Central European Journal of Mathematics

DOI: 10.2478/s11533-007-0032-2 Research article CEJM 5(4) 2007 619–638 Finite-tight sets Liviu C. Florescu∗ “Al. I. Cuz...

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DOI: 10.2478/s11533-007-0032-2 Research article CEJM 5(4) 2007 619–638

Finite-tight sets Liviu C. Florescu∗ “Al. I. Cuza” University, Faculty of Mathematics, Blvd. Carol I, 11, 700506 - Ia¸si, Romˆ ania

Received 19 April 2007; accepted 19 August 2007 Abstract: We introduce two notions of tightness for a set of measurable functions - the ﬁnite-tightness and the Jordan ﬁnite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a ﬁnite family of sets of small measure. In the Euclidean case, the Jordan ﬁnite-tight sets form a subclass of ﬁnite-tight sets for which the ﬁnite family of sets of small measure is composed by d-dimensional intervals. The main result aﬃrms that each tight set H ⊆ W 1,1 for which the set of the gradients ∇H is a Jordan ﬁnite-tight set is relatively compact in measure. This result oﬀers very good conditions to use ﬁber product lemma for obtaining a relaxed lower semicontinuity condition. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: finite-tight set, Jordan finite-tight set, Young measure, w2 - convergence. MSC (2000): Primary: 28A20, 49J45; Secondary: 28A33, 46E27

1

Introduction

In the last years the Young measures constituted the object of an intense research because of their uses to obtain relaxed solutions for the optimization problems but also for the diﬀerential inclusions. To obtain a Young measure as a generalized limit for a minimizing sequence one needs a condition of compactness. The tightness is such a condition; because it can be easily veriﬁed, it represents the main attraction for the space of Young measures. Let (X, .) be a separable Banach space, let (Ω, A, μ) be a ﬁnite positive measure ∗

E-mail: lﬂ[email protected]

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space and let M(X) be the set of all (A − BX ) - measurable mappings u : Ω → X, where BX denotes the Borel σ-algebra on X. A Young measure on X is a positive measure τ : A⊗BX → R+ such that, τ (A×B) = μ(A), for every A ∈ A. Thanks to a theorem of disintegration, one can identify each Young measure τ with an application τ. (τ ≡ τ. ) which, to each t ∈ Ω, associates a probability τt on X (τt ∈ PX ) with a measurability condition: t → τt (C) is (A − BR ) measurable, for any C ∈ BX . Y(X) will denote the space of all Young measures on X. For every bounded (or positive) integrand Ψ : Ω × X → R and for every τ ≡ τ. ∈ Y(X), Ω×X

Ψ(t, x)dτ (t, x) =

Ψ(t, x)dτt (x) dμ(t).

Ω

X

We shall denote the characteristic function of the set A by ½A . A sequence (un )n ⊆ M(X) is stable convergent to a Young measure τ ≡ τ. ∈ Y(X) S (un −→ τ ) if for every A ∈ A and every real bounded continuous mapping f : X → R, ½A (t)f (x)dτ (t, x) = f (x)dτt (x) dμ(t) = lim f (un (t))dμ(t) Ω×X

A

n

X

A

or, equivalently, if for every bounded below l.s.c. integrand Ψ : Ω × X → R, Ψ(t, x)dτ (t, x) ≤ lim inf Ψ(t, un (t))dμ(t). n

Ω×X

Ω

Particularly, if for any t ∈ Ω, τt = δu(t) (where u ∈ M(X) and δu(t) indicates the Dirac S

mass at u(t)), then un −→ τ ≡ δu(.) if and only if (un )n is convergent in measure to u (see deﬁnition of Hoﬀmann-Jørgensen in [9]); we say in this case that the Young measure τ is an elementary Young measure. The application u → δu(.) is an embedding of M(X) in Y(X). Let X, Z be two separable Banach spaces and let the Young measures τ ≡ τ. ∈ Y(X) and σ ≡ σ. ∈ Y(Z) be given; the ﬁber product of τ and σ is the Young measure τ ⊗ σ ≡ (τ ⊗ σ). ∈ Y(X × Z) deﬁned by (τ ⊗ σ)t = τt ⊗ σt , for every t ∈ Ω. In case where τ and σ are elementary Young measures (there exist u ∈ M(X), v ∈ M(Z) such that τt = δu(t) and σt = δv(t) , for every t ∈ Ω), then (τ ⊗ σ)t = δ(u(t),v(t)) , for every t ∈ Ω. In the relaxed control theory an important tool is the ﬁber product lemma (see theorem 2.3.1 from [2]): Theorem 1.1 (ﬁber product lemma). Let X and Z be two separable Banach spaces and let (un )n ⊆ M(X), (vn )n ⊆ M(Z), u ∈ M(X), τ ≡ τ. ∈ Y(Z). Assume that (i) (un )n is convergent in measure to u; S (ii) vn −→ τ . S Then (un , vn ) −→ δu(.) ⊗ τ. ∈ Y(X × Z). Particularly, this result is useful in case where Ω ⊆ Rd , (un )n is a minimizing sequence for an optimization problem and, for every n ∈ N, vn = ∇un is the gradient of un ; if

