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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005
Some operator inequalities for unitarily invariant norms*†
Cristina Cano, Irene Mosconi‡ and Demetrio Stojanoff§
‡ Partially supported by Universidad Nac. del Comahue
§ Partially supported CONICET (PIP 4463/96), UNLP 11 X350 and ANPCYT (PICT03-09521)
Abstract:
Let be the algebra of bounded operators on a complex separable Hilbert space
. Let
be a unitarily invariant norm defined on a norm ideal
. Given two positive invertible operators
and
, we show that
,
. This extends Zhang's inequality for matrices. We prove that this inequality is equivalent to two particular cases of itself, namely
and
. We also characterize those numbers
such that the map
given by
is invertible, and we estimate the induced norm of
acting on the norm ideal
. We compute sharp constants for the involved inequalities in several particular cases.
Keywords and phrases: Positive matrices; Inequalities; Unitarily invariant norm.
AMS Subject Classification: Primary 47A30, 47B15.
1. INTRODUCTION
Let![H](/img/revistas/ruma/v46n1/1a0626x.png)
![L(H)](/img/revistas/ruma/v46n1/1a0627x.png)
![H](/img/revistas/ruma/v46n1/1a0628x.png)
![S ∈ L(H)](/img/revistas/ruma/v46n1/1a0629x.png)
![T ∈ L(H)](/img/revistas/ruma/v46n1/1a0630x.png)
![]() | (1) |
![T ↦-→ ST S -1 + S -1T S](/img/revistas/ruma/v46n1/1a0632x.png)
![S ∈ L(H)](/img/revistas/ruma/v46n1/1a0633x.png)
![S ∈ L(H)](/img/revistas/ruma/v46n1/1a0634x.png)
![T ∈ L(H),](/img/revistas/ruma/v46n1/1a0635x.png)
![k = 0, 1,2](/img/revistas/ruma/v46n1/1a0636x.png)
![]() | (2) |
![P,Q ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a0638x.png)
![T ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a0639x.png)
![k ∈ (- 2,2]](/img/revistas/ruma/v46n1/1a0640x.png)
![]() | (3) |
for every unitarily invariant norm on
.
![I](/img/revistas/ruma/v46n1/1a0644x.png)
![L(H)](/img/revistas/ruma/v46n1/1a0645x.png)
![9 ⋅9](/img/revistas/ruma/v46n1/1a0646x.png)
![k ∈ (- 2,2]](/img/revistas/ruma/v46n1/1a0647x.png)
![]() | (4) |
![]() | (5) |
![]() | (6) |
We give a proof of inequality (6), using a technical result about unitarily invariant norms, which allows us to obtain a reduction to the matricial case. In this case, we use a result of Bhatia and Parthasarathy [4], and some properties of the Hadamard product of matrices. This result was previously proved for in [2], for not necessarily positive
,
and
. We study the operators associated to the three mentioned inequalities, and their restriction as operators on the norm ideal
. We compute their spectra and, in some cases, their reduced minimum moduli (also called conorms). The rest of the paper deals with the estimation of sharp constants for inequality (5), with respect to the usual norm of
. We get the optimal constant, if one restricts to operators
. Using the notion of Hadamard index for positive matrices, studied in [7], we compute, for a fixed
, the constant
![-1 -1 + M (S, k) = max{C ≥ 0 : ∥ST S + S T S + kT ∥ ≥ C ∥T ∥ for every T ∈ L(H) },](/img/revistas/ruma/v46n1/1a0660x.png)
for (see Proposition 5.6). Finally, we give some partial results for
, in lower dimensions, showing numerical estimates of sharp constants. For
and
, we characterize the best intervals
such that the inequality (6) holds in
for every
.
In section 2, we fix several notations and state some preliminary results. We expose with some detail the theory of unitarily invariant norms defined on norm ideals of , proving some technical results in this area. In section 3, we show the equivalence of the mentioned inequalities and we give the proof of (6). In section 4, we study the associated operators. In section 5, we describe the theory of Hadamard index, and we use it to obtain a description of the constant
. In section 6 we study the case of matrices of lower dimensions.
We wish to acknowledge Prof. G. Corach who shared with us fruitful discussions concerning these matters.
2. PRELIMINARIES
Let be a separable Hilbert space, and
be the algebra of bounded linear operators on
. We denote
the ideal of compact operators,
the group of invertible operators,
the set of hermitian operators,
the set of positive definite operators,
the unitary group, and
the set of invertible positive definite operators.
