Nonisospectral flows on semiinfinite unitary block Jacobi matrices
      It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately, then the corresponding operator $\textbf{J}(t)$ satisfies the g...
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| author | Mokhonko, A. A. Мохонько, О. А. |
| author_facet | Mokhonko, A. A. Мохонько, О. А. |
| author_sort | Mokhonko, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:47:27Z |
| description |       It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately,
then the corresponding operator $\textbf{J}(t)$ satisfies the generalized
Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$,
where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix.
      The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$.
It is mapped into the unitary operator of multiplication $L(t)$ in the isomorphic space $L^2({\mathbb T}, d\rho)$, where ${\mathbb T} = {z: |z| = 1}$.
This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation.
A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented.
      The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices
(from self-adjoint case on ${\mathbb R}$)
and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory). |
| first_indexed | 2026-03-24T02:37:37Z |
| format | Article |
| fulltext |
UDС 517.53 + 517.91
O. A. Mokhon’ko (Kyiv Nat. Taras Shevchenko Univ.)
NONISOSPECTRAL FLOWS ON SEMIINFINITE
UNITARY BLOCK JACOBI MATRICES
НЕIЗОСПЕКТРАЛЬНI ПОТОКИ
НА НАПIВНЕСКIНЧЕННИХ УНIТАРНИХ
БЛОЧНИХ ЯКОБIЄВИХ МАТРИЦЯХ
It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite
block Jacobi matrix J(t) vary appropriately, then the corresponding operator J(t) satisfies the generalized
Lax equation
•
J(t) = Φ(J(t), t) + [J(t), A(J(t), t)], where Φ(λ, t) is a polynomial in λ and λ̄ with
t-dependent coefficients and A(J(t), t) = Ω + I +
1
2
Ψ is a skew-symmetric matrix.
The operator J(t) is analyzed in the space C⊕C2 ⊕C2 ⊕ . . . . It is mapped into the unitary operator
of multiplication L(t) in the isomorphic space L2(T, dρ), where T = {z : |z| = 1}. This fact enables
one to construct an efficient algorithm for solving the block lattice of differential equations generated
by the Lax equation. A procedure that allows one to solve the corresponding Cauchy problem by the
Inverse-Spectral-Problem method is presented.
The article contains examples of block difference-differential lattices and the corresponding flows that
are analogues of the Toda and van Moerbeke lattices (from self-adjoint case on R) and some notes about
applying this technique for Schur flow (unitary case on T and OPUC theory).
Доведено, що у випадку, коли спектр та спектральна мiра унiтарного оператора, породженого
напiвнескiнченною блочною якобiєвою матрицею J(t), змiнюються заданим чином, вiдповiдний
оператор J(t) задовольняє узагальнене рiвняння Лакса
•
J(t) = Φ(J(t), t) +
[
J(t), A(J(t), t)
]
, де
Φ(λ, t) є полiномом по λ та λ̄ з коефiцiєнтами, що залежать вiд t, i A(J(t), t) = Ω + I +
1
2
Ψ —
деяка кососиметрична матриця.
Оператор J(t) аналiзується у просторi C ⊕ C2 ⊕ C2 ⊕ . . . . Вiн вiдображається в унiтарний
оператор множення L(t) в iзоморфному просторi L2(T, dρ), де T = {z : |z| = 1}. Це дає можливiсть
побудувати ефективний алгоритм розв’язування блочного ланцюжка диференцiальних рiвнянь, що
породжується рiвнянням Лакса. У статтi наведено процедуру, що дозволяє розв’язувати вiдповiдну
задачу Кошi методом оберненої спектральної задачi.
Розглянуто приклади блочних диференцiально-рiзницевих ланцюжкiв та вiдповiдних їм потокiв,
що є аналогами ланцюжкiв Тоди та Ван Мербека (у самоспряженому випадку на R), а також деякi
зауваження стосовно застосування цiєї технiки до потоку Шура (унiтарний випадок на T та OPUC
теорiя).
1. Introduction. This article is the next logical step in developing the theory of
difference-differential lattices of equations generated by various forms of Lax equati-
on
•
J(t) = Φ(J(t), t) + [J(t), A(J(t), t)] of the following type. It is required that
J(t) : l2 → l2 can be mapped into the operator L(t) of multiplication by independent
variable in separable Hilbert space L2(C, dρ). Probability measure dρ has an infinite
compact support and is defined on the Borelean σ-algebra B(C). In the whole arti-
cle it is assumed that all operators are bounded. These restrictions define the class of
difference-differential lattices of equations that can be integrated by the method presented
here.
This work is based on numerous results by Yu. Berezansky, N. Dudkin, M. Shmoish,
L. Golinskii. And it became possible because of advance in OPUC theory (see related
c© O. A. MOKHON’KO, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 521
522 O. A. MOKHON’KO
article by M. J. Cantero, L. Moral, L. Velázquez [1] and Simon’s works [2 – 4]) and
CMV matrices theory (see [5, 6]).
In [7 – 9] Yu. M. Berezansky developed an approach to Cauchy problem for Toda
lattice on semiaxis and other similar difference-differential lattices. The author used
a number of results from spectral theory of classical Jacobi matrices. The main idea
in these works is as follows. Solution u(t), t ∈ [0,∞), was attached in a very si-
mple manner to a bounded self-adjoint Jacobi matrix J(t). At some restrictions for
initial difference-differential lattice the evolution of spectral measure dρ(λ; t) of the
corresponding operator J(t) could be found for initial spectral measure dρ(λ; 0) of any
pre-given J(0) that corresponds to initial condition u(0). Measure dρ(λ; 0) was built
using Direct-Spectral-Problem. Final result u(t) was obtained as the set of entries J(t)
that was reconstructed from dρ(λ; t) using Inverse-Spectral-Problem for ordinary Jacobi
matrices.
Later in [10 – 13] this method was extended for nonisospectral equations. In this
case the spectrum of J(t) varies with time t (in the case of Toda lattice the spectrum
is always the same — this is isospectral lattice). And in [14] the case of unbounded
selfadjoint J(t) was investigated.
In all previously mentioned articles J(t) was a self-adjoint operator in ordinary `2.
So the support of its spectral measure (spectrum σ(J(t))) always laid on R. Currently
considerable attention is paid to the theory of Orthogonal Polynomials on the Unit Circle
(OPUC) mostly because of the influence of Simon’s books (see. [2, 3]). These polynomi-
als are connected with five-diagonal operators in ordinary `2 (instead of three-diagonal
ones in self-adjoint case, see [1]). At the assumption that J(t) is unitary operator one can
build the analogous theory of difference-differential lattices with σ(J(t)) concentrated
on the unit circle T = {z ∈ C : |z| = 1}. First results in this direction were obtained by
L. Golinskii in [15]. Toda lattice is replaced with Schur flow here.
In mentioned above articles all operators were considered in ordinary `2 = C1⊕C1⊕
⊕ . . . . Recently Yu. Berezansky and M. Dudkin noticed that five-diagonal matrices
in OPUC can be concidered as ordinary three-diagonal block Jacobi matrices (see
[16]). Moreover this structure is absolutely natural (and arises in much simpler way as
essential construction) from slightly more general point of view (see [17]). J(t) must be
considered as Jacobi matrix in l2 = C1 ⊕ C2 ⊕ C3 ⊕ . . . if it is
normal operator. In particular it can be unitary. In this case it should be considered in
subspace l2,u = C1 ⊕ C2 ⊕ C2 ⊕ . . . ⊂ l2. Then one has Jacobi matrix for which it is
possible to apply a wide range of ideas developed in the last 150 years for ordinary
Jacobi matrices. The origins of this fruitful idea can be found in Mark Krein’s
article [18].
