Then, for a unitary γ ∈ C , the following identities hold for the matrix-Laurent polynomial solutions of (3.22) defined in (3.56) and (3.57):
(γ) (γ) ∗ (γ) (γ) ∗ k+1 P± (z,k,k0)Q± (1/z,k,k0) + Q± (z,k,k0)P± (1/z,k,k0) = 2(−1) Im, (3.89) (γ) (γ) ∗ (γ) (γ) ∗ k R± (z,k,k0)S± (1/z,k,k0) + S± (z,k,k0)R± (1/z,k,k0) = 2(−1) Im, (3.90) (γ) (γ) ∗ (γ) (γ) ∗ P± (z,k,k0)S± (1/z,k,k0) + Q± (z,k,k0)R± (1/z,k,k0) =0, (3.91) (γ) (γ) ∗ (γ) (γ) ∗ R± (z,k,k0)Q± (1/z,k,k0) + S± (z,k,k0)P± (1/z,k,k0) =0. (3.92)
Proof. For the case k = k0, each of the equations (3.89)–(3.92) follows from (3.56) and (3.57). The induction argument described in [19, Lemma 3.2] then suffices to establish (3.89)–(3.92) when k 6= k0. As already noted, the proof in [19, Lemma 3.2] treats the special case when γ = Im for solutions defined by (3.56). In all cases under consideration, the proof involves a number of cases all following a similar pattern. We outline one of these cases for a solution of (3.22) defined in (3.57). Suppose equations (3.89)–(3.92) hold when k ∈ Z is even. Then utilizing (3.22) together with (3.8) and (3.9), one computes
(γ) (γ) ∗ (γ) (γ) ∗ P− (z, k +1, k0)Q− (1/z, k +1, k0) + Q− (z, k +1, k0)P− (1/z, k +1, k0) −1 (γ) (γ) ∗ (γ) (γ) ∗ ∗ −1 = ρk+1αk+1 P− (z,k,k0)Q− (1/z,k,k0) + Q− (z,k,k0)P− (1/z,k,k0) αk+1ρk+1 −1 (γ) (γ) ∗ (γ) (γ) ∗ −1 + ρ R− (z,k,k0)S− (1/z,k,k0) + S− (z,k,k0)R− (1/z,k,k0) ρ e k+1 k+1 e −1 (γ) (γ) ∗ (γ) (γ) ∗ ∗ −1 + zρ R− (z,k,k0)Q− (1/z,k,k0) + S− (z,k,k0)P− (1/z,k,k0) αk ρ e k+1 e 0 k+1 −1 (γ) (γ) ∗ (γ) (γ) ∗ −1 −1 + ρ αk0 P− (z,k,k0)S− (1/z,k,k0) + Q− (z,k,k0)R− (1/z,k,k0) ρ z ke+1 ke+1 k+1 −1 ∗ −1 −2 (k+1)+1 = 2(−1) ρk+1αk+1αk+1ρk+1 − ρk+1 = 2(−1) Im. (3.93) e e Similarly, one checks all remaining cases at the point k + 1. Then by inverting the matrix T(z, k) e e e and utilizing (3.22) in the form P (z, k − 1, k ) P (z,k,k ) − 0 = T(z, k)−1 − 0 , (3.94) R (z, k − 1, k ) R (z,k,k ) − 0 − 0 where −1 ∗ −1 −ρk αk zρk −1 −1 −1 k odd, z ρk −ρk αk! T −1 (z, k) = (3.95) −1 −1 −ρkeαk eρk −1 −1 ∗ k even, ρk −ρk αk! e e one verifies the equations (3.89)–(3.92) at the point k − 1. Similarly, one verifies (3.89)–(3.92) at the points k + 1 and k − 1 under the assumption that k ∈ Z is odd.