Given an operator ,
denotes the range of
,
the nullspace of
,
the spectrum of
,
the adjoint of
,
the modulus of
,
the spectral radius of
, and
the spectral norm of
. Given a closed subspace
of
, we denote by
the orthogonal projection onto
.
When , we shall identify
with
,
with
, and we use the following notations:
for
,
for
,
for
, and
for
. A norm
in
is called unitarily invariant if
for every
and
.
Remark 2.1. The notion of unitarily invariant norms can be defined also for operators on Hilbert spaces. We give some basic definitions (see Simon's book [17]): Let . Then also
. We denote by
, the sequence of eigenvalues of
, taken in non increasing order and with multiplicity. If
, we take
for
. The numbers
are called the singular values of
.
Denote by the set of complex sequences which converge to zero. Consider
the set of sequences with finite non zero entries. For
, denote
. A gauge symmetric function (or symmetric norm) is a map
which satisfy the following properties:
• is a norm on
,
• for every
,
•and is invariant under permutations. We say that
is normalized if
. For
, define
![g(a) = sup g (a1,...,an,0,...) ∈ ℝ ∪ {+ ∞}. n∈ℕ](/img/revistas/ruma/v46n1/1a06142x.png)
A unitarily invariant norm in is a map
given by
,
, where
is a symmetric norm. The set
![I = Ig = {A ∈ L0(H) : 9A9 < ∞}](/img/revistas/ruma/v46n1/1a06148x.png)
![L(H)](/img/revistas/ruma/v46n1/1a06149x.png)
![9 ⋅9](/img/revistas/ruma/v46n1/1a06150x.png)
![(I,9 ⋅ 9)](/img/revistas/ruma/v46n1/1a06151x.png)
- If
, then
, and
for every
.
- If
has finite rank, then
, because
.
- If
, then
.
- Given
and
such that
and
.
- For every
and
, there exists a finite rank operator
such that
.
![p](/img/revistas/ruma/v46n1/1a06170x.png)
![p1∕p 9 A9p = tr(∣A ∣ )](/img/revistas/ruma/v46n1/1a06171x.png)
![1 ≤ p ≤ ∞](/img/revistas/ruma/v46n1/1a06172x.png)
![∥ ⋅ ∥(k)](/img/revistas/ruma/v46n1/1a06173x.png)
![k ∈ ℕ](/img/revistas/ruma/v46n1/1a06174x.png)
![∥ ⋅ ∥(1)](/img/revistas/ruma/v46n1/1a06175x.png)
![9 ⋅9 ∞](/img/revistas/ruma/v46n1/1a06176x.png)
![L0(H)](/img/revistas/ruma/v46n1/1a06177x.png)
![△](/img/revistas/ruma/v46n1/1a06178x.png)
Proposition 2.2. Let be an unitarily invariant norm on an ideal
. Let
be a increasing net of projections in
which converges strongly to the identity (i.e.,
for every
). Then
![N N PFT --→ T and PF T PF -- → T for every T ∈ I . F∈F F∈F](/img/revistas/ruma/v46n1/1a06185x.png)
Proof. By Remark 2.1, for every , there exists a finite rank operator
such that
. For every
and every projection
, it holds that
. In particular,
for every
. Hence, we can assume that
. Given
, denote
. Since
and
![T - PF TPF = QF T + TQF - QF T QF , F ∈ F,](/img/revistas/ruma/v46n1/1a06198x.png)
it suffices to prove that . Fix
. Note that
. Therefore
![∥(T *QF T )1∕2x∥2 = 〈(T *QF T)x, x〉 ≤ ∥(T*QF T)x∥ ∥x∥ --→ 0 . F ∈F](/img/revistas/ruma/v46n1/1a06202x.png)
This implies that , and all these operators act on the fixed finite dimensional subspace
, where the convergence of operators in every norm (included
) is equivalent to the SOT (or strong) convergence.
![9 ⋅9](/img/revistas/ruma/v46n1/1a06207x.png)
![I ⊆ L(H)](/img/revistas/ruma/v46n1/1a06208x.png)
![L(H ⊕ H)](/img/revistas/ruma/v46n1/1a06209x.png)
![2 × 2](/img/revistas/ruma/v46n1/1a06210x.png)
![L(H)](/img/revistas/ruma/v46n1/1a06211x.png)
![L(H)2 ×2](/img/revistas/ruma/v46n1/1a06212x.png)
![I 2](/img/revistas/ruma/v46n1/1a06213x.png)
![L(H ⊕ H)](/img/revistas/ruma/v46n1/1a06214x.png)
![9 ⋅9](/img/revistas/ruma/v46n1/1a06215x.png)
![g](/img/revistas/ruma/v46n1/1a06216x.png)
- Let
, and define
as any of the following matrices
Then,
, and
if and only if
.