The main aim of this article is to show that the described above approach gives
results that are fully compatible with already obtained results (see [2, 3, 1, 15]). We
restrict ourselves with the unitary case. However most of the results can be formally
copied for more general case of normal J(t) (because all the proofs have algebraic
taste and does not depend on space structure). Now we are about to give a mechanism
of solving quite general lattice of block difference-differential equations and show that
already known results can be easily obtained as particular samples.
It is worth stressing that the described approach (use of block three-diagonal Jacobi
matrices in block spaces) allows to obtain in simple algebraic way a wide range of
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 523
well-known and completely new results which otherwise (if considered in ordinary `2)
would be technically complicated. This method reveals algebraic structure of spaces and
operators and in particular gives the hope to make OPUC much simpler. A series of
articles is planned by the author on this subject in the nearest future.
2. Common notes. We shall start with definitions of the main objects that are used
in this article: spaces, equations and operators. In the next section we shall formulate
the main result. Last sections are devoted to the explanation why the result is as it is
shown here and why is it convenient just in this form. All the proofs are contained in
Section 4.
The article is synchronized with [11]. So it is very simple to compare old and new
results if one has the two articles at hand simultaneously. Remind that one of the goals of
this work is to show that the corresponding well-known results (see [11, 1, 15]) naturally
embed into the new theory and are really simple here. Thus the article is organized in
such a way that the comparison of new and old results is as convenient as possible. The
last section will contain samples that show how one can construct the embedding of
theories.
Consider three-diagonal block Jacobi matrix
J(t) =
b0(t) c0(t) ·
a0(t) b1(t) c1(t) ·
a1(t) b2(t) c2(t) ·
· · · · ·
(1)
in the space
l2 = H0 ⊕H1 ⊕H2 ⊕ . . . , H0 = C, Hn = C2, n ≥ 1. (2)
l2 is Hilbert space with natural scalar product: for f, g ∈ l2 with coordinates in the
standard orthonormal basis
e0 =
(
1,
(
0
0
)
,
(
0
0
)
, . . .
)
,
en,1 =
0,
(
0
0
)
, . . . ,
(
0
0
)
,
(
1
0
)
n
,
(
0
0
)
. . .
, (3)
en,2 =
0,
(
0
0
)
, . . . ,
(
0
0
)
,
(
0
1
)
n
,
(
0
0
)
. . .
,
the norm and scalar product are defined as follows:
‖f‖2l2 =
∞∑
n=0
‖fn‖2Hn
, (f, g)l2 =
∞∑
n=0
(fn, gn)Hn
.
Here ‖fn‖2Hn
= ‖fn‖2C2 = |fn,1|2 + |fn,2|2,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
524 O. A. MOKHON’KO
(fn, gn)Hn
= (fn, gn)C2 = fn,1 · gn,1 + fn,2 · gn,2.
Let J in standard orthonormal basis have the following view:
J(t) =
b0;22 c0;21 c0;22
a0;12 b1;11 b1;12 0 0
0 b1;21 b1;22 c1;21 c1;22
a1;11 a1;12 b2;11 b2;12 0 0
0 0 b2;21 b2;22 c2;21 c2;22
a2;11 a2;12 b3;11 b3;12 0 0
0 0 b3;21 b3;22 c3;21 c3;22
(4)
All the entries are assumed to be continuously differentiable functions on the interval
[0, T ]. Matrix J(t) generates the corresponding operator J(t). The next step we need
is to apply [16] (Theorem 2). So we make assumptions for the entries of J(t) that
are described in this theorem. Assume that J(t) is unitary operator (in particular, it
is bounded, defined on the whole l2 and operators an(t), bn(t), cn(t) are uniformly
bounded). Additionally we assume that the entries of J(t) have the following properties
(see [16], formula (42)):
a0;12 > 0, a1;11 > 0, a2;11 > 0, a3;11 > 0, . . . ,
c0;22 > 0, c1;22 > 0, c2;22 > 0, c3;22 > 0, . . . .
(5)
Now it is necessary to cite [16] (Lemma 5).
Denote by lfin the linear subset of finite vectors in l2 (i.e., vectors that have only
finite number of non-zero coordinates) and by l′fin denote the corresponding conjugate
space.
Lemma 1. Let ϕ(z) = (ϕn(z))∞n=0, ϕn(z) ∈ Hn, z ∈ T, be a fixed solution from
l′fin of the following system with boundary condition ϕ0(z) ≡ ϕ0 ∈ C :
(Jϕ(z))n = an−1ϕn−1(z) + bnϕn(z) + cnϕn+1(z) = zϕn(z),
(J+ϕ(z))n = c∗n−1ϕn−1(z) + b∗nϕn(z) + a∗nϕn+1(z) = z̄ϕn(z),
n ∈ N0, ϕ−1(z) ≡ 0.
(6)
Then this solution exists for all ϕ0 and has the form for all n ∈ N
ϕn(z) = Qn(z)ϕ0 =
(
Qn;1(z), Qn;2(z)
)T
ϕ0,
where Qn;1 and Qn;2 are polynomials in z and z̄ of the following form:
Qn;1(z) = ln;1z̄
n + qn;1(z), Qn;2(z) = ln;2z
n + qn;2(z).
Here ln;1 > 0, ln;2 > 0, Q0(z) ≡ 1 and qn;1(z), qn;2(z) are linear combinations of 1,
z, z̄, . . . , zn−1, z̄n−1 for qn;1(z) and 1, z, z̄, . . . , zn−1, z̄n−1, z̄n for qn;2(z).
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 525
This lemma will be very important for us a bit later: this is conceptual point that
glues difference-differential lattices of equations with spectral theory technique. Now
we follow [16] (Theorem 2). Take any complex number ϕ0 ∈ C and build the solution
ϕ(z) = (ϕn(z))∞n=0, ϕn(z) ∈ Hn of equations (6) using Lemma 1 [16] (Theorem 2)
says that the mapping
̂ : l2 ⊃ lfin −→ L2(T, dρ(z)),
f = (fn)∞n=0 7−→ f0 +
∞∑
n=1
(
Qn;1(z)fn;1 +Qn;2(z)fn;2
)
after closure by continuity is unitary mapping between l2 and L2(T, dρ(z)). Here ρ is
probability spectral measure of J . Moreover the following Parseval equality takes place:
∀f, g ∈ lfin
(f, g)l2 =
∫
T
f̂(z) · ĝ(z)dρ(z), (Jf, g)l2 =
∫
T
z · f̂(z) · ĝ(z)dρ(z).
Explicit substitution of the elements of standard orthonormal basis (3) reveals that
Fourier transform ̂ maps them to Qn;α(z), n ∈ N, α = 1, 2 and Q0,α(z) ≡ 1. The last
statement of [16] (Theorem 2) claims that Qn;α(z), n ∈ N, α = 1, 2 and Q0,α(z) ≡ 1
constitute orthonormal basis of L2(T, dρ). Thus we can jump to two conclusions: first
under Fourier transform operator J maps to operator L of multiplication by independent
variable in L2(T, dρ) :
l2
J−−−−→ l2
̂
y ŷ
L2(T, dρ) −−−−→
L
L2(T, dρ)
and second: matrices of J (in standard orthonormal basis) and L (in Qn;α(z), n ∈ N,
α = 1, 2 and Q0,α(z) ≡ 1) coincide: they both are equal to J. Second conclusion is
of great importance for us. It says that there will be no need to make any changes
to the coefficients of difference-differential equations while passing from initial task
formulation (that is being performed in l2) to the space L2(T, dρ) where it can be solved
by using the explicit sense of L (see first conclusion).
Mentioned above [16] (Theorem 2) solves the Direct-Spectral-Problem. The corollary
of [16] (Theorem 2) (see the same page: corollary is unnumbered) solves Inverse-
Spectral-Problem. It says that if we apply [16] (Theorem 1) to measure dρ then we
reconstruct the original matrix J.
