The result is related to Korányi’s work on monotone matrix functions of several variables

http://jipam.vu.edu.au/

Volume 4, Issue 5, Article 88, 2003

INTERPOLATION FUNCTIONS OF SEVERAL MATRIX VARIABLES

Y. AMEUR KARLSROGATAN83A

S-752 39 UPPSALA, SWEDEN. yacin@math.uu.se

Received 16 March, 2003; accepted 14 November, 2003 Communicated by F. Hansen

ABSTRACT. An interpolation theorem of Donoghue is extended to interpolation of tensor prod- ucts. The result is related to Korányi’s work on monotone matrix functions of several variables.

Key words and phrases: Interpolation, Tensor product, Hilbert space, Korányi’s theorem.

2000 Mathematics Subject Classification. Primary: 46B70, Secondary: 46C15, 47A80.

1. STATEMENT ANDPROOF OF THEMAIN RESULT

Recall the definition of an interpolation function (of one variable). LetA∈M_n(C) :=L(`ⁿ₂) be a positive definite matrix. A real functionhdefined onσ(A)is said to belong to the classC_A of interpolation functions with respect toAif

(1.1) T ∈M_n(C), T^∗T ≤1, T^∗AT ≤A

imply

(1.2) T^∗h(A)T ≤h(A).

(Here A ≤ B means that B −A is positive semidefinite). By Donoghue’s theorem (cf. [4, Theorem 1], see also [1, Theorem 7.1]), the functions inC_A are precisely those representable in the form

(1.3) h(λ) =

Z

[0,∞]

(1 +t)λ

1 +tλ dρ(t), λ∈σ(A),

for some positive Radon measure ρ on the compactified half-line [0,∞]. Thus, by Löwner’s theorem (see [6] or [3]), C_A is precisely the set of restrictions toσ(A)of the positive matrix monotone functions onR+, in the sense thatA, B ∈M_n(C)positive definite andA≤Bimply h(A)≤h(B). Before we proceed, it is important to note that

(1.4) h∈CA implies h¹² ∈CA

ISSN (electronic): 1443-5756

c 2003 Victoria University. All rights reserved.

037-03

because the functionλ 7→ λ¹² is matrix monotone and the class of matrix monotone functions is a semi-group under composition.

Given two positive definite matricesA_i ∈ M_ni(C), define the classC_A1,A2 of interpolation functions with respect toA₁, A₂as the set of functionshdefined onσ(A₁)×σ(A₂)having the following property:

(1.5) T_i ∈M_ni(C) T_i^∗T_i ≤1 T_i^∗A_iT_i ≤A_i, i= 1,2 imply

(1.6) (T1⊗T2)^∗h(A1, A2)(T1 ⊗T2)≤h(A1, A2).

(Here (cf. [8])

h(A₁, A₂) = X

(λ1,λ2)∈σ(A1)×σ(A2)

h(λ₁, λ₂)E_λ1 ⊗F_λ2,

whereE,F are the spectral resolutions ofA₁,A₂).

Note that ifh =h₁⊗h₂ is an elementary tensor whereh_i ∈C_Ai, thenh∈ C_A1,A2, because then (1.5) yields

(T₁⊗T₂)^∗h(A₁, A₂)(T₁⊗T₂) = (T₁^∗h₁(A₁)T₁)⊗(T₂^∗h₂(A₂)T₂)

≤h₁(A₁)⊗h₂(A₂) =h(A₁, A₂), i.e. (1.6) holds. Since by (1.3) each function

λ7→ (1 +t)λ 1 +tλ

is inC_Afor anyA, and since the classC_A1,A2 is a convex cone, closed under pointwise conver- gence, it follows that functions of the type

(1.7) h(λ1, λ2) =

Z

[0,∞]²

(1 +t₁)λ₁ 1 +t₁λ₁

(1 +t₂)λ₂

1 +t₂λ₂ dρ(t1, t2),

where ρ is a positive Radon measure on [0,∞]² are in C_A1,A2 for all A₁, A₂. We have thus proved the easy part of our main theorem:

Theorem 1.1. Let h be a real function defined onσ(A₁)×σ(A₂). Then h ∈ C_A1,A2 iffh is representable in the form (1.7) for some positive Radon measureρ.

It remains to prove “⇒”. Let us make some preliminary observations:

(i) ([2, Lemma 2.2]) The classC_A1,A2 is unitarily invariant in the sense that if A₁ andA₂ are unitarily equivalent toA⁰₁ andA⁰₂ respectively, thenh∈CA1,A2 impliesh ∈C_A⁰

1,A⁰₂. (Indeed,

h(U₁^∗A₁U₁, U₂^∗A₂U₂) = (U₁ ⊗U₂)^∗h(A₁, A₂)(U₁⊗U₂) for all unitariesU₁, U₂).