- Under the mentioned identification,
.
![A = [aij],B = [bij] ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a06226x.png)
![A ∘ B](/img/revistas/ruma/v46n1/1a06227x.png)
![[aijbij]](/img/revistas/ruma/v46n1/1a06228x.png)
![A ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a06229x.png)
![ΦA : Mn( ℂ) → Mn( ℂ)](/img/revistas/ruma/v46n1/1a06230x.png)
![]() | (7) |
![9 ⋅9](/img/revistas/ruma/v46n1/1a06232x.png)
![Mn( ℂ)](/img/revistas/ruma/v46n1/1a06233x.png)
![ΦA](/img/revistas/ruma/v46n1/1a06234x.png)
![]() | (8) |
The following result collects two classical results of Schur about Hadamard (or Schur) products of positive matrices (see [16]), and a generalization of the second one for unitarily invariant norms, proved by Ando in [1, Proposition 7.7] .
Proposition 2.4 (Schur). Let![A ∈ Mn( ℂ)+](/img/revistas/ruma/v46n1/1a06236x.png)
![B ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a06237x.png)
- If
then also
.
- Denote by
. Then
(9) on
.
![k ∈ ℝ](/img/revistas/ruma/v46n1/1a06245x.png)
![9 ⋅9](/img/revistas/ruma/v46n1/1a06246x.png)
![I ⊆ L(H)](/img/revistas/ruma/v46n1/1a06247x.png)
, for every
and
.
, for every
and
.
, for every
and
.
![L(H ⊕ H) ~= L(H)2 ×2](/img/revistas/ruma/v46n1/1a06257x.png)
![P,Q ∈ Gl(H)+](/img/revistas/ruma/v46n1/1a06258x.png)
![T ∈ I](/img/revistas/ruma/v46n1/1a06259x.png)
![[ ] [ ] S1 = P 0 and T1 = 0 T ∈ I2 . 0 Q 0 0](/img/revistas/ruma/v46n1/1a06260x.png)
Then
![[ ] -1 - 1 0 P T Q- 1 + P -1T Q + kT S1T1S 1 + S1 T1S1 + kT1 = 0 0 .](/img/revistas/ruma/v46n1/1a06261x.png)
Therefore, as for every
, then
![9P TQ -1 + P -1TQ + kT 9 ≥ (2 + k) 9 T9 , T ∈ I.](/img/revistas/ruma/v46n1/1a06264x.png)
This shows . The same arguments using
show
. □
Remark 3.2. As said in the Introduction, the inequality 2 of Theorem 3.1 was proved, for the usual norm, by Corach-Porta-Recht in [6] with (and
not necessarily positive), and by Ameur Seddik in [15] with
. The inequality 1 of Theorem 3.1 was proved, in the finite dimensional case, by X. Zhan in [18], for
. In the rest of this section, we give a proof of inequality 2 of Theorem 3.1 for
in the general setting.
Lemma 3.3. Let ,
and
. Let
be given by
![λ λ Cn(k, λ)ij = -------i-j------, 1 ≤ i,j ≤ n . kλiλj + λ2i + λ2j](/img/revistas/ruma/v46n1/1a06277x.png)
Then for every
.
![Cn(k, λ)ij = (λiλ-j1 + λ-i1λj + k)- 1](/img/revistas/ruma/v46n1/1a06280x.png)
![x + x-1 ≥ 2](/img/revistas/ruma/v46n1/1a06281x.png)
![x ∈ ℝ +](/img/revistas/ruma/v46n1/1a06282x.png)
![k > - 2](/img/revistas/ruma/v46n1/1a06283x.png)
![- 2 < k ∈ ℝ](/img/revistas/ruma/v46n1/1a06284x.png)
![n λ ∈ ℝ +](/img/revistas/ruma/v46n1/1a06285x.png)
![Zn(k, λ) ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a06286x.png)
![]() | (10) |
satisfies for every
and
if and only if
On the other hand, if , then the matrix
By Propposition 2.4,
for every
. □
Theorem 3.4. Let and
. Then, for every unitarily invariant norm
on an ideal
, and for every
,
![9kT + ST S- 1 + S -1T S9 ≥ (k + 2) 9 T 9 .](/img/revistas/ruma/v46n1/1a06301x.png)
Proof. We follow the same steps as in [6]. By the spectral theorem, we can suppose that is finite, since
can be approximated in norm by operators
such that each
is finite. We can suppose also that
, by choosing an adequate net of finite rank projections
which converges strongly to the identity and replacing
by
. Indeed, the net may be chosen in such a way that
and
for every
. Note that, by Proposition 2.2,
converges to
for every
.
