Discrete self-adjoint Dirac systems and ArovKrein entropy Alexander - - PowerPoint PPT Presentation

Discrete self-adjoint Dirac systems and Arov–Krein entropy Alexander Sakhnovich, University of Vienna Operator Theory and Krein Spaces 2019 This research was supported by the Austrian Science Fund (FWF) under Grant No. P 29177.

Contents

• 1. Discrete and continuous Dirac systems.
• 2. Self-adjoint discrete Dirac systems: Verblunsky-type theorem.
• 3. Christoffel-Darboux formula and asymptotic relations.
• 4. Arov-Krein entropy (sign-indefinite case).
• 5. Self-adjoint discrete Dirac systems: rational Weyl functions.

Discrete and continuous Dirac systems The self-adjoint discrete Dirac system has the form yk+1(z) = (Im + izjCk)yk(z); j := Im1 −Im2

• ,

(1) Ck > 0, CkjCk = j (m = m1 + m2), (2) where Im is the m × m identity matrix. Continuous Dirac system may be rewritten in the form Y ′(x, z) = izjH(x)Y (x, z), H = H∗ > 0, HjH ≡ j, (3) and the analogy between the systems (1), (2) and (3) is clear. Systems (1) and (3) may be also considered as the special cases

• f canonical systems.

We start with a short explanation and references.

Close interconnections between discrete problems and continuous Dirac systems and between the spectral theory of Dirac systems and structured operators follow already from the famous note Continuous analogues of propositions on polynomials orthogonal

• n the unit circle, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955),

by M.G. Krein. Skew-self-adjoint discrete Dirac systems were introduced in M.A. Kaashoek and A.L. Sakhnovich, “Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model”,

• J. Functional Anal. 228 (2005)

Self-adjoint discrete Dirac systems were introduced later in

• B. Fritzsche, B. Kirstein, I. Roitberg and A.L. Sakhnovich,

“Weyl matrix functions and inverse problems for discrete Dirac-type self-adjoint systems: explicit and general solutions”,

• Oper. Matrices 2 (2008)

Some further results and references one can find, for instance, in: A.L. Sakhnovich, L.A. Sakhnovich, and I.Ya. Roitberg, “Inverse problems and nonlinear evolution equations...”, De Gruyter, 2013.

Our recent work A.L. Sakhnovich, New “Verblunsky-type” coefficients of block Toeplitz and Hankel matrices and of corresponding Dirac and canonical systems. J. Approx. Theory 237 (2019)

• n Verblunsky-type coefficients for Toeplitz matrices and discrete

self-adjoint Dirac systems was initiated by the interesting paper

• M. Derevyagin and B. Simanek, Szeg¨
• ’s theorem for a nonclassical

case, J. Funct. Anal. 272 (2017), where indefinite Szeg¨

• limit theorem was proved in a way which

differs from our approach in A.L. Sakhnovich, J. Funct. Anal. 171 (2000). Namely, M. Derevyagin and B. Simanek used the theory of

• rthogonal polynomials, and it was interesting to understand the

analogies. We consider discrete Dirac systems as an alternative (to the famous Szeg˝

• recurrencies and orthogonal polynomials) approach

to the study of the corresponding Toeplitz matrices, which could be especially useful for the case of block Toeplitz matrices.

Recall that the theory of orthogonal polynomials on the unit circle (OPUC) studies interrelations between a measure dτ (or, equivalently, nondecreasing weight function τ(t) on [−π, π]) and positive-definite Toeplitz matrices S(n) of the form S(n) = {sj−i}n

i,j=1,

sk = 1 2π π

−π

eiktdτ(t) (4)

• n one side, and orthogonal trigonometric polynomials generated

by the measure dτ on the other side. Clearly S(n) > 0 if τ has infinite support. It is well known that the orthonormal polynomials Pr(λ) on the unit circle satisfy Szeg˝

• recurrence

Zk+1(λ) = 1

• 1 − |ak|2

1 −ak −ak 1 λ 1

• Zk(λ),

(5) where Zr(λ) := col

• Pr(λ)

λrPr(1/λ)

• , the coefficients {ak} are

so called Verblunsky coefficients and |ak| < 1.

