Toeplitz condition numbers as an $H^{\infty}$ interpolation problem

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Toeplitz condition numbers as an H ^∞ interpolation problem

Rachid Zarouf

To cite this version:

Rachid Zarouf. Toeplitz condition numbers as an

interpolation problem. Journal of Mathematical

Sciences, Springer Verlag (Germany), 2009, 156 (5), pp.819–823. �hal-00468200v2�

TOEPLITZ CONDITION NUMBERS AS AN H^∞ INTERPOLATION PROBLEM

Rachid Zarouf

Abstract. The condition numbersCN(T) =kTk. T⁻¹

of Toeplitz and analyticn×nmatricesT are studied. It is shown that the supremum ofCN(T) over all such matrices withkTk ≤1 and the given minimum of eigenvalues r=min_i=1..n|λi|>0 behaves as the corresponding supremum over alln×nmatrices (i.e., as _r¹ⁿ(Kronecker)), and this equivalence is uniform innandr. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem.

Let H be a Hilbert space of finite dimension n and an invertible operator T acting on H such that k T k≤ 1. We are interested in estimating the norm of the inverse of T :

k T

k .

More precisely, given a family F of n−dimensional operators and a T ∈ F , we set r

(T ) = min

|λ

| > 0,

where {λ

, ..., λ

} = σ(T ) is the spectrum of T . We are looking for “the best possible majorant” Φ

(r) such that

T⁻¹

≤ Φ

(r)

for every T ∈ F , kT k ≤ 1. Let r ∈]0, 1[. This leads us to define the following bound c

(F, r),

c

(F, r) = sup

: T ∈ F , kT k ≤ 1, r

(T) ≥ r The following classical result is attributed to Kronecker ( XIX c.)

Theorem. (Kronecker):

Let F be the set of all n-dimensional operators defined on an euclidean space. Then c

(r) := c

(F, r) = 1

r

Since obviously the upper bound in c

(r) is attained (by a compactness argument), a natural question arises: how to describe the extremal matrices T such that kT k ≤ 1, r

(T ) ≥ r and

T⁻¹

=

. The answer is contained in N. Nikolski [1] in the following form: the case of equality

T⁻¹

= 1

r

occurs for a matrix T with k T k= 1 if and only if:

1

TOEPLITZ CONDITION NUMBERS AS ANH^∞ INTERPOLATION PROBLEM 2

(1) either r = 1 and then T is an arbitrary unitary matrix.

(2) or r < 1, and then the eigenvalues λ

(T) of T are such that

|λ

(T )| = r

and given σ = {λ

, ..., λ

} on the circle, there exists a unique extremal matrix T (up to a unitary equivalence) with the spectrum {λ

, ..., λ

} having the form

T = U + K

where K is a rank one matrix, U is unitary and U and K are both given explicitly. (In fact, T is nothing but the so-called model operator corresponding to the Blaschke product B = Π

b

, see [2] for definitions).

For numerical analysis, the interest is in some classes of structured matrices such as Toeplitz, Hankel etc.... In that note, we are going to focus on the Toeplitz structure. Recall that T is a Toeplitz matrix if and only if there exists a sequence (a

)

such that

T = T

=













a

a

. . a

a

. . . .

. . . . .

. . . . a

a

. . a

a













,

and that T is an analytic Toeplitz matrix if and only if there exists a sequence (a

)

such that

T = T

=













a

0 . . 0 a

. . . .

. . . . .

. . . . 0

a

. . a

a













.

We will denote by T

the set of Toeplitz matrices of size n and T

will be the set of analytic Toeplitz matrices of size n. This leads us to the following questions.

How behave the constants c

(T

, r) and c

(T

, r) when n → ∞ and/or r → 0? Are they uniformly comparable with the Kronecker bound c

(r)? Are there exist Toeplitz matrices among extremal matrices described above? The answers seem not to be obvious, at least the obvious candidates like T =

nk

, where J

is the n−dimensional Jordan matrix, do not lead to the needed uniform (in n and r) equivalence.

For short, we denote,

t

(r) = c

(T

, r) and

t

(r) = c

(T

, r).

Obviously we have,

t

(r) ≤ t

(r) ≤ c

(r) = 1 r

. The following theorem answers the above questions.

Theorem.

1) For all r ∈]0, 1[ and n ≥ 1, 1

2 ≤ r

t

(r) ≤ r

c

(r) = 1 2) For every n ≥ 1

lim

r

t

(r) = lim

r

t

(r) = 1 and for every 0 < r ≤ 1

lim

r

t

(r) = 1.

The proof of the theorem is given in section 2 below.

1. The operator Mn and its commutant

Let M

: (

, < ., . >) −→ (

, < ., . >) be the nilpotent Jordan Block of size n

M

=













0 1 .

. . . .

1 0













.

It is well known that the commutant {M

}

= {A ∈ M

(

) : AM

= M

A} of M

verifies {M

}

= {p (M

) : p ∈ P ol

} ,

where P ol

is the space of analytic polynomials. On the other hand, we can state this fact in the following way. Let

K

= z

H

= Lin 1, z, ..., z

, H

being the standard Hardy space in the disc

= {z : |z| < 1}, and

M

: K

→ K

such that

M

f = P

(zf ), ∀f ∈ K

.