![S](/img/revistas/ruma/v46n1/1a06316x.png)
![ℂn](/img/revistas/ruma/v46n1/1a06317x.png)
![S = diag(λ1,λ2, ...,λn)](/img/revistas/ruma/v46n1/1a06318x.png)
![- 1 -1 k T + ST S + S T S = Bn(k, λ) ∘ T](/img/revistas/ruma/v46n1/1a06319x.png)
![]() | (11) |
Since for every
,
, it follows that, if
, then
for every
. Consider the matrix
given by
. Hence, in order to prove inequality (6) for every
, it suffices to show that
for
and
. By Lemma 3.3,
for every
. Finally, note that
,
Therefore, inequality (6) holds by Eq. (9) in Proposition 2.4. □
As a consequence of this result and Theorem 3.1, we get an infinite dimensional version, for every unitarily invariant norm, of Zhang inequality:
Corollary 3.5. Let and
. Then, for every unitarily invariant norm
on an ideal
, and for every
,
![9P TQ -1 + P -1T Q + kT 9 ≥ (2 + k) 9 T 9 .](/img/revistas/ruma/v46n1/1a06342x.png)
Corollary 3.6. Let and
. Then, for every unitarily invariant norm
on an ideal
, and for every
,
![9ST S + S- 1TS -1 + kT 9 ≥ (2 + k) 9 T 9 .](/img/revistas/ruma/v46n1/1a06348x.png)
4. THE ASSOCIATED OPERATORS
In this section we study the operators associated with the inequalities proved in the previous section. Given![P,Q ∈ Gl(H)+](/img/revistas/ruma/v46n1/1a06349x.png)
![k ∈ ℝ](/img/revistas/ruma/v46n1/1a06350x.png)
![ΥP,Q,k : L(H) → L(H)](/img/revistas/ruma/v46n1/1a06351x.png)
![]() | (12) |
Hence, for every unitarily invariant norm defined on an ideal
, inequality (4) means that
for
,
. Given
and
, define the operators
and
associated with inequalities (6) and (5):
and
. In this section we characterize, for fixed
, those
such that
is invertible. In some cases we estimate, for a given norm on some ideal of
, the induced norms of their inverses.
![u,v ∈ H](/img/revistas/ruma/v46n1/1a06368x.png)
![u ⊗ v ∈ L(H)](/img/revistas/ruma/v46n1/1a06369x.png)
![u ⊗ v(z) = 〈z,v 〉u](/img/revistas/ruma/v46n1/1a06370x.png)
![z ∈ H](/img/revistas/ruma/v46n1/1a06371x.png)
![R(u ⊗ v) = span {u}](/img/revistas/ruma/v46n1/1a06372x.png)
![u ⊗ v ∈ I](/img/revistas/ruma/v46n1/1a06373x.png)
![9 ⋅9](/img/revistas/ruma/v46n1/1a06374x.png)
.
- The map
is sesqui linear.
- If
, then
and
.
![+ P,Q ∈ Gl(H)](/img/revistas/ruma/v46n1/1a06381x.png)
![ΥP,Q = ΥP,Q,0](/img/revistas/ruma/v46n1/1a06382x.png)
![-1 -1 σ(ΥP,Q) = {λμ + λ μ : λ ∈ σ (P) ,μ ∈ σ(Q)}.](/img/revistas/ruma/v46n1/1a06383x.png)
Moreover, has the same spectrum, if it is considered as acting on any norm ideal
associated with a unitarily invariant norm.
Proof. Fix the norm ideal and consider the restriction
. Let
be given by
,
. Note that
. Therefore, by the known properties of the Riesz functional calculus for operators on Banach spaces (in this case, the Banach space is
and the map is
), it suffices to show that
.
Given , denote by
(resp.
) the operator given by
(resp.