Instead of the Szeg¨

• recurrences, one can consider

discrete Dirac systems yk+1(z) = (Im + izjCk)yk(z); Ck > 0, CkjCk = j; (6) k ≥ 0, j := Im1 −Im2

• ,

m := m1 + m2, (7) where Ck are m × m matrices. It is important that relations for Ck in (6) are equivalent to: Ck = DkHk, Dk := diag

• Im1 − ρkρ∗

k

− 1

2 ,

• Im2 − ρ∗

kρk

− 1

2

• ,

Hk := Im1 ρk ρ∗

k

Im2

• (ρ∗

kρk < Im2).

Discrete Dirac systems are also closely related to Toeplitz matrices. We note that Weyl theory of systems (6) is developed for the case m1, m2 ∈ N but Toeplitz matrices appear when m1 = m2. Next, we turn to the case m1 = m2 = p.

Now, we consider self-adjoint discrete Dirac systems (called, for shortness, discrete Dirac systems): yk+1(z) = (Im + izjCk)yk(z); Ck > 0, CkjCk = j, m = 2p. Recall that relations Ck > 0, CkjCk = j are equivalent to: Ck = DkHk, Hk := Ip ρk ρ∗

k

Ip

• (ρ∗

kρk < Ip),

(8) Dk := diag

• Ip − ρkρ∗

k

− 1

2 ,

• Ip − ρ∗

kρk

− 1

2

• .

(9) Hence, matrix Ck satisfying Ck > 0, CkjCk = j is in one to one correspondence with the p × p matrix ρk such that ρ∗

kρk < Ip.

This correspondence is given by (8), (9), and by the equality ρk = Ip

• Ck
• Ip

∗ −1 Ip

• Ck
• Ip

∗ . The matrices ρk are called Verblunsky-type coefficients and there is a one to one correspondence between the sequences of Verblunsky-type coefficients and Dirac systems.

Thus, the one to one correspondence between the sequences of Verblunsky-type coefficients and Dirac systems is clear. It remains to show the one to one correspondence between Dirac systems and Toeplitz matrices, and Verblunsky-type result on one to one correspondence between the sequences of Verblunsky-type coefficients and Toeplitz matrices will follow. For this purpose, we need to consider Toeplitz matrices in greater detail.

Any block Toeplitz S(n) = S(n)∗ satisfies the matrix identity AS(n) − S(n)A∗ = iΠJΠ∗; Π =

• Φ1

Φ2

• ,

where (10) A =

• aj−i

n

i,j=1 ,

ak =      for k > 0 i 2 Ip for k = 0 i Ip for k < 0 , J = Ip Ip

• ;

Φ1 =     Ip Ip · · · Ip     , Φ2 =     s0/2 s0/2 + s−1 · · · s0/2 + s−1 + . . . + s1−n     + iΦ1ν, ν = ν∗; A = A(n), Π = Π(n), Φ1 = Φ1(n), Φ2 = Φ2(n). In this case, the transfer matrix function wA in Lev Sakhnovich form is given by wA(n, z) = I2p − iJΠ(n)∗S(n)−1 A(n) − zInp −1Π(n). (11) The factorization of wA is equivalent to the recovery of Dirac system from S(n).

Let Wk(z) be the normalized fundamental solution of the discrete Dirac system: Wk+1(z) = (I2p + izjCk) Wk(z), W0(z) = I2p. Let S(n) > 0 and recall that the transfer m.-functions wA are given by the formulas wA(k, z) = I2p − iJΠ(k)∗S(k)−1 A(k) − zIkp −1Π(k) (k ≤ n). These wA are uniquely determined by the matrix S(n) (with the p × p blocks) and by the p × p matrix ν = ν∗. The following equality holds for Wk corresponding to S(n) and ν: Wk(−1/z) = z−k(z + i)kK ∗wA(k, −z/2)K, (12) W0(z) := I2p, K := 1 √ 2 Ip −Ip Ip Ip

• .