Lemma 1 ([16], Lemma 5) gives an interesting result (see remarks in the proof of ISP
mentioned above in [16], Theorems 1, 2). Polynomials Qn;α(z), n ∈ N, α = 1, 2 and
Q0,α(z) ≡ 1 can be constructed in the same manner as orthonormal basis was built in
the same situation in [1]. This is particular case of more common construction (see [17])
and that’s the way how [16] (Theorem 1) is being proved. We give only the necessary
brief sketch of this construction.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
526 O. A. MOKHON’KO
Denote by M the set of probability Borel measures on the unit circle T with infinite
support. Take a measure ρ ∈ M. Functions
1, z, z̄ =
1
z
, z2, z̄2 =
1
z2
, . . . (7)
are linearly independent in the space L2(T, dρ). Denote moments of ρ by
tn =
∫
T
zndρ(z), n ∈ Z. (8)
By using standard Gramm – Schmidt orthogonalization procedure construct the followi-
ng orthonormal basis of the space L2(T, dρ) :
P0(z) ≡ 1,
P1,1(z) =
z − (z, P0(z))L2P0(z)∥∥z − (z, P0(z))L2P0(z)
∥∥
L2
=
z −
∫
T
zdρ(z)
1/k1,1
= k1,1(z − t1), k1,1 > 0,
P1,2(z) =
1
z
−
(
1
z
, P0(z)
)
L2
P0(z)−
(
1
z
, P1,1(z)
)
L2
P1,1(z)∥∥∥∥1
z
−
(
1
z
, P0(z)
)
L2
P0(z)−
(
1
z
, P1,1(z)
)
L2
P1,1(z)
∥∥∥∥
L2
= (9)
= k1,2
1
z
+
(
− t−1k1,2 + k2
1,1k1,2t1
(
t−2 − t1t−1
))
− k2
1,1k1,2
(
t−2 − t1t−1
)
z;
Pn,1(z) = kn,1z
n + . . . , kn,1 > 0,
Pn,2(z) = kn,2z
−n + . . . , kn,2 > 0.
The fact is that in this way we actually obtain basis elements Qn;α(z), n ∈ N, α = 1, 2,
and Q0,α(z) ≡ 1. According to [16] (proof of Corollary from Theorem 2) the following
equality holds:
Q0(z) ≡ 1 = P0(z), Qn;1(z) = Pn;1(z), Qn;2(z) = Pn;2(z).
It is worth noting that P0, Pn;α, n ∈ N0, α = 1, 2, are the same polynomials as χn,
n ∈ N0, in [3, p. 442].
Finally from (6) we have the following system of equations:
an−1Pn−1(z) + bnPn(z) + cnPn+1(z) = zPn(z),
c∗n−1Pn−1(z) + b∗nPn(z) + a∗nPn+1(z) = z̄Pn(z).
(10)
In coordinate form it has the following view:(
an−1,11 an−1,12
0 0
)(
Pn−1,1
Pn−1,2
)
+
(
bn,11 bn,12
bn,21 bn,22
)(
Pn,1
Pn,2
)
+
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 527
+
(
0 0
cn,21
)
cn,22
(
Pn+1,1
Pn+1,2
)
= z
(
Pn,1
Pn,2
)
,
(
0 cn−1,21
0 cn−1,22
)(
Pn−1,1
Pn−1,2
)
+
(
bn,11 bn,21
bn,12 bn,22
)(
Pn,1
Pn,2
)
+
+
(
an,11 0
an,12 0
)(
Pn+1,1
Pn+1,2
)
= z
(
Pn,1
Pn,2
)
.
This result will play significant role for us. It gives explicit formulae for entries of
multiplication operators by z (operator L) and by z̄ =
1
z
(operator L∗ = L−1). It is
worth noting that these equations are equivalent to Szegő recursion (this question is not
considered in this article). For now it is sufficient to note that described above ideas
give the possibility to establish a connection with OPUC theory. To make the first step
towards OPUC it is necessary to rewrite entries of J in terms of Verblunsky coefficients
using [1].
The last remark touches the block structure of the spaces and operators used in this
article. Space l2 is built as block space from the most start. Its image under Fourier
transform L2(T; dρ) does not have any block structure. To be accurate it is necessary to
show the image of each cell of l2. This can be done fairly easily.
Introduce the spacesP0 = span{1} = C,Pn,1 = span
{
1, z, z̄, . . . , z(n−1), z−(n−1),
zn
}
, Pn,2 = span
{
1, z, z̄, . . . , z(n−1), z−(n−1), zn, z−n
}
. It is obvious by construction
of elements of orthonormal basis that
Pn,1 = {P0} ⊕ {P1,1} ⊕ {P1,2} ⊕ . . .⊕ {Pn−1,1} ⊕ {Pn−1,2} ⊕ {Pn,1},
Pn,2 = Pn,1 ⊕ {Pn,2}.
(11)
It is quite natural to combine pairs of one-dimensional subspaces in (11). Unite each
pair Pn,1, Pn,2 and construct the vector Pn(z) =
(
Pn,1(z)
Pn,2(z)
)
.
Denote P (z) =
(
Pn(z)
)∞
n=0
. Final answer is as follows:
l2 3 f = (fn)∞n=0 7→ f̂(z) =
∞∑
n=0
(fn, Pn(z))Hn
∈ L2(T, dρ).
3. Main result. Now let us pass to the central result of the article. Consider the
following polynomials in λ :
Φ(λ, t) =
l∑
j=−l
ϕj(t)λj , ϕj(t) ∈ C1
[0,∞)→C, λ ∈ T, (12)
Ψ(λ, t) =
m∑
j=−m
ψj(t)λj , ψj(t) ∈ C1
[0,∞)→C, λ ∈ T. (13)
Denote by D — differential operator
∂
∂λ
and consider operators:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
528 O. A. MOKHON’KO
Ω = Φ(L(t), t)D, Ω̂ = Φ(L∗(t), t)D, Ψ = Ψ(L(t), t), Ξ = −Ω− Ω̂∗ −Ψ,
I =
1
2
Ξ0,1;0,1 Ξ0,1;1,1 Ξ0,1;1,2 Ξ0,1;2,1 Ξ0,1;2,2
0
1
2
Ξ1,1;1,1 Ξ1,1;1,2 Ξ1,1;2,1 Ξ1,1;2,2
0 0
1
2
Ξ1,2;1,2 Ξ1,2;2,1 Ξ1,2;2,2
0 0 0
1
2
Ξ2,1;2,1 Ξ2,1;2,2
0 0 0 0
1
2
Ξ2,2;2,2
. (14)
Consider the following differential equation:
d
dt
L(t) = Φ(L(t), t) +
[
L(t),Ω + I +
1
2
Ψ
]
. (15)
Here [A,B] = AB −BA. This Lax equation is equivalent to the following differential-
difference chain of equations in matrix-variables an, bn, cn :
ȧn(t) = (an · Ωn,n + bn+1 · Ωn+1,n)+
+(Ω̂∗n−1,n+1 · cn−1 + Ω̂∗n,n+1 · bn + Ω̂∗n+1,n+1 · an)+
+
an ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
+
[(
Ξ∗n−1,n+1 · cn−1 + Ξ∗n,n+1 · bn + Ξ∗n+1,n+1 · an
)
−
−1
2
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)
· an
]
+
+ (an ·Ψn,n + bn+1 ·Ψn+1,n) + Φn+1,n,
ḃn(t) = (an−1 · Ωn−1,n + bn · Ωn,n + cn · Ωn+1,n)+
+(Ω̂∗n−1,n · cn−1 + Ω̂∗n,n · bn + Ω̂∗n+1,n · an)+
+
an−1 · {−Ω− Ω̂∗ −Ψ}n−1,n + bn ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
=
+
{−Ω∗ − Ω̂−Ψ∗}n−1,n · cn−1 +
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· bn
+
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 529
+(an−1 ·Ψn−1,n + bn ·Ψn,n + cn ·Ψn+1,n) + Φn,n, (16)
ċn(t) = (an−1 · Ωn−1,n+1 + bn · Ωn,n+1 + cn · Ωn+1,n+1)+
+(Ω̂∗n,n · cn + Ω̂∗n+1,n · bn+1 + Ω̂∗n+2,n · an+1)+
+
[
(an−1 · Ξn−1,n+1 + bn · Ξn,n+1 + cn · Ξn+1,n+1)−
−1
2
cn ·
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)]
+
+
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· cn
+
+ (an−1 ·Ψn−1,n+1 + bn ·Ψn,n+1 + cn ·Ψn+1,n+1) + Φn,n+1.