),
. By Remark 2.1, these operators are bounded. If
, then
, and similarly for
. Hence
and
. Note that
. Therefore
![σ(AP,Q) ⊆ σ(LP )σ(RQ -1) ⊆ {λμ- 1 : λ ∈ σ (P ), μ ∈ σ(Q)}.](/img/revistas/ruma/v46n1/1a06407x.png)
Given ,
and
, let
be unit vectors such that
and
. Such vectors exist because
and
are selfadjoint operators. Consider the rank one operator
. Then, by Remark 4.1,
. Hence
![-1 - 1 - 1 9(AP,Q - λμ IdI)x ⊗ y9 = 9P x ⊗ Q y - λx-1⊗ μ y9 -1 - 1 ≤ 9(P x - λx) ⊗ Q y 9 + 9 λx ⊗ (Q y - μ y)9 = ∥P x - λx ∥∥Q - 1y ∥ + ∥λx∥ ∥Q -1y - μ-1y ∥ < (∥Q -1y∥ + ∥λx ∥) ɛ ≤ (∥Q -1∥ + ∥P ∥) ɛ .](/img/revistas/ruma/v46n1/1a06418x.png)
Therefore , because
. This shows that
, and the proof is complete. □
Corollary 4.3. Let and
. Then
is invertible if and only if
.
Proof. Just note that Id
. Then apply Proposition 4.2. □
Remark 4.4. Let be a unitarily invariant norm defined on a norm ideal
. By Remark 2.1,
for
and
. Given a linear operator
, we denote by
the induced norm:
![9 Υ9 = sup{9 Υ(T )9 : T ∈ I,9T 9 = 1}.](/img/revistas/ruma/v46n1/1a06435x.png)
By a standard argument and using the continuity of "taking inverse", one can show that, for
fixed, the map
is continuous.
Proposition 4.5. Let and
. Let
be a unitarily invariant norm defined on a norm ideal
. Then
. Moreover, if
, then
.
Proof. The inequality follows from Corollary 3.5. Suppose that is an eigenvalue for both
and
. Let
be unit vectors such that
and
. Consider
. Then, since
and
, it is easy to see that
. Hence,
in this case. An easy consequence of spectral theory is that every
such that
can be arbitrarily approximated by positive invertible operators such that
is a isolated point of their spectra, hence an eigenvalue. Applying this, jointly for
and
, for some
, and using the fact that the map
is continuous, the proof is completed. □
Corollary 4.6. Let and
. Let
be a unitarily invariant norm defined on a norm ideal
. Then
. If there exists
such that also
, then also
.
Proof. The first case follows applying Proposition 4.5 with . Note that the hypothesis
becomes obvious. For the second, take
and
. Note that
. □
Remark 4.7. The Forbenius norm works on the ideal of Hilbert Schmit operators, which is a Hilbert space with this norm. In this case, the operator
defined in Eq. (12) is positive, so that
. Therefore, Proposition 4.2 gives the sharp constant for inequality (4) for this norm. Observe that
if and only if
.
5. SHARP CONSTANTS AND HADAMARD INDEX
Preliminary results. In this subsection, we shall give a brief exposition of the definitions and results of the theory of Hadamard index, which we shall use in the rest of the section. All the results are taken from [7].
Denote by and
, the matrix with all its entries equal to
. Given
and
a norm on
, we define the
-index of
by
![I(N, A) = max{ λ ≥ 0 : N (A ∘ B) ≥ λN (B) ∀ B ∈ P (n)}](/img/revistas/ruma/v46n1/1a06492x.png)
and the minimal index of by
![I(A) = max { λ ≥ 0 : A ∘ B ≥ λB ∀ B ∈ Mn( ℂ)+ } = max { λ ≥ 0 : A - λP ≥ 0 }.](/img/revistas/ruma/v46n1/1a06494x.png)
The index relative to the spectral norm on will be denoted by
, and the index relative to the 2-norm
tr
,
, is denoted by
.
Proposition 5.1. Let . Then
if and only if
.
![( a b ) A = ¯ ∈ P2 b c](/img/revistas/ruma/v46n1/1a06505x.png)
![]() | (13) |
![A ∈ Mn( ℂ)+](/img/revistas/ruma/v46n1/1a06507x.png)
![Aii ⁄= 0](/img/revistas/ruma/v46n1/1a06508x.png)
- There exist
with nonnegative entries such that
.