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(un )n is convergent in measure to u and (∇un )n is stable convergent to τ ≡ τ. ∈ Y(Rmd ) then, for every bounded below l.s.c. integrand Ψ : Ω × Rm × Rmd → R, Ψ(t, u(t), y)dτ (t, y) ≤ lim inf Ψ (t, un (t), ∇un (t)) dμ(t). n

Ω×Rmd

Ω

Definition 1.2. Let X be a separable Banach space and let KX be the family of all compacts in (X, || · X ); a set H ⊆ M(X) is a tight set if, for any ε > 0, there exists K ∈ KX such that, for every u ∈ H, μ u−1 (X \ K) < ε. A sequence (un )n ⊆ M(X) is a tight sequence if H = {un : n ∈ N} is a tight set. Particularly, if X is a Euclidean space, H ⊆ M(X) is a tight set if and only if, for any ε > 0 there is a k > 0 such that μ({t ∈ Ω : u(t)X > k}) < ε, for every u ∈ H. In this case every bounded set H ⊆ L1 (Ω, X) is a tight set; indeed, if M = supu∈H u1 < +∞ then, for every k > 0, μ(uX > k) ≤ Mk . The interest for tightness is motivated by the following theorem (see theorem 4.3.5 of [4]). Theorem 1.3 (Prohorov). A set H ⊆ M(X) is relatively stable compact (sequentially stable compact) in Y(X) if and only if it is a tight set. If X is a Euclidean space and H is a bounded subset of L1 (Ω, X) then H is a tight set and thus each net (sequence) has a subnet (subsequence) stable convergent to a Young measure. For all concepts and results about Young measures used in this paper one can consult [4] and [11]. We recall that a set H ⊆ L1 (Ω, X) is absolutely continuous if udμ = 0. lim sup μ(E)→0 u∈H

E

H is uniformly integrable if lim sup

t→+∞ u∈H

(u≥t)

udμ = 0.

H is uniformly integrable if and only if it is absolutely continuous and bounded in L1 (Ω, X). Biting lemma is a very general result of weak compactness on L1 (see [1], [4]). Theorem 1.4 (Biting Lemma). Let X be a separable Banach space and let (un )n be a bounded sequence in L1 (Ω, X). There exist a subsequence (unk )k∈N and a decreasing sequence of “bits” (Bp )p∈N ⊆ A with μ(Bp ) ↓ 0 such that the sequence (½Ω \ B · unk )k∈N is k uniformly integrable. If X is reflexive then (½Ω \ B · unk )k∈N has a subsequence weakly convergent to a k mapping u ∈ L1 (Ω, X).

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Using Prohorov’s theorem and biting lemma for the bounded sequences of L1 , Valadier and Saadoune obtained one of the most complete results of compacity in L1 (Ω, X) (see [10]); we give here the vectorial variant of this result, theorem 6.1.11 from [4]. Theorem 1.5. Let X be a separable Banach space and let (un )n be a bounded sequence in L1 (Ω, X) which is a tight sequence also; then there exist a subsequence (un )n of (un )n , a function u ∈ L1 (Ω, X), a decreasing sequence (Bp )p ⊆ A with μ(Bp ) → 0 and a Young measure τ ≡ τ. ∈ Y(X) satisfying: (i) (½Ω \ B un )n is uniformly integrable; n (ii) for every p ∈ N, the restriction un |Ω\Bp converge weakly to u|Ω\Bp ; (iii) (un )n is stably convergent to τ ; (iv) for a.e. t ∈ Ω, τt has a barycenter and bar(τt ) ≡ X xdτt (x) = u(t). These last two results, in which the condition of L1 -boundedness of sequences plays an important role, found many applications in the control problems (see [11]) as well as in the study of variational limits for second order evolution inclusions (for example see [3]) . In the present paper we obtain alternative results to biting lemma and to ValadierSaadoune’s theorem for some classes of unbounded sequences: ﬁnite-tight sequences and Jordan ﬁnite-tight sequences. First, we give an alternative to the biting lemma and then one to the Valadier and Saadoune’s theorem for ﬁnite-tight sequences. Finite-tightness locates the great growths of a set of measurable mappings on a ﬁnite family of sets of small measure. In the Euclidean case, the Jordan ﬁnite-tight sets form a subclass of the family of ﬁnite-tight sets for which the ﬁnite family of sets of small measure is composed of ddimensional intervals. The main result aﬃrms that each tight set H ⊆ W 1,1(Ω, Rm ) for which the set of the gradients ∇H is a Jordan ﬁnite-tight set is relatively compact in measure. This result oﬀers very good conditions to use ﬁber product lemma for obtaining a relaxed lower semicontinuity condition. The ﬁrst attempt for a study of ﬁnite-tightness was made in [8] where the results were exposed in the particular case where Ω = [0, 1], X = Rd and the ﬁnite-tight sets were what here we will call the Jordan ﬁnite-tight sets; an application to limit of solutions of one dimensional heat equation is given in [8].