The Cauchy problem for the differential equation (15) can be stated as follows.
Suppose we have bounded unitary block Jacobi matrix L0 with entries an;11 > 0,
cn,22 > 0. Find L(t), t ∈ [0;T ], with continuously differentiable entries such that: L(t)
is a solution of (15) for t ∈ [0, T ] where T depends only on initial condition L0 and
functions Φ, Ψ (see (12), (13)) and
L(0) = L0. (17)
Here we introduce the following algorithm that solves the described above Cauchy
problem.
Algorithm. Let ρ(·, 0) be the spectral measure of the Jacobi matrix L0. It is built
using Direct-Spectral-Problem discussed in Section 2. Denote byM = supp ρ(·, 0) ⊂ T.
Consider the Cauchy problem
dλ(t)
dt
= Φ(λ(t), t), λ(0) = µ, µ ∈M, t ≥ 0. (18)
From the standard theory of differential equations it is well known that one can choose
T > 0 such that for every µ ∈ M there exists unique solution λ(·, µ) of the Cauchy
problem (18) defined on the interval [0, T ]. We suppose that polynomial Φ(λ, t) is such
that |λ(t)| = 1 ∀t ∈ [0;T ].
For every fixed t ∈ [0, T ] consider the mapping
ωt : M −→ T,
µ 7−→ λ(t, µ)
(19)
and construct the following measure (mapping step):
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530 O. A. MOKHON’KO
ρ̃(∆, t) = ρ(ω−1
t (∆), 0), ∆ ∈ B(T). (20)
Here ω−1
t (∆) is full preimage of the set ∆ under the mapping ωt. Let us consider the
following partial differential equation:
∂s(λ, t)
∂λ
Φ(λ, t) +
∂s(λ, t)
∂t
= Ψ(λ, t)s(λ, t), s(λ, 0) = 1, λ ∈ T, t ≥ 0. (21)
Let s(λ, t) be its nonnegative solution. Build the final measure tranformation (multipli-
cation step):
ρ(∆, t) =
∫
∆
s(λ, t)dρ̃(λ, t), ∆ ∈ B(T). (22)
The last step is to reconstruct L(t) from its spectral measure ρ(·, t), t ∈ [0, T ] by solving
the Inverse-Spectral-Problem discussed in Section 2. Briefly recall the corresponding
algorithm.
Consider the following family of functions:
1, z, z̄ =
1
z
, z2, z̄2 =
1
z2
, . . . . (23)
Build orthonormal basis P0(z, t), P1,1(z, t), P1,2(z, t), P2,1(z, t), P2,2(z, t), . . . of the
space L2(T, dρ(·, t)) (using standard Schmidt orthogonalization procedure). L(t) is
operator of multiplication by independent variable in the space L2(T, dρ(·, t)). Thus its
entries (that are the desired solution) can be found as:
Lj,α;k,β(t) =
∫
T
λPk,β(λ, t)Pj,α(λ, t)dρ(λ, t).
Theorem 1. A solution of the Cauchy problem (15), (17) exists and can be found
using the described above algorithm.
4. Proof. Here we restrict ourselves with existence theorem only. Uniqueness
theorem can be found in [11]. The aim now is to prove that if we take the described
above measure transformation then operator of multiplication by independent variable
satisfies differential equation (15) with initial condition (17).
Let F (λ, t) ∈ C1(T× [0, T ] → C) and consider the following function:
f(t) =
∫
T
F (λ, t)dρ(λ, t) =
∫
T
F (λ, t)s(λ, t)dρ̃(λ, t) =
=
∫
ω−1
t (T)
F (ωt(µ), t)s(ωt(µ), t)dρ(µ, 0) =
=
∫
T
F (λ(t, µ), t)s(λ(t, µ), t)dρ(µ, 0). (24)
Using (18) and (21) obtain the formula for df/dt :
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 531
df
dt
=
∫
T
{(
∂F (λ(t, µ), t)
∂λ
∂λ(t, µ)
∂t
+
∂F (λ(t, µ), t)
∂t
)
s(λ(t, µ), t) +
+
(
∂s(λ(t, µ), t)
∂λ
∂λ(t, µ)
∂t
+
∂s(λ(t, µ), t)
∂t
)
F (λ(t, µ), t)
}
dρ(µ, 0) =
=
∫
T
{(
∂F (λ(t, µ), t)
∂λ
· Φ(λ(t, µ), t) +
∂F (λ(t, µ), t)
∂t
)
s(λ(t, µ), t) +
+
(
∂s(λ(t, µ), t)
∂λ
· Φ(λ(t, µ), t) +
∂s(λ(t, µ), t)
∂t
)
F (λ(t, µ), t)
}
dρ(µ, 0) =
=
∫
T
{(
∂F (λ(t, µ), t)
∂λ
· Φ(λ(t, µ), t) +
∂F (λ(t, µ), t)
∂t
)
s(λ(t, µ), t) +
+Ψ(λ(t, µ), t)s(λ(t, µ), t)F (λ(t, µ), t)
}
dρ(µ, 0) =
=
∫
T
{
∂F (λ, t)
∂λ
· Φ(λ, t) +
∂F (λ, t)
∂t
+ Ψ(λ, t)F (λ, t)
}
dρ(λ, t).
Final result:
d
dt
∫
T
F (λ, t)dρ(λ, t) =
=
∫
T
{
∂F (λ, t)
∂λ
· Φ(λ, t) +
∂F (λ, t)
∂t
+ Ψ(λ, t)F (λ, t)
}
dρ(λ, t). (25)
The next step is to take left-hand side of (15) and obtain its right-hand side. So it is
necessary to compute
d
dt
Lj,α;k,β(t).
Let P0(·, t), P1,1(·, t), P1,2(·, t), . . . be the elements of orthonormal basis of the space
L2(T, dρ(·, t)) according to (9). Consider two operators: operator of multiplication and
differentiation operator in L2
(
T, dρ(·, t)
)
:
L(t) : L2(T, ρ(·, t)) −→ L2(T, ρ(·, t)),
f(z) 7−→ z · f(z),
(26)
D(t) : C∞ −→ L2(T, ρ(·, t)),
f(z, t) 7−→ df(z, t)
dz
.
(27)
By applying (25) with F (λ, t) = λPk,β(λ, t)Pj,α(λ, t) one can find the expression for
derivative of the coordinate Lj,α;k,β :
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532 O. A. MOKHON’KO
d
dt
Lj,α;k,β(t) =
=
∫
T
{
∂
∂λ
(
λPk,β(λ, t)Pj,α(λ, t)
)
· Φ(λ, t) +
∂
∂t
(
λPk,β(λ, t)Pj,α(λ, t)
)
+
+Ψ(λ, t)
(
λPk,β(λ, t)Pj,α(λ, t)
)}
dρ(λ, t) df= J1 + J2 + J3. (28)
Simplify each components J1, J2, J3 one-by-one:
J3 =
{
L(t)Ψ(L(t), t)
}
j,α;k,β
= {LΨ}j,α;k,β , (29)
J1 =
∫
T
∂
∂λ
(
λPk,β(λ, t)Pj,α(λ, t)
)
· Φ(λ, t)dρ(λ, t) =
=
∫
T
Pk,β(λ, t)Pj,α(λ, t)Φ(λ, t)dρ(λ, t)+
+
∫
T
λ
∂Pk,β(λ, t)
∂λ
Pj,α(λ, t)Φ(λ, t)dρ(λ, t)+
+
∫
T
λPk,β(λ, t)
∂Pj,α(λ, t)
∂λ
· Φ(λ, t)dρ(λ, t) =
=
{
Φ(L(t), t) + L(t)Φ(L(t), t)D(t) +D∗(t)Φ(L(t), t)L(t)
}
j,α;k,β
=
=
{
Φ + LΩ + Ω̂∗L
}
j,α;k,β
. (30)
Here Ω = Φ(L(t), t)D(t), Ω̂ = Φ(L∗(t), t)D(t).