The next lemma introduces the half-lattice resolvent operators for U(γ) ; this appears to be a ±,k0 new result: MINIMALRANKDECOUPLINGOFCMVOPERATORS 23
m×m Lemma 3.13. Let z ∈ C\(∂D ∪{0}) and fix k0 ∈ Z. Then, for a unitary γ ∈ C , the resolvent (U(γ) − zI)−1 for the unitary CMV operator U(γ) on ℓ2([k , ±∞) ∩ Z)m is given in terms of its ±,k0 ±,k0 0 2 m matrix representation in the standard basis of ℓ ([k0, ±∞) ∩ Z) by
(γ) (γ) ′ ∗ −P+ (z,k,k0)U+ (1/z,¯ k , k0) , 1 k < k′ and k = k′ odd, (U(γ) − zI)−1 = k, k′ ∈ [k , ∞) ∩ Z, (3.96) +,k0 (γ) (bγ) ′ ∗ 0 2z U (z,k,k0)P (1/z,¯ k , k0) , + + k > k′ and k = k′ even, b (γ) (γ) where P+ (z,k,k0) is defined in (3.56), and U+ (z,k,k0) is defined in (3.72);
b (γ) (γ) ′ ∗ −U− (z,k,k0)P− (1/z,¯ k , k0) , 1 k < k′ and k = k′ odd, (U(γ) − zI)−1 = k, k′ ∈ (−∞, k ] ∩ Z, (3.97) −,k0 (bγ) (γ) ′ ∗ 0 2z P− (z,k,k0)U− (1/z,¯ k , k0) , k > k′ and k = k′ even, b (γ) (γ) where P− (z,k,k0) is defined in (3.57), and U− (z,k,k0) is defined in (3.72).
Proof. We begin by noting that the followingb equations hold for k ∈ Z:
(γ) (γ) ∗ (γ) (γ) ∗ R± (z,k,k0)U± (1/z,k,k¯ 0) + V± (z,k,k0)P± (1/z,k,k¯ 0) =0, (3.98) P (γ)(z,k,k )U (γ)(1/z,k,k¯ )∗ + U (γ)(z,k,k )P (γ)(1/z,k,k¯ )∗ = 2(−1)k+1I . (3.99) ± 0 b± 0 b± 0 ± 0 m b b (γ) These equations are a consequence of (3.72), (3.73), (3.89), (3.92), and the fact that m± (z, k0) are matrix-valued Caratheodory functions and hence satisfy the property given in (A.9). ′ Z (γ) ′ For k, k ∈ [k0, ∞) ∩ , let G+ (z,k,k , k0) be defined by
(γ) (γ) ′ ∗ ′ ′ (γ) ′ −P+ (z,k,k0)U+ (1/z,¯ k , k0) , k
(γ) (γ) ′ ′ U − zI G (z, ·, k , k )=2z∆ ′ , k ∈ [k , ∞) ∩ Z. (3.101) +,k0 + 0 k 0 ′ ′ ′ Assume that k ∈ [k0, ∞) ∩ Z is odd. Then, for ℓ ∈ ([k0, ∞) ∩ Z)\{k , k +1}, note that
U(γ) − zI G(γ)(z, ·, k′, k ) (ℓ)= V(γ) W(γ) − zI G(γ)(z, ·, k′, k ) (ℓ)=0. (3.102) +,k0 + 0 +,k0 +,k0 + 0 24 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Next, by (3.98), (3.99), note that