![B ∈ Mn( ℂ)+](/img/revistas/ruma/v46n1/1a06513x.png)
![bij ≥ 0](/img/revistas/ruma/v46n1/1a06514x.png)
![i,j](/img/revistas/ruma/v46n1/1a06515x.png)
![J0](/img/revistas/ruma/v46n1/1a06516x.png)
![핀n = {1,2,...,n}](/img/revistas/ruma/v46n1/1a06517x.png)
![I(sp,B) = I(sp,BJ0) = I(BJ0)](/img/revistas/ruma/v46n1/1a06518x.png)
![I(sp,B) = min{ I(sp,B ) : J ⊆ 𝕀 J n](/img/revistas/ruma/v46n1/1a06519x.png)
![I(sp,B ) = I(B ) }. J J](/img/revistas/ruma/v46n1/1a06520x.png)
![□](/img/revistas/ruma/v46n1/1a06521x.png)
Let ,
and
. We consider
![( ) Λx = λiλj + --1-- ∈ Mn( ℂ)+ and λiλj ij](/img/revistas/ruma/v46n1/1a06525x.png)
![]() | (14) |
Proposition 5.4. Let ,
. Consider the matrix
defined before. Then
if and only
. More precisely,
- If
(i.e. #L = 1), then
and
.
- If
, then the image of
is the subspace generated by the vectors
and
.
- If
, say
, denote
. Then
(15) - If
then
, because
.
![positive](/img/revistas/ruma/v46n1/1a06547x.png)
![T](/img/revistas/ruma/v46n1/1a06548x.png)
![#L > 2](/img/revistas/ruma/v46n1/1a06549x.png)
![I(Λx) = 0](/img/revistas/ruma/v46n1/1a06550x.png)
![E (x, k) = Λ + kP n x](/img/revistas/ruma/v46n1/1a06551x.png)
![+ En(x, k) ∈ Mn( ℂ)](/img/revistas/ruma/v46n1/1a06552x.png)
![Λx + kP ≥ 0](/img/revistas/ruma/v46n1/1a06553x.png)
![k ≥ - I(Λx)](/img/revistas/ruma/v46n1/1a06554x.png)
![k ≥ 0](/img/revistas/ruma/v46n1/1a06555x.png)
![]() | (16) |
Hence, applying Proposition 5.4, we get formulas for in any case.
We shall compute using Theorem 5.3. Hence, we shall use the principal minors of
, which are matrices of the same type. Let
,
and
the induced vectors. Then
![En(x, k)J = E ∣J∣(xJ,k) = (Λx)J + kPJ = ΛxJ + kPJ .](/img/revistas/ruma/v46n1/1a06564x.png)
Claim: If then,
.
Indeed, by Proposition 5.4, , because
. Note that every
must satisfy
, because
span
. By Proposition 5.4,
, where
. As all entries of
are strictly positive, if
and
, then
. Therefore, the Claim follows from Theorem 5.2.
Hence, if , then
. If
, let
such that
. By Theorem 5.2 there exists a vector
such that
. Let
and
Easy computations show that, if we denote
, then
and, by Theorem 5.2,
. Moreover, by equations (15) and (16),
![(λi + λi )2 I(sp, En(x, k)J) = I(En(x, k)J) = ---1--2-22- + k = I(E0) = I(sp,E0). 1 + λi1λ i2](/img/revistas/ruma/v46n1/1a06591x.png)
Therefore, in order to compute using Theorem 5.3, we have to consider only the diagonal entries of
and some of its principal minors of size
. If
and
then, by equations (13) and (16),
![I(E0) = I(sp,E0) ⇐ ⇒ I(sp,Λ{λi,λj}) = I(Λ{λi,λj}) ⇐ ⇒ λ λ + -1--≤ min{ λ2 + -1 , λ2 + -1}. i j λiλj i λ2i j λ2j](/img/revistas/ruma/v46n1/1a06597x.png)
![λi < λj](/img/revistas/ruma/v46n1/1a06598x.png)
![λ2≤ -1--≤ λ2 i λiλj j](/img/revistas/ruma/v46n1/1a06599x.png)
![λi < 1 < λj](/img/revistas/ruma/v46n1/1a06600x.png)
![]() | (17) |
where and
![{ (λ + λ )2 1 } M2 = min --i---2j2-+ k : λi < 1 < λj and λ2i ≤ -----≤ λ2j . 1 + λiλj λiλj](/img/revistas/ruma/v46n1/1a06603x.png)
Now we can characterize the sharp constant for inequality (5), if we consider only operators
.