We used the following facts:
Φ(λ, t) =
l∑
j=−l
ϕj(t)λj =
l∑
j=−l
ϕj(t)λ̄j =
l∑
j=−l
ϕj(t)
1
λj
= Φ
(
1
λ
, t
)
.
The second fact is that L(t) is unitary operator in L(T, dρ(·, t)), so L∗(t) = L−1(t).
And inverse operator L−1(t) is obviously the operator of multiplication by 1/λ :
J2 =
∫
T
∂
∂t
(
λPk,β(λ, t)Pj,α(λ, t)
)
dρ(λ, t) =
=
∫
T
λ
∂Pk,β(λ, t)
∂t
Pj,α(λ, t)dρ(λ, t) +
∫
T
λPk,β(λ, t)
∂Pj,α(λ, t)
∂t
dρ(λ, t) =
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 533
=
(
∂Pk,β
∂t
;
1
λ
Pj,α
)
L2
+
(
λPk,β ;
∂Pj,α
∂t
)
L2
.
Denote by
I : C∞ −→ L2(T, ρ(·, t)),
f(z, t) 7−→ df(z, t)
dt
.
(31)
Operator J2 can be represented as:
J2 = (LIPk,β , Pj,α)L2 + (I∗LPk,β , Pj,α)L2
. (32)
Thus
{J2}j,α;k,β = {LI + I∗L}j,α;k,β . (33)
Finally we have the following expression:
d
dt
L = Φ + LΩ + Ω̂∗L+ LI + I∗L+ LΨ. (34)
To obtain from this formula (15) it is necessary to express I through Ω and Ψ. We shall
synchronize our notations and proof with the one from [11] (Lemma 2). Note that in
this article operator J is self-adjoint and the main space is ordinary `2. We shall use
block-matrix ideology in our case to make the reasonings as close to [11] as possible.
Lemma 2 [analogue of [11], Lemma 2]. Denote by:
Ij,α;k,β =
(
∂Pk,β
∂t
;Pj,α
)
L2
(35)
and Ξ = −Ω− Ω̂∗ −Ψ.
The following matrix equalities hold:
1) Ij;k = 0 for j > k;
2) Ik;k =
1
2
Ξk,1;k,1 Ξk,1;k,2
0
1
2
Ξk,2;k,2
, k > 0, I0;0 =
1
2
Ξ0;0;
3) Ij;k = Ξj;k for j < k.
Note that in [11] ak = ck, Θ = Ψ, Ω∗ = Ω̂.
Proof. Since
∂Pk,β
∂t
∈ Pk,β it is obvious that
Ij,α;k,β = 0 for j > k and for j = k, α = 2, β = 1. (36)
All the other cases must be examined in the following specific way. We shall
differentiate the equality
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
534 O. A. MOKHON’KO∫
T
Pk,β(λ, t)Pj,α(λ, t)dρ(λ, t) = δk,jδα,β .
To do this we use formula (25) where F (λ, t) = Pk,β(λ, t)Pj,α(λ, t) :
0 =
d
dt
∫
T
Pk,β(λ, t)Pj,α(λ, t)dρ(λ, t) =
=
∫
T
{
∂
∂λ
(
Pk,β(λ, t)Pj,α(λ, t)
)
· Φ(λ, t) +
∂
∂t
(
Pk,β(λ, t)Pj,α(λ, t)
)
+
+Ψ(λ, t)
(
Pk,β(λ, t)Pj,α(λ, t)
)}
dρ(λ, t) =
=
∫
T
Φ(λ, t)
∂
∂λ
Pk,β(λ, t)Pj,α(λ, t)dρ(λ, t) +
+
∫
T
Φ(λ, t)Pk,β(λ, t)
∂
∂λ
Pj,α(λ, t)dρ(λ, t) +
+
∫
T
∂
∂t
Pk,β(λ, t)Pj,α(λ, t)dρ(λ, t) +
∫
T
Pk,β(λ, t)
∂
∂t
Pj,α(λ, t)dρ(λ, t) +
+
∫
T
Ψ(λ, t)Pk,β(λ, t)Pj,α(λ, t)dρ(λ, t) =
=
{
Φ(L(t), t)D(t)
}
j,α;k,β
+
{
Φ(L∗(t), t)D(t)
}
k,β;j,α
+
+ Ij,α;k,β + Ik,β;j,α +
{
Ψ(L(t), t)
}
j,α;k,β
.
Thus the following equality takes place:
Ij,α;k,β + Ik,β;j,α =
= −
[
{Φ(L(t), t)D(t)}j,α;k,β + {Φ(L∗(t), t)D(t)}k,β;j,α + {Ψ(L(t), t)}j,α;k,β
]
.
Thus we have
I + I∗ = −Ω− Ω̂∗ −Ψ. (37)
If j < k or (j = k, β = 2, α = 1) then Ik,β;j,α = 0 and we obtain
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 535
Ij,α;k,β = −
[
{Φ(L(t), t)D(t)}j,α;k,β+{Φ(L∗(t), t)D(t)}k,β;j,α+{Ψ(L(t), t)}j,α;k,β
]
.
The last option left is j = k; α = β. First let us show that Ij,α;j,α ∈ R.
Let α = 1 (option α = 2 is analyzed in the same manner). As we saw in (9)
Pj,1 = kj,1λ
j + . . . where kj,1 > 0. It is obvious that
∂
∂t
Pj,1(λ, t) = c0P0 + c1,1P1,1 +
+ c1,2P1,2 + . . . + cj,1Pj,1. That’s why
(
∂
∂t
Pj,1(·, t), Pj,1(·, t)
)
L2
= cj,1. Compare
coefficients at λj .We have
dkj,1(t)
dt
= cj,1(t)kj,1(t).Here kj,1(t) is real-valued non-zero
function, so cj,1(t) is real-valued too. So Ij,α;k,β + Ik,β;j,α = 2 · Ij,α;k,β , j = k, α = β.
Finally we obtain the following expression for Ij,α;j,α, j ∈ N0, α = 1, 2:
Ij,α;j,α =
= −1
2
[{
Φ(L(t), t)D(t)
}
j,α;j,α
+
{
Φ(L∗(t), t)D(t)
}
j,α;j,α
+
{
Ψ(L(t), t)
}
j,α;j,α
]
.
Thus we have the following view of I :
I =
1
2
Ξ0,1;0,1 Ξ0,1;1,1 Ξ0,1;1,2 Ξ0,1;2,1 Ξ0,1;2,2
0
1
2
Ξ1,1;1,1 Ξ1,1;1,2 Ξ1,1;2,1 Ξ1,1;2,2
0 0
1
2
Ξ1,2;1,2 Ξ1,2;2,1 Ξ1,2;2,2
0 0 0
1
2
Ξ2,1;2,1 Ξ2,1;2,2
0 0 0 0
1
2
Ξ2,2;2,2
. (38)
The lemma is proved.
Actually in this lemma the key role for us has formula (37). It allows us to finish
the proof of (15). The following lemma makes it obvious.