U(γ) − zI G(γ)(z, ·, k′, k ) (k′) +,k0 + 0 (γ) (γ) ′ ′ U − zI G (z, ·, k , k0) (k + 1) +,k0 + ! V(γ) W(γ) − zI G(γ)(z, ·, k′, k ) (k′) +,k0 +,k0 + 0 = (γ) (γ) (γ) ′ ′ V W − zI G (z, ·, k , k0) (k + 1) +,k0 +,k0 + ! (γ) ′ (γ) ′ ∗ (γ) ′ ′ ′ −zR+ (z, k , k0)U+ (1/z,¯ k , k0) G+ (z, k , k , k0)) = Θ(k + 1) (γ) ′ (γ) ′ ∗ − z (γ) ′ ′ zV+ (z, k +1, k0)P+ (1/z,¯ k , k0) ! G+ (z, k +1, k , k0))! b (γ) ′ (γ) ′ ∗ (γ) ′ ′ ′ bzV+ (z, k , k0)P+ (1/z,¯ k , k0) G+ (z, k , k , k0)) = Θ(k + 1) (γ) ′ (γ) ′ ∗ − z (γ) ′ ′ zV+ (z, k +1, k0)P+ (1/z,¯ k , k0) ! G+ (z, k +1, k , k0))! b U (γ)(z,k,k )P (γ)(1/z,k,k¯ )∗ + P (γ)(z,k,k )U (γ)(1/z,k,k¯ )∗ = z + b0 + 0 + 0 + 0 0 ′ 2zb(−1)k +1I 2zI b = m = m . (3.103) 0 0 ′ Hence, when k ∈ [k0, ∞) ∩ Z is odd, (3.101) is a consequence of (3.102) and (3.103). ′ ′ ′ Assume that k ∈ [k0, ∞) ∩ Z is even. Then, for ℓ ∈ ([k0, ∞) ∩ Z)\{k − 1, k }, note that (3.102) holds. Again, by (3.98) and (3.99), note that
U(γ) − zI G(γ)(z, ·, k′, k ) (k′ − 1) +,k0 + 0 (γ) (γ) ′ ′ U − zI G (z, ·, k , k0) (k ) +,k0 + ! V(γ) W(γ) − zI G(γ)(z,·, k′, k ) (k′ − 1) +,k0 +,k0 + 0 = (γ) (γ) (γ) ′ ′ V W − zI G (z, ·, k , k0) (k ) +,k0 +,k0 + ! (γ) ′ (γ) ′ ∗ (γ) ′ ′ ′ −zR+ (z, k − 1, k0)U+ (1/z,¯ k− 1, k0) G+ (z, k − 1, k , k0)) = Θ(k ) (γ) ′ (γ) ′ ∗ − z (γ) ′ ′ zV+ (z, k , k0)P+ (1/z,¯ k , k0) ! G+ (z, k , k , k0)) ! b (γ) ′ (γ) ′ ∗ (γ) ′ ′ ′ −zR+b(z, k − 1, k0)U+ (1/z,¯ k − 1, k0) G+ (z, k − 1, k , k0)) = Θ(k ) (γ) ′ (γ) ′ ∗ − z (γ) ′ ′ −zR+ (z, k , k0)U+ (1/z,¯ k , k0) ! G+ (z, k , k , k0)) ! b 0 = −z b P (γ)(z,k,k )U (γ)(1/z,k,k¯ )∗ + U (γ)(z,k,k )P (γ)(1/z,k,k¯ )∗ + 0 + 0 + 0 + 0 0 0 = ′ = . (3.104) 2z(−1)k +2I b 2zI b m m ′ Hence, when k ∈ [k0, ∞) ∩ Z is even, (3.101) is a consequence of (3.102) and (3.104). The proof of (3.97) is omitted here for brevity, but follows a line of reasoning similar that just completed for the proof of (3.96) by using (3.72), (3.73), (3.98), and (3.99).