Proposition 5.6. Given and
, denote by
the greatest number such that
for every
. Then
where
and
![{ 2 } M2(S, k) = inf (λ-+-μ)--+ k : λ, μ ∈ σ(S),λ < μ and λ2 ≤ 1--≤ μ2 . 1 + λ2μ2 λμ](/img/revistas/ruma/v46n1/1a06613x.png)
In particular, if , then
.
Proof. We shall use the same steps as in the proof of Theorem 3.4 (and [6]). By the spectral theorem, we can suppose that is finite, since
can be approximated in norm by operators
such that
is a finite subset of
,
for all
and
is dense in
. So
(and
) converge to
(resp.
).
We can also suppose that , by choosing an adequate net of finite rank projections
which converges strongly to the identity and replacing
by
. Indeed, the net may be chosen in such a way that
and
for all
. Note that for every
,
converges to
.
Finally, we can suppose that is diagonal by a unitary change of basis in
. In this case, if
are the eigenvalues of
(with multiplicity) and
, then
. Note that all our reductions (unitary equivalences and compressions) preserve the fact that
. Now the statement follows from formula (17). If
then
, since
is the infimum of the empty set. Note that
is attained at
, because the map
is decreasing on
. □
6. NUMERICAL RESULTS
Let![S ∈ Gl(n)+](/img/revistas/ruma/v46n1/1a06653x.png)
![k ∈ (- 2,+∞)](/img/revistas/ruma/v46n1/1a06654x.png)
![{ } Nn(S, k) = max c ≥ 0 : ∥kT + ST S + S -1T S-1∥ ≥ c∥T∥, T ∈ Mn( ℂ) .](/img/revistas/ruma/v46n1/1a06655x.png)
![k ≤ 2](/img/revistas/ruma/v46n1/1a06656x.png)
![Nn(S, k) ≥ k + 2](/img/revistas/ruma/v46n1/1a06657x.png)
![+ S ∈ Gl(n)](/img/revistas/ruma/v46n1/1a06658x.png)
![-1 λ ∈ σ (S) ∩ σ (S )](/img/revistas/ruma/v46n1/1a06659x.png)
![Nn(S, k) = k + 2](/img/revistas/ruma/v46n1/1a06660x.png)
![S](/img/revistas/ruma/v46n1/1a06661x.png)
![Nn(S, k) = min λ∈ σ(S)λ2 + λ-2 + k](/img/revistas/ruma/v46n1/1a06662x.png)
![S = diag(λ)](/img/revistas/ruma/v46n1/1a06663x.png)
![λ = (λ1,λ2,...,λn) ∈ ℝn>0](/img/revistas/ruma/v46n1/1a06664x.png)
![k T + ST S + S -1TS -1 = E (λ,k) ∘ T n](/img/revistas/ruma/v46n1/1a06665x.png)
![E (λ,k) ∈ M (ℂ)+ n n](/img/revistas/ruma/v46n1/1a06666x.png)
![G ∈ Mn( ℂ)h](/img/revistas/ruma/v46n1/1a06667x.png)
![]() | (18) |
. Then, for
,
. Denote by
the map given by
, for
. We conclude that
.
Remark 6.1. There exists an extensive bibliography concerning methods for computing the norm of a Hadamard multiplier like . The oldest result in this direction is Schur Theorem (Proposition 2.4) for the positive case. We have applied this result in the proof of Theorem 3.4, but it is not useful in this case, because
. The most general result is 1983's Haagerup theorem [10], which gives a complete characterization, but it is not effective. There exist also several fast algorithms (see, for example, [9]) which allow to make numerical experimentation for this problem. For example, we have observed that the behavior of the map
, for any fixed
, is chaotic. But, as a great number of examples suggest, it seems that
if and only if
. Note that these cases are exactly those considered in Corollary 4.6.
Cowen and others [8] and [9] proved the following result for hermitian matrices:
Theorem 6.2. Let![B = (bij) ∈ Mn( ℂ)h](/img/revistas/ruma/v46n1/1a06683x.png)
![0 < b11 = max bjj 1≤j≤n](/img/revistas/ruma/v46n1/1a06684x.png)
![B](/img/revistas/ruma/v46n1/1a06685x.png)
![B](/img/revistas/ruma/v46n1/1a06686x.png)
![n - 1](/img/revistas/ruma/v46n1/1a06687x.png)
.