Lemma 3. The following equality takes place:
d
dt
L(t) = Φ(L(t), t) +
[
L(t),Ω + I +
1
2
Ψ
]
. (39)
Proof. Taking into account that λ ·Ψ(λ, t) = Ψ(λ, t) · λ⇒ LΨ = ΨL we obtain
d
dt
L = Φ + LΩ + Ω̂∗L+ LI + I∗L+ LΨ =
=
(
Φ + LΩ + Ω̂∗L
)
+ (LI + I∗L) +
(
1
2
(LΨ) +
1
2
(ΨL)
)
=
= Φ + L
(
Ω + I +
1
2
Ψ
)
+
(
Ω̂∗ + I∗ +
1
2
Ψ
)
L =
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
536 O. A. MOKHON’KO
= Φ + L
(
Ω + I +
1
2
Ψ
)
+
(
Ω̂∗ + (−Ω− Ω̂∗ −Ψ− I) +
1
2
Ψ
)
L =
= Φ + L
(
Ω + I +
1
2
Ψ
)
−
(
Ω + I +
1
2
Ψ
)
L = Φ +
[
L,Ω + I +
1
2
Ψ
]
.
For algebraic purposes this result is the most convenient one. Note that in [11] the
same one-dimensional result was obtained in much more complicated and obscure way.
5. Difference-differential lattices. We follow [11] and give coordinate-wise
interpretation of (15). This section is devoted to the proof of (16). It is important
for numerical applications and also gives the possibility (by choosing appropriate Φ and
Ψ) to obtain different matrix flows. This is obviously interesting in comparison with
e.g. Schur flow (see [15]).
The idea is to establish a connection between difference-differential lattices and Lax
equations (this section), Lax equations and spectral measures (previous section), spectral
measures and block Jacobi matrices (see [16, 17]), block Jacobi matrices and OPUC
theory (in particular with Verblunsky coefficients and their flows, Szegő recursion etc.
— see further papers). First we need another lemma.
Lemma 4 [analogue of [11], Lemma 3]. Denote by
Ej,α;k,β =
∫
T
λPk,β(λ, t)
∂Pj,α(λ, t)
∂t
dρ(λ, t) =
(
LPk,β ; IPj,α
)
L2
,
Êj,α;k,β =
∫
T
λ
∂Pk,β(λ, t)
∂t
Pj,α(λ, t)dρ(λ, t) =
(
LIPk,β ;Pj,α
)
L2
.
The following equalities take place:
(a) Ej,k = 0, Êj,k = {LΞ}j,k, j < k − 1;
(b) Ek−1,k =
1
2
Ξk−1,1;k−1,1 0
Ξk−1,1;k−1,2
1
2
Ξk−1,2;k−1,2
· ck−1,
Êk−1,k = {L · Ξ}k−1,k −
1
2
ck−1 ·
(
Ξk,1;k,1 0
0 Ξk,2;k,2
)
;
(c) Ek,k = {Ξ∗}k−1,k · ck−1 +
1
2
Ξk,1;k,1 0
Ξk,1;k,2
1
2
Ξk,2;k,2
· bk,
Êk,k = ak−1 · Ξk−1,k + bk ·
1
2
Ξk,1;k,1 Ξk,1;k,2
0
1
2
Ξk,2;k,2
;
(d) Ek+1,k = {Ξ∗ · L}k+1,k −
1
2
(
Ξk+1,1;k+1,1 0
0 Ξk+1,2;k+1,2
)
· ak,
Êk+1,k = ak ·
1
2
Ξk,1;k,1 Ξk,1;k,2
0
1
2
Ξk,2;k,2
;
(e) Ej,k = {Ξ∗ · L}j,k, Êj,k = 0, j > k + 1.
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 537
Proof. J is the matrix of multiplication operator L. So the following holds:
E∗j;k =
(
zPk,1
zPk,2
)
·L2
(
∂Pj,1
∂t
;
∂Pj,2
∂t
)
=
=
(
c̄∗k−1 ·
(
Pk−1,1
Pk−1,2
)
+ b̄∗k ·
(
Pk,1
Pk,2
)
+ ā∗k ·
(
Pk+1,1
Pk+1,2
))
·L2
(
∂Pj,1
∂t
;
∂Pj,2
∂t
)
=
= c∗k−1 ·
(
Pk−1,1
Pk−1,2
)
·L2
(
∂Pj,1
∂t
;
∂Pj,2
∂t
)
+ b∗k ·
(
Pk,1
Pk,2
)
·L2
(
∂Pj,1
∂t
;
∂Pj,2
∂t
)
+
+a∗k ·
(
Pk+1,1
Pk+1,2
)
·
(
∂Pj,1
∂t
;
∂Pj,2
∂t
)
=
= c∗k−1 · Ik−1,j + b∗k · Ik,j + a∗k · Ik+1,j .
Finally we obtain
Ej,k = I∗k−1,j · ck−1 + I∗k,j · bk + I∗k+1,j · ak = {I∗L}j,k. (40)
In the same manner we obtain the corresponding matrix representation for Êj,α;k,β :
Êj,α;k,β =
∫
T
λ
∂Pk,β(λ, t)
∂t
Pj,α(λ, t)dρ(λ, t) =
=
∫
T
∂Pk,β(λ, t)
∂t
1
λ
Pj,α(λ, t)dρ(λ, t) =
(
∂Pk,β(λ, t)
∂t
; λ̄Pj,α(λ, t)
)
L2
.
Thus (the same way as in the previous case)
Êj;k =
(
z̄Pj,1
z̄Pj,2
)
·L2
(
∂Pk,1
∂t
;
∂Pk,2
∂t
)
=
=
(
āj−1 ·
(
Pj−1,1
Pj−1,2
)
+ b̄j ·
(
Pj,1
Pj,2
)
+ c̄j ·
(
Pj+1,1
Pj+1,2
))
·L2
(
∂Pk,1
∂t
;
∂Pk,2
∂t
)
=
= aj−1 · Ij−1,k + bj · Ij,k + cj · Ij+1,k.
Final result:
Êj;k = aj−1 · Ij−1,k + bj · Ij,k + cj · Ij+1,k = {LI}j,k. (41)
Formulae (40), (41) allow us to write out the following expressions for Ej,k, Êj,k :
(a) Ej,k = 0, Êj,k = aj−1 · Ij−1,k + bj · Ij,k + cj · Ij+1,k, j < k − 1;
(b) Ek−1,k = I∗k−1,k−1 · ck−1, Êk−1,k = ak−2 · Ik−2,k + bk−1 · Ik−1,k + ck−1 · Ik,k;
(c) Ek,k = I∗k−1,k · ck−1 + I∗k,k · bk, Êk,k = ak−1 · Ik−1,k + bk · Ik,k;
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
538 O. A. MOKHON’KO
(d) Ek+1,k = I∗k−1,k+1 · ck−1 + I∗k,k+1 · bk + I∗k+1,k+1 · ak, Êk+1,k = ak · Ik,k;
(e) Ej,k = I∗k−1,j · ck−1 + I∗k,j · bk + I∗k+1,j · ak, Êj,k = 0, j > k + 1.
Consider the element Ek+1,k (here calculations are the most complicated; all other
coefficients are obtained in the same manner):
Ek+1,k = I∗k−1,k+1 · ck−1 + I∗k,k+1 · bk + I∗k+1,k+1 · ak =
= −
(
Ω∗k−1,k+1 + Ω̂k−1,k+1 + Ψ∗k−1,k+1
)
· ck−1−
−
(
Ω∗k,k+1 + Ω̂k,k+1 + Ψ∗k,k+1
)
· bk+
+
1
2
[
−Ωk+1,1;k+1,1 − Ω̂∗k+1,1;k+1,1 −Ψk+1,1;k+1,1
]
0[
−Ωk+1,1;k+1,2 − Ω̂∗k+1,1;k+1,2 −Ψk+1,1;k+1,2
] 1
2
[Ξk+1,2;k+1,2]
· ak =
= (−{Ω∗ · L}k+1,k + Ω∗k+1,k+1 · ak)+
+(−{Ω̂ · L}k+1,k + Ω̂k+1,k+1 · ak) + (−{Ψ∗ · L}k+1,k + Ψ∗k+1,k+1 · ak)+
+
1
2
[
−Ωk+1,1;k+1,1 − Ω̂∗k+1,1;k+1,1 −Ψk+1,1;k+1,1
]
0[
−Ωk+1,1;k+1,2 − Ω̂∗k+1,1;k+1,2 −Ψk+1,1;k+1,2
] 1
2
[Ξk+1,2;k+1,2]
· ak.