Before stating our final result for the full-lattice resolvent of U, let us recall the definition of, and some facts about, the matrix-valued Wronskian, defined in [19], for two Cm×m-valued sequences Uj (z, ·), j =1, 2. First, the Wronskian is defined for k ∈ Z, z ∈ C\{0}, by
W (U1(1/z, k),U2(z, k)) (−1)k+1 = U (1/z, k)∗U (z, k) − (V∗U (1/z, · ))(k)∗(V∗U (z, · ))(k) , (3.105) 2 1 2 1 2 MINIMALRANKDECOUPLINGOFCMVOPERATORS 25
where V is defined in (3.15). It is shown in [19, Lemma 3.1] when UUj(z, ·)= zUj(z, ·), and hence ∗ V Uj (z, ·)= Vj (z, ·), j =1, 2, where U is viewed as a difference expression rather than as an operator 2 m×m acting on ℓ (Z) , that the Wronskian of Uj (z, ·), j = 1, 2, is independent of k ∈ Z. Moreover, (γ) (γ) (γ) for P+ (z, ·, k0) and Q+ (z, ·, k0) defined in (3.56), and for U± (z, ·, k0) defined in (3.74), with k, k0 ∈ Z, z ∈ C\{0}, as a consequence of (3.56), (3.57), (3.74), (3.75), and property (A.9), we see that (γ) (γ) W P+ (1/z,k,k0),Q+ (z,k,k0) = Im, (3.106) (γ) (γ) (γ) (γ) W U+ (1/z,k,k0),U− (z,k,k0) = M− (z, k0) − M+ (z, k0). (3.107) (γ) (γ) For notational simplicity, we abbreviate the Wronskian of U+ and U− by (γ) (γ) (γ) W (z, k0)= −W U+ (1/z,k,k0),U− (z,k,k0) . (3.108) (γ) Then, using (3.77), (3.81), and (3.107), one analytically continues W (z, k0) to z = 0 and obtains (γ) (γ) (γ) Z C W (z, k0)= M+ (z, k0) − M− (z, k0), k ∈ , z ∈ . (3.109) (γ) Moreover, one verifies the following symmetry property of the Wronskian W (z, k0), for k ∈ Z, z ∈ C (γ) (γ) −1 (γ) (γ) (γ) −1 (γ) M+ (z, k0)W (z, k0) M− (z, k0)= M− (z, k0)W (z, k0) M+ (z, k0). (3.110) Then, using (3.74), (3.75), (3.89), (3.92), (3.107), (3.109), and following the steps in the proof for [19, Lemma 3.2] for the special case when γ = Im, we find that
(γ) (γ) −1 (γ) ∗ U+ (z,k,k0)W (z, k0) U− (1/z,k,k0) (γ) (γ) −1 (γ) ∗ k+1 − U− (z,k,k0)W (z, k0) U+ (1/z,k,k0) = 2(−1) Im, (3.111) (γ) (γ) −1 (γ) ∗ V+ (z,k,k0)W (z, k0) U− (1/z,k,k0) (γ) (γ) −1 (γ) ∗ − V− (z,k,k0)W (z, k0) U+ (1/z,k,k0) =0. (3.112) Then, using (3.111) and (3.112), and following the steps in the proof for [19, Lemma 3.3] for the special case when γ = Im, we obtain the next result for the resolvent of the full-lattice operator U. m×m Lemma 3.14. Let z ∈ C\(∂D ∪{0}), fix k0 ∈ Z, and let γ ∈ C be unitary. Then the resolvent (U − zI)−1 of the unitary CMV operator U on ℓ2(Z)m is given, for k, k′ ∈ Z, in terms of its matrix representation in the standard basis of ℓ2(Z)m by
(γ) (γ) −1 (γ) ′ ∗ U− (z,k,k0)W (z, k0) U+ (1/z, k , k0) , ′ or ′ odd U −1 ′ 1 k < k k = k , ( − zI) (k, k )= (γ) (γ) −1 (γ) ′ ∗ (3.113) 2z U (z,k,k0)W (z, k0) U− (1/z, k , k0) , + k > k′ or k = k′ even, Moreover, since 0 ∈ C\σ(U), (3.113) analytically extends to z =0.