- If
and
, then
.
- For
, it holds that
.
![λ = (λ ,λ ,...,λ ) ∈ ℝn 1 2 n >0](/img/revistas/ruma/v46n1/1a06695x.png)
![1 ≥ λ ≥ λ ≥ ⋅⋅⋅ ≥ λ > 0 1 2 n](/img/revistas/ruma/v46n1/1a06696x.png)
![1 ≤ λ1 ≤ λ2 ≤ ⋅⋅⋅ ≤ λn.](/img/revistas/ruma/v46n1/1a06697x.png)
![-1 f (t) = t + t](/img/revistas/ruma/v46n1/1a06698x.png)
![(1,+ ∞)](/img/revistas/ruma/v46n1/1a06699x.png)
![(0,1)](/img/revistas/ruma/v46n1/1a06700x.png)
![G](/img/revistas/ruma/v46n1/1a06701x.png)
![0 < g11 = m1≤ajx≤n gjj](/img/revistas/ruma/v46n1/1a06702x.png)
![n > 2](/img/revistas/ruma/v46n1/1a06703x.png)
![G](/img/revistas/ruma/v46n1/1a06704x.png)
Proposition 6.3. Let . Suppose that
, or
S. Then
![Nn(S, 0) = min λ2 + λ- 2 . λ∈ σ(S)](/img/revistas/ruma/v46n1/1a06708x.png)
Proof. Suppose that with
or
. Let
as in equation (18), for
. Then
. Hence, since
, can apply Theorem 6.2. Then, in order to prove that
, it suffices to verify the inequality
Note that
![λ2 λ2 λ2 λ2λ1 C = (--4-1--)2 + --4-1-----4-2---- 2(--------)2 λ 1 + 1 (λ 1 + 1) (λ2 + 1) λ1λ2 + 1 2 2 2 6 2 4 4 4 6 6 = λ1(λ2---λ1)(λ1-+-λ2)(--1 +-λ1λ2-+-2λ1λ-2 --2λ-2 --λ-1λ2 +-λ1λ2) . (λ41 + 1)2(λ42 + 1)(1 + λ21λ22)2](/img/revistas/ruma/v46n1/1a06718x.png)
Straightforward computations with show that
, since the polynomial
in this case. A similar analysis shows that still
for
. The result follows by applying Theorem 6.2. □
It was proved by Kwong (see [13]) that, if , then the matrix
defined in Eq. (10), is positive semidefinite in the following cases:
and
,
and
, and
and
. Therefore, by the proof of Theorem 3.4, inequality (6) holds in
in these cases, for every unitarily invariant norm. Note that the proof Theorem 3.1 does not give similar estimates for the inequalities (4) and (5), because one needs to duplicate dimensions.
A numerical approach suggests that these intervals are optimal, both for the positivity of the matrix , defined in Eq. (11), and for inequality (6). In the case of
, if
and
then, using symbolic computation with the software Mathematica, one obtains a kind of "proof" of the fact that
for every
if and only if
The
principal sub matrices of
have the form
, and they live in
for every
. Therefore,
![+ 3 B3(k, λ) ∈ M3( ℂ) for every λ ∈ ℝ>0 ⇐ ⇒ - 2 < k ≤ 8.](/img/revistas/ruma/v46n1/1a06745x.png)
Likewise, for the matrix case, it suffices to study
, for
, and one obtains similar results.
Denote by the maximum number
such that inequality (6) holds in
for the spectral norm. By the preceding comments, and the proof of Theorem 3.4,
![k2 = + ∞, k3 ≥ 8, k4 ≥ 4, and kn ≥ 2 for n ≥ 5.](/img/revistas/ruma/v46n1/1a06752x.png)
Computer experimentation using the softwares Mathematica and Matlab suggests that, also in this case, ,
, and
, for
. In other words, inequality (6) holds for
in the same intervals as it holds that
for every
.
References
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Cristina Cano
Depto. de Matemática,
FaEA-UNC,
Neuquén, Argentina.
cbcano@uncoma.edu.ar
Irene Mosconi
Depto. de Matemática,
FaEA-UNC,
Neuquén, Argentina.
imosconi@uncoma.edu.ar
Demetrio Stojanoff
Depto. de Matemática,
FCE-UNLP, La Plata, Argentina
and IAM-CONICET.
demetrio@mate.unlp.edu.ar
Recibido: 3 de junio de 2005
Aceptado: 21 de noviembre de 2005