TOEPLITZ CONDITION NUMBERS AS ANH^∞ INTERPOLATION PROBLEM 4

Then the matrix of M

in the orthonormal basis of K

, B

=

1, z, ..., z

is exactly M

, and hence {p (M

) , p ∈ P ol

} = {M

}

= {M

}

The following straightforward link between n × n analytic Toeplitz matrices and {M

}

is well known.

Lemma 1.

T

= {M

}

.

Let

φ(z) =

k≥0

φ(k)z ˆ

. Then,

φ (M

) =

n−1

X

k=0

φ(k)M ˆ

=















φ(0) ˆ φ(1) ˆ .

. . .

. . .

φ(n ˆ − 1) . . φ(1) ˆ φ(0) ˆ















.

Conversely, if A =













a

0 . . 0 a

. . . .

. . . . .

. . . . 0

a

. . a

a













∈ T

then A =

k=0

a

z

(M

) .

We also need the Schur-Caratheodory interpolation theorem (1912), which also can be considered as a partial case of the commutant lifting theorem of Sarason and Sz-Nagy-Foias (1968) see [2] p.230 Theorem 3.1.11.

Proposition 2.

The following are equivalent.

i) T is an n × n analytic Toeplitz matrix.

ii) There exists g ∈ H

such that T = g (M

).

Moreover

k T k= inf {k g k

: g ∈ H

(

) , g (M

) = T }

= min {k g k

: g ∈ H

(

) , g (M

) = T } , where k g k

= sup

|g(z)|.

2. Proof of the theorem

Lemma 3.

Let T be an invertible analytic Toeplitz matrix of size n × n (which means that there exists f ∈ P ol

⊂ H

such that T = f (M

)). Then

T

= inf {k g k

: g, h ∈ H

, f g + z

h = 1} .

Proof.

Since T

belongs also to {M

}

, there exists g ∈ P ol

⊂ H

such that T

= g (M

). This implies in particular that

(f g) (M

) = I

, which means that f g − 1 annihilates M

. That means that

f g − 1

is a multiple of z

in H

. Conversely, if g ∈ H

verifies the above Bezout equation with h ∈ H

then g (M

) = T

.

But by Proposition 1.2 we have

= inf

k g k

: g ∈ H

, g (M

) = T

, and hence

T⁻¹

= inf {k g k

: g, h ∈ H

, f g + z

h = 1} .

Proof of the theorem.

First, we prove that for every r ∈]0, 1[ there exists an analytic n × n Toeplitz matrix T

such that

1 ) kT

k ≤ 1, 2) σ (T

) = {r} , 3)

T_r⁻¹

≥

− 1.

Indeed, let

b

(z) = r − z

1 − rz ∈ H

be the Blaschke factor corresponding to r. The H

calculus of M

tells us that the operator T

:= b

(M

)

satisfies property 1):

kT

k ≤ kb

k

= 1.

On the other hand, by the spectral mapping theorem

σ (T

) = {b

(σ (M

))} = {b

(0)} = {r} .

In particular this proves that T

is invertible. Finally, using Lemma 2.3, we get

= inf {k g k

: g, h ∈ H

, b

g + z

h = 1} =

TOEPLITZ CONDITION NUMBERS AS ANH^∞ INTERPOLATION PROBLEM 6

= inf

1 − z

h b

∞

: h ∈ H

, r

h(r) = 1

=

= inf {k 1 − z

h k

: h ∈ H

, r

h(r) = 1} . But if h ∈ H

and r

h(r) = 1, we have

k 1 − z

h k

≥k h k

−1 and

k h k

≥ |h(r)| = 1 r

, which gives

k 1 − z

h k

≥ 1 r

− 1.

Therefore

T

≥ 1 r

− 1, which completes the proof of property 3) of T

.

Now we obtain

1 − r

≤ r

k≤ r

t

(r) ≤ r

t

(r) ≤ r

c

(r) = 1 for every r ∈]0, 1[. On the other hand, we have

T_r⁻¹

kT

k ≥ 1 and hence

≥ 1 kT

k ≥ 1.

As a result for all r ∈]0, 1[,

r

≥ r

, and combining with the previous estimate, we obtain

1

2 ≤ max(r

, 1 − r

) ≤ r

≤ r

t

(r) ≤ r

t

(r) ≤ r

c

(r) = 1,

which completes the first claim of the theorem. The second claim follows from the previous inequalities.

Remark.

It should be mention that we have not obtained an explicit formula for t

(r). Regarding the description of extremal matrices (for the quantity c

(r)) mentioned in the Introduction, it seems likely that t

(r) < c

(r) =

. In the same spirit, it would be of interest to know the limits lim

(inf

r

t

(r)) and lim

(inf

r

t

(r)) .

References

[1] N.Nikolski, Condition Numbers of Large Matrices and Analytic Capacities, St. Petersburg Math. J., 17 (2006), 641-682.

[2] N.Nikolski, Operators, Function, and Systems: an easy reading , Vol.1, Amer. Math. Soc. Monographs and Surveys, 2002.

Equipe d’Analyse & G´eom´etrie, Institut de Math´ematiques de Bordeaux, Universit´e Bordeaux, 351 Cours de la Lib´eration, 33405 Talence, France.

E-mail address : rzarouf@math.u-bordeaux1.fr