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 � I m 1 � 0 y k +1 ( z ) = ( I m + i zjC k ) y k ( z ); j := , (1) 0 − I m 2 C k > 0 , C k jC k = j ( m = m 1 + m 2 ) , (2) where I m is the m × m identity matrix. Continuous Dirac system may be rewritten in the form H = H ∗ > 0 , Y ′ ( x , z ) = i zjH ( x ) Y ( x , z ) , 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 of 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 on 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) on Verblunsky-type coefficients for Toeplitz matrices and discrete self-adjoint Dirac systems was initiated by the interesting paper M. Derevyagin and B. Simanek, Szeg¨ o’s theorem for a nonclassical case , J. Funct. Anal. 272 (2017), where indefinite Szeg¨ o 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 orthogonal polynomials, and it was interesting to understand the analogies. We consider discrete Dirac systems as an alternative (to the famous Szeg˝ o 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 d τ (or, equivalently, nondecreasing weight function τ ( t ) on [ − π, π ]) and positive-definite Toeplitz matrices S ( n ) of the form � π s k = 1 S ( n ) = { s j − i } n e i kt d τ ( t ) i , j =1 , (4) 2 π − π on one side, and orthogonal trigonometric polynomials generated by the measure d τ on the other side. Clearly S ( n ) > 0 if τ has infinite support. It is well known that the orthonormal polynomials P r ( λ ) on the unit circle satisfy Szeg˝ o recurrence � 1 � � λ � 1 − a k 0 Z k +1 ( λ ) = � Z k ( λ ) , (5) − a k 1 0 1 1 − | a k | 2 � � λ r P r (1 /λ ) where Z r ( λ ) := col P r ( λ ) , the coefficients { a k } are so called Verblunsky coefficients and | a k | < 1.
Instead of the Szeg¨ o recurrences, one can consider discrete Dirac systems y k +1 ( z ) = ( I m + i zjC k ) y k ( z ); C k > 0 , C k jC k = j ; (6) � I m 1 � 0 k ≥ 0 , j := , m := m 1 + m 2 , (7) 0 − I m 2 where C k are m × m matrices. It is important that relations for C k in (6) are equivalent to: �� � � − 1 � � − 1 2 , I m 1 − ρ k ρ ∗ I m 2 − ρ ∗ C k = D k H k , D k := diag k ρ k , 2 k � I m 1 � ρ k ( ρ ∗ H k := k ρ k < I m 2 ) . ρ ∗ I m 2 k Discrete Dirac systems are also closely related to Toeplitz matrices. We note that Weyl theory of systems (6) is developed for the case m 1 , m 2 ∈ N but Toeplitz matrices appear when m 1 = m 2 . Next, we turn to the case m 1 = m 2 = p.
Now, we consider self-adjoint discrete Dirac systems (called, for shortness, discrete Dirac systems): y k +1 ( z ) = ( I m + i zjC k ) y k ( z ); C k > 0 , C k jC k = j , m = 2 p . Recall that relations C k > 0 , C k jC k = j are equivalent to: � I p � ρ k ( ρ ∗ C k = D k H k , H k := k ρ k < I p ) , (8) ρ ∗ I p k �� � � − 1 � � − 1 2 , I p − ρ k ρ ∗ I p − ρ ∗ D k := diag k ρ k . (9) 2 k Hence, matrix C k satisfying C k > 0, C k jC k = j is in one to one correspondence with the p × p matrix ρ k such that ρ ∗ k ρ k < I p . This correspondence is given by (8), (9), and by the equality � � � ∗ � − 1 � � � � � � ∗ . ρ k = 0 0 0 0 I p C k I p I p C k I p The matrices ρ k 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 ∗ = i Π J Π ∗ ; Π = Φ 1 Φ 2 , where (10) 0 for k > 0 � 0 � � � n i I p A = i , j =1 , a k = , J = ; a j − i 2 I p for k = 0 I p 0 i I p for k < 0 I p s 0 / 2 s 0 / 2 + s − 1 I p ν = ν ∗ ; Φ 1 = , Φ 2 = + i Φ 1 ν, · · · · · · s 0 / 2 + s − 1 + . . . + s 1 − n I p A = A ( n ) , Π = Π( n ) , Φ 1 = Φ 1 ( n ) , Φ 2 = Φ 2 ( n ) . In this case, the transfer matrix function w A in Lev Sakhnovich form is given by w A ( n , z ) = I 2 p − i J Π( n ) ∗ S ( n ) − 1 � � − 1 Π( n ) . A ( n ) − zI np (11) The factorization of w A is equivalent to the recovery of Dirac system from S ( n ).
Let W k ( z ) be the normalized fundamental solution of the discrete Dirac system: W k +1 ( z ) = ( I 2 p + i zjC k ) W k ( z ) , W 0 ( z ) = I 2 p . Let S ( n ) > 0 and recall that the transfer m.-functions w A are given by the formulas w A ( k , z ) = I 2 p − i J Π( k ) ∗ S ( k ) − 1 � � − 1 Π( k ) A ( k ) − zI kp ( k ≤ n ) . These w A 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 W k corresponding to S ( n ) and ν : W k ( − 1 / z ) = z − k ( z + i ) k K ∗ w A ( k , − z / 2) K , (12) � I p � 1 − I p W 0 ( z ) := I 2 p , K := √ . (13) I p I p 2 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 ≤ ∞ .
� π s k = 1 S ( n ) = { s j − i } n e i kt d τ ( t ) , i , j =1 , (14) Recall 2 π − π � 1 � � λ � 1 − a k 0 Z k +1 ( λ ) = � Z k ( λ ) . (15) − a k 1 0 1 1 − | a k | 2 Verblunsky’s theorem. There is a one to one correspondence between the sequences { s k } k ≥ 0 such that S ( n ) > 0 for all n > 0 (or equivalently measures d τ with infinite support) and sequences (of Verblusky coefficients) { a k } k ≥ 0 , where | a k | < 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 { s k } 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 { s k } k ≥ 0 and sequences { ρ k } k ≥ 0 of Verblunsky-type coefficients. We note that we deal with the general block Toeplitz matrices.
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