Note that
Ω∗k+1,k+1 · ak =
(
Ωk+1,1;k+1,1 Ωk+1,2;k+1,1
Ωk+1,1;k+1,2 Ωk+1,2;k+1,2
)(
ak;11 ak;12
0 0
)
=
=
(
Ωk+1,1;k+1,1 0
Ωk+1,1;k+1,2 Ωk+1,2;k+1,2
)(
ak;11 ak;12
0 0
)
.
So we obtain the following result (overline can be stripped because Ij,α;j,α ∈ R):
Ek+1,k = −{Ω∗ · L+ Ω̂ · L+ Ψ∗ · L}k+1,k−
−1
2
([
−Ωk+1,1;k+1,1 − Ω̂∗k+1,1;k+1,1 −Ψk+1,1;k+1,1
]
0
0 [Ξk+1,2;k+1,2]
)
· ak =
= {Ξ∗ · L}k+1,k −
1
2
(
Ξ∗k+1,1;k+1,1 0
0 Ξ∗k+1,2;k+1,2
)
· ak.
Compare this result with analogous one-dimensional result contained in [11] (Lemma 3,
formula (d)).
Formula (d) for Ej,k at j > k + 1 is being calculated in the same way (its proof is
part of the proof for Ek+1,k). Consider the element Êk−1,k :
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 539
Êk−1,k = ak−2 · Ik−2,k + bk−1 · Ik−1,k + ck−1 · Ik,k =
= ak−2 · (−Ωk−2,k − Ω̂∗k−2,k −Ψk−2,k)+
+bk−1 · (−Ωk−1,k − Ω̂∗k−1,k −Ψk−1,k)+
+ck−1 ·
1
2
Ξk,1;k,1 Ξk,1;k,2
0
1
2
Ξk,2;k,2
=
= (−{LΩ}k−1,k + ck−1 · Ωk,k) + (−{LΩ̂∗}k−1,k + ck−1 · Ω̂∗k,k)+
+(−{LΨ}k−1,k + ck−1 ·Ψk,k)+
+ck−1 ·
1
2
[
−Ωk,1;k,1 − Ω̂∗k,1;k,1 −Ψk,1;k,1
] [
−Ωk,1;k,2 − Ω̂∗k,1;k,2 −Ψk,1;k,2
]
0
1
2
[
−Ωk,2;k,2 − Ω̂∗k,2;k,2 −Ψk,2;k,2
]
.
Analogously
ck−1 · Ωk,k =
(
0 0
ck;21 ck;22
)(
Ωk,1;k,1 Ωk,1;k,2
Ωk,2;k,1 Ωk,2;k,2
)
.
So
ck−1 · Ik,k + ck−1 · Ωk,k + ck−1 · Ω̂∗k,k + ck−1 ·Ψk,k =
= ck−1 ·
1
2
[
Ωk,1;k,1 + Ω̂∗k,1;k,1 + Ψk,1;k,1
]
0
Ik,2;k,1
1
2
[
Ωk,2;k,2 + Ω̂∗k,2;k,2 + Ψk,2;k,2
]
.
From (36) we have Ik,2;k,1 = 0, thus
Êk−1,k = {L · Ξ}k−1,k −
1
2
ck−1 ·
(
Ξk,1;k,1 0
0 Ξk,2;k,2
)
.
The lemma is proved.
Lemma 5 [analogue of [11], step 1]. The non-trivial entries an, bn, cn of Jacobi
matrix L(t) satisfy the formulae (16).
Proof. Using Lemma 2 and Lemma 4 it is easy to verify that
ȧn(t) = L̇n+1,n(t) = {J1(t) +
(
Ê(t) + E(t)
)
+ J3(t)}n+1,n =
= {Φ + LΩ + Ω̂∗L}n+1;n +
an ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
540 O. A. MOKHON’KO
+
[
{Ξ∗ · L}n+1,n −
1
2
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)
· an
]
+ {LΨ}n+1;n =
= Φn+1,n + (an · Ωn,n + bn+1 · Ωn+1,n)+
+
(
Ω̂∗n−1,n+1 · cn−1 + Ω̂∗n,n+1 · bn + Ω̂∗n+1,n+1 · an
)
+
+
an ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
+
[
(Ξ∗n−1,n+1 · cn−1 + Ξ∗n,n+1 · bn + Ξ∗n+1,n+1 · an)−
−1
2
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)
· an
]
+
+(an ·Ψn,n + bn+1 ·Ψn+1,n).
Now verify the same for bn(t) :
ḃn(t) = L̇n,n(t) =
{
J1(t) +
(
Ê(t) + E(t)
)
+ J3(t)
}
n,n
=
= {Φ + LΩ + Ω̂∗L}n,n +
an−1 · Ξn−1,n + bn ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
+
{Ξ∗}n−1,n · cn−1 +
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2 · bn
+ {LΨ}n,n =
= {Φ}n,n + (an−1 · Ωn−1,n + bn · Ωn,n + cn · Ωn+1,n) +
+
(
Ω̂∗n−1,n · cn−1 + Ω̂∗n,n · bn + Ω̂∗n+1,n · an
)
+
+
an−1 · {−Ω− Ω̂∗ −Ψ}n−1,n + bn ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
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NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 541
+
{−Ω∗ − Ω̂−Ψ∗}n−1,n · cn−1 +
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· bn
+
+
{
1
2
LΨ +
1
2
ΨL
}
n,n
=
= {Φ}n,n + (an−1 · Ωn−1,n + bn · Ωn,n + cn · Ωn+1,n)+
+
(
Ω̂∗n−1,n · cn−1 + Ω̂∗n,n · bn + Ω̂∗n+1,n · an
)
+
+
an−1 · {−Ω− Ω̂∗ −Ψ}n−1,n + bn ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
+
{− Ω∗ − Ω̂−Ψ∗
}
n−1,n
· cn−1 +
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· bn
+
+
{
1
2
(an−1 ·Ψn−1,n + bn ·Ψn,n + cn ·Ψn+1,n) +
+
1
2
(Ψn,n−1 · cn−1 + Ψn,n · bn + Ψn,n+1 · an)
}
n,n
=
= {Φ}n,n + (bn · Ωn,n + cn · Ωn+1,n) +
(
Ω̂∗n−1,n · cn−1 + Ω̂∗n,n · bn + Ω̂∗n+1,n · an
)
+
+
an−1 · {−Ω̂∗}n−1,n + bn ·
1
2
Ξn,1;n,1 Ξn,1;n,2
0
1
2
Ξn,2;n,2
+
+
{− Ω∗ − Ω̂−Ψ∗
}
n−1,n
· cn−1 +
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· bn
+
+
{
1
2
(bn ·Ψn,n + cn ·Ψn+1,n) +
1
2
(Ψn,n−1 · cn−1 + Ψn,n · bn + Ψn,n+1 · an)
}
n,n
.
The lemma is proved.