(13) Using (12) one can show one to one correspondence between {S(n), ν} and Dirac system on the intervals 1 ≤ k ≤ n and 1 ≤ k ≤ ∞.

Recall S(n) = {sj−i}n

i,j=1,

sk = 1 2π π

−π

eiktdτ(t), (14) Zk+1(λ) = 1

• 1 − |ak|2

1 −ak −ak 1 λ 1

• Zk(λ).

(15) Verblunsky’s theorem. There is a one to one correspondence between the sequences {sk}k≥0 such that S(n) > 0 for all n > 0 (or equivalently measures dτ with infinite support) and sequences (of Verblusky coefficients) {ak}k≥0, where |ak| < 1. Verblunsky coefficients and Verblunsky’s theorem play a fundamental role in the theory of orthogonal polynomials (see, e.g. B. Simon “Orthogonal polynomials on the unit circle”). Our Verblunsky-type theorem. There is a one to one correspondence between the sequences of p × p blocks {sk}k≥0 such that S(n) > 0 for all n > 0 (complemented by some p × p matrix ν = ν∗) and discrete Dirac systems or, equivalently, between such sequences {sk}k≥0 and sequences {ρk}k≥0 of Verblunsky-type coefficients. We note that we deal with the general block Toeplitz matrices.

Christoffel-Darboux formula and asymptotic relations The discussed above results from J. Approx. Theory 237 (2019) were developed further (jointly with B. Fritzsche, B. Kirstein,

• I. Roitberg) in several papers. Here, we will discuss the latest

results from:

• A. Sakhnovich, Discrete self-adjoint Dirac systems: asymptotic

relations, Weyl functions and Toeplitz matrices, arXiv:1912.05213 Recall that Wk(z) is the normalized fundamental solution of the discrete Dirac system: Wk+1(z) = (I2p + izjCk) Wk(z), W0(z) = I2p. It is easy to see that we have Christoffel-Darboux formula

N

• k=0

c(z, ζ)kWk(ζ)∗CkWk(z) (16) = i1 + zζ ζ − z

• c(z, ζ)N+1WN+1(ζ)∗jWN+1(z) − j
• ,

where c(z, ζ) = (1 + zζ)−1. Here, Wk(ζ)∗jWk(z) are analogs of reproducing kernels and their asymptotics is important.

Recall: J = Ip Ip

• ,

K =

1 √ 2

Ip −Ip Ip Ip

• . Since we have

Wk(ζ)∗jWk(z) = −(1 + iζ)k(1 − iz)kKM(k, −2ζ, −2z)K ∗, (17) where M(k, ζ, z) := J + i(z − ζ)JjΠ(k)∗ × (Ikp + ζA(k)∗)−1S(k)−1(Ikp + zA(k))−1Π(k)jJ, we will study the asymptotics of M(k, ζ, z) = {Mij((k, ζ, z))}2

i,j=1

when k → ∞. First, we consider the p × p block M22(k, ζ, ζ) = i(ζ − ζ) × Φ1(k)∗ Ikp + ζA(k)∗−1S(k)−1 Ikp + ζA(k) −1Φ1(k). (18) Recall that Φ1(k)∗ =

• Ip,

. . . , Ip

• ,

Φ1(k)∗ Ikp + ζA(k)∗−1 =

• 1 − (i/2)ζ

−1 ×

• Ip,

2+iζ 2−iζ ,

. . . ,

• 2+iζ

2−iζ

k−1 Ip

• .

We will need some auxiliary definitions and results.