Appendix A. Basic Facts on Caratheodory and Schur Functions In this appendix we summarize a few basic facts on matrix-valued Caratheodory and Schur functions used throughout this manuscript. (For the analogous case of matrix-valued Herglotz functions we refer to [48] and the extensive list of references therein.) We denote by D and ∂D the open unit disk and the counterclockwise oriented unit circle in the complex plane C, D = {z ∈ C | |z| < 1}, ∂D = {ζ ∈ C | |ζ| =1}. (A.1) 26 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Moreover, we denote as usual Re(A) = (A + A∗)/2 and Im(A) = (A − A∗)/(2i) for square matrices A with complex-valued entries. N −1 Definition A.1. Let m ∈ and F±, Φ+, and Φ− be m × m matrix-valued analytic functions in D. (i) F+ is called a Caratheodory matrix if Re(F+(z)) ≥ 0 for all z ∈ D and F− is called an anti- Caratheodory matrix if −F− is a Caratheodory matrix. (ii)Φ+ is called a Schur matrix if kΦ+(z)kCm×m ≤ 1, for all z ∈ D.Φ− is called an anti-Schur matrix −1 if Φ− is a Schur matrix. Theorem A.2. Let F be an m × m Caratheodory matrix, m ∈ N. Then F admits the Herglotz representation ζ + z F (z)= iC + dΩ(ζ) , z ∈ D, (A.2) D ζ − z I∂ C = Im(F (0)), dΩ(ζ) = Re(F (0)), (A.3) D I∂ where dΩ denotes a nonnegative m × m matrix-valued measure on ∂D. The measure dΩ can be reconstructed from F by the formula 1 θ2+δ Ω Arc eiθ1 ,eiθ2 = lim lim dθ Re F rζ , (A.4) δ↓0 r↑1 2π Iθ1+δ where iθ1 iθ2 Arc e ,e = ζ ∈ ∂D | θ1 <θ ≤ θ2 , θ1 ∈ [0, 2π), θ1 <θ2 ≤ θ1 +2π. (A.5) Conversely, the right-hand side of equation with ∗ and a finite nonnegative (A.2) C = C dΩ m × m matrix-valued measure on ∂D defines a Caratheodory matrix. We note that additive nonnegative m×m matrices on the right-hand side of (A.2) can be absorbed into the measure dΩ since ζ + z dµ0(ζ) =1, z ∈ D, (A.6) D ζ − z I∂ where dθ dµ (ζ)= , ζ = eiθ, θ ∈ [0, 2π) (A.7) 0 2π denotes the normalized Lebesgue measure on the unit circle ∂D. Given a Caratheodory (resp., anti-Caratheodory) matrix F+ (resp. F−) defined in D as in (A.2), one extends F± to all of C\∂D by ζ + z C D ∗ F±(z)= iC± ± dΩ±(ζ) , z ∈ \∂ , C± = C±. (A.8) D ζ − z I∂ In particular, ∗ F±(z)= −F±(1/z) , z ∈ C\D. (A.9)
Of course, this continuation of F±|D to C\D, in general, is not an analytic continuation of F±|D. Next, given the functions F± defined in C\∂D as in (A.8), we introduce the functions Φ± by −1 Φ±(z) = [F±(z) − Im][F±(z)+ Im] , z ∈ C\∂D. (A.10)
We recall (cf., e.g., [78, p. 167]) that if ±Re(F±) ≥ 0, then [F± ± Im] is invertible. In particular, −1 Φ+|D and [Φ−] |D are Schur matrices (resp., Φ−|D is an anti-Schur matrix). Moreover, −1 F±(z) = [Im − Φ±(z)] [Im +Φ±(z)], z ∈ C\∂D. (A.11) Acknowledgments. Fritz Gesztesy would like to thank all organizers of the 14th International Conference on Difference Equations and Applications (ICDEA 2008), for their kind invitation and MINIMALRANKDECOUPLINGOFCMVOPERATORS 27
the stimulating atmosphere created during the meeting. In addition, he is particularly indebted to Mehmet Unal¨ for the extraordinary hospitality extended to him during his ten day stay in Istanbul in July of 2008.
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Department of Mathematics & Statistics, University of Missouri, Rolla, MO 65409, USA E-mail address: [email protected] URL: http://web.umr.edu/~sclark/index.html 30 S. CLARK, F. GESZTESY, AND M. ZINCHENKO
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA E-mail address: [email protected] URL: http://www.math.caltech.edu/~maxim