(ii) ([2, Lemma 2.1]) The classC_A1,A2 respects compressions to invariant subspaces in the sense that iff ∈C_A1,A2 andA⁰₁,A⁰₂are compressions ofA₁,A₂respectively to invariant subspaces, thenh∈C_A⁰

1,A⁰₂. (Indeed,

(E⊗F)h(A₁, A₂)(E⊗F) = (E⊗F)h(EA₁E, F A₂F)(E⊗F)

wheneverE,F are orthogonal projections commuting withA₁,A₂respectively).

(iii) If λ^∗₂ is any (fixed) eigenvalue of A₂ and the function h_λ^∗

2 : σ(A₁) → R is defined by h_λ^∗₂(λ₁) = h(λ₁, λ^∗₂), then

h(A1, λ^∗₂Fλ^∗₂) = X

λ1∈σ(A1)

h(λ1, λ^∗₂)(Eλ1 ⊗Fλ^∗₂)

=



 X

λ1∈σ(A1)

hλ^∗₂(λ1)Eλ1



⊗Fλ^∗₂

=hλ^∗₂(A1)⊗Fλ^∗₂.

(iv) By symmetry, of course (with fixedλ^∗₁inσ(A1)andhλ^∗₁(λ2) =h(λ^∗₁, λ2)), h(λ^∗₁E_λ^∗

1, A₂) = E_λ^∗

1 ⊗h_λ^∗

1(A₂).

Lemma 1.2. Leth∈C_A1,A2 and letλ^∗₁,λ^∗₂be fixed eigenvalues ofA₁andA₂respectively. Then h

1 2

λ^∗₁ ∈C_A2 andh

1 2

λ^∗₂ ∈C_A1.

Proof. By symmetry of the problem, it suffices to prove the statement abouth

1 2

λ^∗₂. Ifh ∈C_A1,A2, then by (iii),

h(A₁, λ^∗₂F_λ^∗₂) = h_λ^∗₂(A₁)⊗F_λ^∗₂. Let f₂^∗ be a fixed non-zero vector in the range of F_λ^∗

2 and put c = (F_λ^∗

2f₂^∗, f₂^∗) > 0. Put T₂ =F_λ^∗

2 and letT₁ be any matrix fulfillingT₁^∗T₁ ≤ 1andT₁^∗A₁T₁ ≤ A₁; then plainly T₁, T₂ satisfy condition (1.5). Thus, sinceh∈C_A1,λ^∗

2F_λ∗

2, we get from (1.6)

((T₁⊗T₂)^∗h(A₁, λ^∗₂F_λ^∗₂)(T₁⊗T₂)(f₁⊗f₂^∗), f₁⊗f₂^∗)−(h(A₁, λ^∗₂F_λ^∗₂)(f₁⊗f₂^∗), f₁⊗f₂^∗)

=c((T₁^∗h_λ^∗

2(A₁)T₁f₁, f₁)−(h_λ^∗

2(A₁)f₁, f₁))≤0, f₁ ∈M_n1(C).

This yields T₁^∗h_λ^∗

2(A₁)T₁ ≤ h_λ^∗

2(A₁), T₁ ∈ M_n1(C), i.e. h_λ^∗

2 ∈ C_A1. In view of (1.4), h

1 2

λ^∗₂ ∈

C_A1.

Leth be a fixed function in the classCA1,A2. Replacing the matricesA1, A2 byc1A1, c2A2

for suitable constantsc1, c2 >0, we can assume without loss of generality that

(1.8) (1,1)∈σ(A₁)×σ(A₂).

DefineC to be theC^∗-algebra of continuous functions [0,∞] → Cwith the supremum norm, and denote (for fixedλ∈R+) bye_λ the function

e_λ(t) = (1 +t)λ

1 +tλ ∈C, t∈[0,∞].

Let two finite-dimensional subspacesV1, V2 be defined by

Vi = span{eλi :λi ∈σ(Ai)} ⊂C, i= 1,2.

Then (1.8) yields that the unit1 = e₁(t)∈ C belongs toV₁∩V₂. For fixedλ^∗_i ∈ σ(A_i), define two linear functionals

φ_λ^∗

1 :V₂ →C, φ_λ^∗

2 :V₁ →C by

φ_λ^∗

1



 X

λ2∈σ(A₂)

a_λ2e_λ2



= X

λ2∈σ(A₂)

a_λ2h_λ^∗

1(λ₂)¹²,

and

φ_λ^∗₂



 X

λ1∈σ(A1)

a_λ1e_λ1



= X

λ1∈σ(A1)

a_λ1h_λ^∗₂(λ₁)¹² respectively. We then have the following lemma:

Lemma 1.3. The functionalφ_λ^∗

1 is positive onV₂ in the sense that ifu ∈ V₂ satisfiesu(t) ≥ 0 for allt >0, thenφ_λ^∗

1(u)≥0. Similarly,φ_λ^∗

2 is a positive functional onV₁.