• Definition. A p × p matrix function ϕ(λ) holomorphic in the

lower complex half-plane C− is called a Weyl function for Dirac system if the inequality

∞

• k=0

[iϕ(z)∗ Ip]q(z)kKWk(z)∗CkWk(z)K ∗ −iϕ(z) Ip

• < ∞,

holds for q(z) := (|z2| + 1)−1. Such Weyl function exists, is unique (limit point case) and belongs to Herglotz class. The function ω(z) = −ϕ(−z/2) belongs to Herglotz class in C+ and admits Herglotz representation ω(ζ) = βζ + γ + ∞

−∞

1 + tζ (t − ζ)(1 + t2)dτ(t) (ζ ∈ C+). (19) If τ ′ (the positive semi-definite derivative of the absolutely continuous part of τ) satisfies the Szeg˝

• condition

∞

−∞

(1 + t2)−1 ln

• det τ ′(t)
• > −∞.

(20) there is is a unique factorization τ ′(t) = Gτ(t)∗Gτ(t), where G(t) are boundary values of G(z) ∈ H (for H see the next frame).

The p × p matrix function G(z) (z ∈ C+) belongs to H if the entries of G

• ζ0λ−ζ0

λ−1

• (where ζ0 ∈ C+) belong to the Hardy class

H2(D) and det

• G
• ζ0λ−ζ0

λ−1

• is an outer function.

Recall that M22(k, ζ, z) = i(z − ζ) × Φ1(k)∗ Ikp + ζA(k)∗−1S(k)−1 Ikp + zA(k) −1Φ1(k). and τ ′(t) = Gτ(t)∗Gτ(t). We showed in AS, USSR-Izv. 30 (1988) that lim

k→∞ M22(k, ζ, ζ) = (1/2π)

• Gτ(ζ)∗Gτ(ζ)

−1 (ζ ∈ C+). (21) The following inequality is valid as well: lim

k→∞ M22(k, ζ, ζ)−1 = 0

(ζ ∈ C+). (22) From (21) and (22) we obtain the asymptotics of M.

Recall that ω(z) = −ϕ(−z/2), where ϕ is the Weyl function of Dirac system, that τ comes from the Herglotz representation of ω and τ ′(t) = Gτ(t)∗Gτ(t), and that the analog M of reproducing kernel is given by: M(k, ζ, z) := J + i(z − ζ)JjΠ(k)∗ × (Ikp + ζA(k)∗)−1S(k)−1(Ikp + zA(k))−1Π(k)jJ. Asymptotical theorem. Let a p × p matrix ν = ν∗ and a sequence {sk}k≥0 such that S(n) = {sj−i}n

i,j=1 > 0 (n ∈ N) be

given, and assume that Szeg¨

• condition is fulfilled.

Then we have (uniformly with respect to z and ζ on any compact

• n C+):

lim

n→∞ M(n, ζ, z) = 1

2π −iω(ζ) Ip

• Gτ(ζ)−1 (Gτ(z)∗)−1

iω(z)∗ Ip

• .

Interesting asymptotical relations appear also in the indefinite interpolation problems. In the famous Arov-Krein note “ The problem of finding the minimum entropy in indeterminate problems of continuation” Funktsional. Anal. i Prilozhen. 15 (1981), the authors studied the entropy functional E(ω, λ) = − 1 4π 2π |eiθ − λ|−2(1 − | λ|2) ln det

• µ′(θ)
• dθ

where λ belongs to the unit circle D and ω belongs to the Caratheodory class and admits representation ω(λ) = iν + 1 2π 2π eiθ + λ eiθ − λdµ(θ) (ν = ν∗). (23) Similar functionals are of interest in the case of Krein-Langer generalized Caratheodory classes Cκ.

In the case ω⋆ ∈ Cκ we put (see I. Roitberg and AS, IEOT 91, Pa. 50 (2019)): E⋆(ϕ, λ) = − 1 4π 2π |eiθ − λ|−2(1 − | λ|2) ln det

• ℜ
• ω⋆(eiθ)
• dθ.