Similarly obtain analogous results for cn :
ċn(t) = L̇n,n+1(t) = {J1(t) +
(
Ê(t) + E(t)
)
+ J3(t)}n,n+1 =
= {Φ + LΩ + Ω̂∗L}n,n+1 +
[
{L · Ξ}n,n+1 −
1
2
cn ·
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)]
+
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
542 O. A. MOKHON’KO
+
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· cn
+ {LΨ}n,n+1 =
=
(
an−1 · Ωn−1,n+1 + bn · Ωn,n+1 + cn · Ωn+1,n+1
)
+
+
(
Ω̂∗n,n · cn + Ω̂∗n+1,n · bn+1 + Ω̂∗n+2,n · an+1
)
+
+
[(
an−1 · Ξn−1,n+1 + bn · Ξn,n+1 + cn · Ξn+1,n+1
)
−
−1
2
cn ·
(
Ξn+1,1;n+1,1 0
0 Ξn+1,2;n+1,2
)]
+
+
1
2
Ξn,1;n,1 0
Ξn,1;n,2
1
2
Ξn,2;n,2
· cn
+
+
(
an−1 ·Ψn−1,n+1 + bn ·Ψn,n+1 + cn ·Ψn+1,n+1
)
+ Φn,n+1.
6. Samples. To obtain difference-differential flows like Schur Flow it is sufficient to
overwrite the coefficients of multiplication operator L (matrix J) in terms of Verblunsky
coefficients. The reader can find particular example in Leonid Golinskii article [15] of
how to obtain Schur flow from the appropriate Lax equation.
Example 1. Let Φ(λ, t) ≡ 0, Ψ(λ, t) = λ+
1
λ
. Then we obtain the case described
in [15] by L. Golinskii.
Proof. At Φ(λ, t) ≡ 0 we have Ω = Φ(L(t), t)D(t) = O, Ω̂ = Φ(L∗(t), t)D(t) =
= O, Ψ(L(t), t) = L + L∗, Ξ = −Ω − Ω̂∗ − Ψ = −Ψ = −L − L∗. Substitute this
into (39):
d
dt
L(t) = Φ(L(t), t) +
[
L(t),Ω + I +
1
2
Ψ
]
= [L,B],
whereB = I+
1
2
Ψ =
(L+ L∗)− − (L+ L∗)+
2
.Compare this with [15] (formulae (1.21),
(1.22)). This equation corresponds to Schur flow:
α′n(t) = (1− |αn|2)(αn+1(t)− αn−1(t)), t > 0. (42)
Example 2. Let Φ(λ, t) ≡ 0, Ψ(λ, t) = λ. Then we obtain two-dimensional
analogue for unitary case of the Toda lattice (that originally was built in one-dimensional
case `2 = C⊕ C⊕ C⊕ . . . for self-adjoint L).
Proof. At Φ(λ, t) ≡ 0 we have Ω = Φ(L(t), t)D(t) = O, Ω̂ = Φ(L∗(t), t)D(t) =
= O, Ψ(L(t), t) = L(t), Ξ = −Ω− Ω̂∗ −Ψ = −Ψ = −L. Substitute this into (39):
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
NONISOSPECTRAL FLOWS ON SEMIINFINITE UNITARY BLOCK JACOBI MATRICES 543
d
dt
L(t) = Φ(L(t), t) +
[
L(t),Ω + I +
1
2
Ψ
]
= [L,A],
where
A =
0 −1
2
c0;10 −1
2
c0;11 0 0
1
2
a0;01 0 −1
2
b1;01 0 0
0
1
2
b1;10 0 −1
2
c1;10 −1
2
c1;11
0
1
2
a1;00
1
2
a1;01 0 −1
2
b2;01
0 0 0
1
2
b2;10 0
.
Note that A is not uniquely determined: differential equation doesn’t change if we
replace A with A+ T where T is an arbitrary operator that commutes with L.
Now if we re-write A and L in terms of Verblunsky coefficients
L = C({αn}) =
ᾱ0 ᾱ1ρ0 ρ0ρ1
ρ0 −ᾱ1α0 −α0ρ1 0 0
0 ᾱ2ρ1 −ᾱ2α1 ᾱ3ρ2 ρ2ρ3
ρ1ρ2 −α1ρ2 −ᾱ3α2 −α2ρ3
0 0 ᾱ4ρ3 −ᾱ4α3
(43)
then we obtain “Toda” flow for unitary case:
α′n(t) = (|αn|2 − 1)αn−1. (44)
There are many ways how to prove this. The simplest one is to modify slightly [15]
(Theorem 2). Similarly the next example is obtained.
Example 3. Let Φ(λ, t) ≡ 0, Ψ(λ, t) = λ2. Then we obtain the analog for Kac –
van Moerbeke lattice.
Proof. Recall that classical Kac – van Moerbeke lattice for self-adjoint L has the
following view:
ẋn(t) = xn(xn+1 − xn−1), n = 0, 1, . . . , x−1 = 0. (45)
In our case Lax equation has the same form as in previous example with A = I +
1
2
Ψ
where Ψ = L2 and Ξ = −L2. In terms of Verblunsky coefficients Kac – van Moerbeke
flow is as follows:
α′n(t) =
(
1− |αn|2
)(
αn+1ᾱnαn−1 − αn−2 + |αn−1|2(αn + αn−2)
)
. (46)
The described above theory gives the possibility to build entire families of different
flows. And this is the object of further investigations.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
544 O. A. MOKHON’KO
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Received 03.07.07,
after revision — 21.12.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
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| id | umjimathkievua-article-3173 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:37Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/06/245fb5c4e409919bbbb97f8ede14fa06.pdf |
| spelling | umjimathkievua-article-31732020-03-18T19:47:27Z Nonisospectral flows on semiinfinite unitary block Jacobi matrices Неізоспектральні потоки на напівнескінченних унiтарних блочних якобієвих матрицях Mokhonko, A. A. Мохонько, О. А. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately, then the corresponding operator $\textbf{J}(t)$ satisfies the generalized Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$, where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$. It is mapped into the unitary operator of multiplication $L(t)$ in the isomorphic space $L^2({\mathbb T}, d\rho)$, where ${\mathbb T} = {z: |z| = 1}$. This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation. A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices (from self-adjoint case on ${\mathbb R}$) and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory). &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Доведено, що у випадку, коли спектр та спектральна мiра унітарного оператора, породженого напівнескінченною блочною якобієвою матрицею $J(і)$, змінюються заданим чином, відповідний оператор $\textbf{J}(t)$ задовольняє узагальнене рівняння Лакса $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$, де $\Phi(\lambda, t)$ є поліномом по $\lambda$ та $\overline{\lambda}$ з коефіцієнтами, що залежать від $t$, і $A(J(t), t) = \Omega + I + \frac12 \Psi$ — деяка кососиметрична матриця. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Оператор $J(t$) аналізується у просторі ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$. Він відображається в унітарний оператор множення $L(t)$ в ізоморфному просторі $L^2({\mathbb T}, d\rho)$, де ${\mathbb T} = {z: |z| = 1}$. Це дає можливість побудувати ефективний алгоритм розв'язування блочного ланцюжка диференціальних рівнянь, що породжується рівнянням Лакса. У статті наведено процедуру, що дозволяє розв'язувати відповідну задачу Коші методом оберненої спектральної задачі. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Розглянуто приклади блочних диференціально-різницевих ланцюжків та відповідних їм потоків, що є аналогами ланцюжків Тоди та Ван Мербека (у самоспряженому випадку на ${\mathbb R}$), а також деякі зауваження стосовно застосування цієї техніки до потоку Шура (унітарний випадок на ${\mathbb T}$ та OPUC теорія). Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3173 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 521–544 Український математичний журнал; Том 60 № 4 (2008); 521–544 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3173/3093 https://umj.imath.kiev.ua/index.php/umj/article/view/3173/3094 Copyright (c) 2008 Mokhonko A. A. |
| spellingShingle | Mokhonko, A. A. Мохонько, О. А. Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title | Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title_alt | Неізоспектральні потоки на напівнескінченних унiтарних блочних якобієвих матрицях |
| title_full | Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title_fullStr | Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title_full_unstemmed | Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title_short | Nonisospectral flows on semiinfinite unitary block Jacobi matrices |
| title_sort | nonisospectral flows on semiinfinite unitary block jacobi matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3173 |
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