Proof of Lemma 1.3. This follows from Lemma 1.2 and Lemma 7.1 of [1].

Proof of Theorem 1.1. Consider now the bilinear form φ:V₁×V₂ →C defined by

(1.9) φ



 X

λ1∈σ(A1)

aλ1eλ1, X

λ2∈σ(A2)

aλ2eλ2





= X

(λ^∗₁,λ^∗₂)∈σ(A₁)×σ(A₂)

φ_λ^∗

1



 X

λ2∈σ(A₂)

a_λ2e_λ2



φ_λ^∗

2



 X

λ1∈σ(A₁)

a_λ1e_λ1



.

By Lemma 1.3,φis positive onV₁×V₂in the sense thatu_i ∈V_i,u_i ≥0impliesφ(u₁, u₂)≥0.

Hence (since theV_i’s contain the function1),

(1.10) kφk= sup{|φ(u₁, u₂)|:u_i ∈V_i,ku_ik∞≤1, i= 1,2}=φ(1,1).

Nowφlifts to a linear functional

φ˜:V₁⊗V₂ →C, which is positive onV₁⊗V₂, because

kφk˜ =kφk=φ(1,1) = ˜φ(1).

The Hahn–Banach theorem yields an extensionΦ :C⊗C =C([0,∞]²)→Cofφ˜of the same norm. Thus the positivity ofφ˜yields

kΦk=kφk˜ = ˜φ(1) = Φ(1),

i.e. Φis a positive functional onC([0,∞]²). Hence, the Riesz representation theorem provides us with a positive Radon measureρon[0,∞]²such that

(1.11) Φ(u) =

Z

[0,∞]²

u(t₁, t₂)dρ(t₁, t₂), u∈C([0,∞]²).

A simple rewriting yields that (1.9) equals X

(λ^∗₁,λ^∗₂)∈σ(A₁)×σ(A₂)



a_λ^∗

1a_λ^∗

2h(λ^∗₁, λ^∗₂) + X

(λ1,λ2)6=(λ^∗₁,λ^∗₂)

a_λ1a_λ2h(λ^∗₁, λ₂)¹²h(λ₁, λ^∗₂)¹²



.

Inserting the latter expression into (1.11) yields h(λ^∗₁, λ^∗₂) =φ(λ^∗₁, λ^∗₂)

= Φ(e_λ^∗

1 ⊗e_λ^∗

2)

= Z

[0,∞]²

(1 +t₁)λ^∗₁ 1 +t₁λ^∗₁

(1 +t₂)λ^∗₂

1 +t₂λ^∗₂ dρ(t1, t2).

Sinceλ^∗₁,λ^∗₂are arbitrary, the theorem is proved.

Remark 1.4. It is easy to modify the above proof to obtain a representation theorem for in- terpolation functions of more than two matrix variables (where the latter set of functions is interpreted in the obvious way).

2. KORÁNYI’STHEOREM

Consider the class of functions which are monotone according to the definition of Korányi [8]¹,A₁ ≤A⁰₁andA₂ ≤A⁰₂ imply

(2.1) h(A⁰₁, A⁰₂)−h(A⁰₁, A₂)−h(A₁, A⁰₂)−h(A₁, A₂)≥0.

The functions

h_t(λ) = (1 +t)λ 1 +tλ

are monotone of one variable(0≤t≤ ∞), whence withh_t1t2 =h_t1 ⊗h_t2 (cf. [8, p. 544]), ht1t2(A⁰₁, A⁰₂)−ht1t2(A⁰₁, A2)−ht1t2(A1, A⁰₂)−ht1t2(A1, A2)

= (h_t1(A⁰₁)−h_t1(A₁))⊗(h_t2(A⁰₂)−h_t2(A₂))≥0, i.e. ht1t2 is monotone. Since the class of monotone functions of two variables is closed under pointwise convergence, the latter inequality can be integrated, which yields that all functions of the form (1.7) are monotone. Hence we have proved the easy half of the following theorem of A. Korányi, cf. [8, Theorem 4], cf. also [9].

Theorem 2.1. Lethbe a positive function onR²+. Assume that(a)the first partial derivatives and the mixed second partial derivatives ofhexist and are continuous. Thenhis monotone iff his representable in the form (1.7) for some positive Radon measureρon[0,∞]².

Remark 2.2. According to Korányi the differentiability condition(a)was imposed “in order to avoid lengthy computations which are of no interest for the main course of our investigation”

([8, bottom of p. 541]).

Let us denote a function h defined on R²+ an interpolation function if h ∈ C_A1,A2 for any positive matrices A₁, A₂. Theorem 1.1 and Theorem 2.1 then yield the following corollary, which nicely generalizes the one-variable case.

Corollary 2.3. The set of interpolation functions coincides with the set of monotone functions satisfying(a).

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1A different definition of monotonicity of several matrix variables was recently given by Frank Hansen in [7].

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