When the matrices S(n) = {sj−i}n

i,j=1 have κ negative eigenvalues

starting from some n0, ω⋆(λ) = −iϕ

• − iλ − 1

λ + 1 ∗ = s0 2 − iν +

∞

• k=1

skλk, and λ = 0 one can use entropy functionals in order to derive Szeg˝

• limit theorem in the sign-indefinite case.

Returning to the Dirac systems, we note that the Taylor representation of the Weyl function iϕ

• iλ − 1

λ + 1

• = s0

2 + iν +

∞

• k=1

s−kzk establishes one to one correspondence between Weyl functions and the sets {sk}k≥0 ∪ {ν} such that the matrices S(n) and ν satisfy S(n) = {sj−i}n

i,j=1 > 0,

ν = ν∗. If ϕ(z) is rational, we recover (from the minimal realization of ϕ) a triple {A, S0, Π0} of two r × r (r ∈ N) matrices A and S0 and an r × 2p matrix Π0 =

• ϑ1

ϑ2

• such that

det(A) = 0, S0 > 0, AS0 − S0A∗ = iΠ0JΠ∗

0,

J := Ip Ip

• .

The generalized B¨ acklund-Darboux transformation (GBDT) of the trivial Dirac system determined by {A, S0, Π0} gives us explicit expression for the Dirac system with the Weyl function ϕ. The corresponding block Toeplitz matrix is block diagonal plus block semi-separable.

Explicit solutions. More precisely triples {A, S0, Π0} determine Dirac systems with rational Weyl functions: Πk+1 = Πk + iA−1Πkj, (24) Sk+1 = Sk + A−1Sk(A∗)−1 + A−1ΠkΠ∗

k(A∗)−1,

(25) Ck := Im + Π∗

kS−1 k Πk − Π∗ k+1S−1 k+1Πk+1.

(26) The corresponding block Toeplitz matrices are given by the formulas ν = ℑ

• Ip + Fϑ2
• ,

s0 = 2ℜ

• Ip + Fϑ2
• ,

(27) s−k = FUk−1(U − In)ϑ2 (k ≥ 1); (28) where F := 2iϑ∗

1S−1 0 (

A + iIn)−1,

• A := A + iϑ2(ϑ2 − ϑ1)∗S−1

0 ,

U = ( A − iIn)( A + iIn)−1, and det( A + In) = 0.

Some remarks on the factorization of wA and transformation into Dirac system Recall : wA(n, λ) = I2p − iJΠ(n)∗S(n)−1 A(n) − λInp −1Π(n). When S(k) > 0, we have tk :=

• . . .

Ip

• S(k)−1

. . . Ip ∗ > 0. (29) Introduce also p × p matrices Xk and Yk by the equalities

• Xk

Yk

• =
• . . .

Ip

• S(k)−1

Φ1(k) Φ2(k)

• .

(30) The factorisation formula for wA (L.A. Sakhnovich, 1976) in our case takes the form wA(n, λ) = wn(λ)wA(n − 1, λ) (n > 1), (31) wn(λ) := I2p − i i 2 − λ −1 J X ∗

n

Y ∗

n

• t−1

n

• Xn

Yn

• .

(32) We rewrite (31) as Dirac system Wn(z) = (Im + izjCn)Wn−1(z).

The transformation of the formulas wA(n, λ) = wn(λ)wA(n − 1, λ), (33) wn(λ) := I2p − i ((i/2) − λ)−1 J

• Xn

Yn ∗ t−1

n

• Xn

Yn

• (34)

into Dirac system follows via the relations Wk(−1/λ) = λ−k(λ + i)kK ∗wA(k, −λ/2)K, W0(λ) := I2p, K := 1 √ 2 Ip −Ip Ip Ip

• ,

Ck := 2K ∗β(k)∗β(k)K − j, β(k) := t−1/2

k+1

• Xk+1

Yk+1

• . (35)

Namely, from (33)–(35) we obtain Wn(z) = (Im + izjCn−1)Wn−1(z), Cn−1 > 0, Cn−1jCn−1 = j (n > 0).