2

Finite-tight sets

Let (Ω, A, μ) be a ﬁnite positive measure space, let X be a separable Banach space and let M(X) be the space of all (A − BX )-measurable mappings u : Ω → X; let us denote by KX the family of all compacts in X. For every u ∈ M(X) and every k > 0 we denote the set {t ∈ Ω : u(t)X > k} by (u > k).

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Definition 2.1. A subset H ⊆ M(X) is a finite-tight set if, for every ε > 0, there exist a compact K ∈ KX and a ﬁnite subfamily Af ⊆ A with μ(A) < ε, for every A ∈ Af , such that, for every u ∈ H, there exists Au ∈ Af with u−1 (X \ K) ⊆ Au . A sequence (un )n ⊆ M(X) is a finite-tight sequence if the set H = {un : n ∈ N} is a ﬁnite-tight set. Proposition 2.2. Let μ∗ : 2Ω → R+ be the outer measure engendered by μ (μ∗ (A) = inf{μ(C) : C ∈ A, A ⊆ C}, for every A ⊆ Ω). H ⊆ M(X) is a finite-tight set if and only if, for every ε > 0, there exist a compact K ∈ KX and a finite

cover {H1 , . . . , Hp } of H, such that, for every i ∈ {1, . . . , p}, u−1 (X \ K) < ε. μ∗ u∈Hi

In the case where H is countable we can use the measure μ instead of μ∗ . Proof. Let H ⊆ M(X) be a ﬁnite-tight set; for every ε > 0 there exist K ∈ KX and a ﬁnite subfamily Af = {A1 , . . . , Ap } ⊆ A with μ(Ai ) < ε, for every i = 1, . . . , p, such that, for any u ∈ H, there exists Au ∈ Af with u−1 (X \ K) ⊆ Au . For every i = 1, . . . , p, let Hi = {u ∈ H : u−1 (X \ K) ⊆ Ai }; then {H1 , . . . , Hp } is the required cover of H. Conversely, let K ∈ KX and let {H 1 , . . . , Hp } be a ﬁnite cover of H such that, for u−1 (X \ K) < ε. Then for every i there exists Ai ∈ A such every i = 1, . . . , p, μ∗ u∈Hi −1 that u∈Hi u (X \ K) ⊆ Ai and μ(Ai ) < ε; therefore Af = {A1 , . . . , Ap } is the required ﬁnite subfamily of A. Remark 2.3. (i) If X is a Euclidean space then H ⊆ M(X) is a ﬁnite-tight set if and only if for every ε > 0 there exist k > 0 and a ﬁnite subfamily Af ⊆ A with μ(A) < ε, for all A ∈ Af such that for every u ∈ H there exists Au ∈ Af with {t ∈ Ω : u(t) > k} = (u > k) ⊆ Au . A bounded sequence in (L∞ (Ω, X), · ∞ ) is obviously a ﬁnite-tight sequence. (ii) According to deﬁnition 2.1, for every u ∈ H, μ(u−1(X \ K)) < ε; thus any ﬁnitetight set is a tight set. The following example show that the converse is not valid. Example 2.4. Let n =

∞

an,p · 2p−1 be the binary form of the natural number n; thus

p=1

for every p ∈ N∗ , an,p ∈ {0, 1}, there exists ln ∈ N such that an,ln = 1 and, for every p > ln , an,p = 0. For all n, p ∈ N∗ we deﬁne kn,p = an,p + an,p+1 · 2 + · · · + an,2p−1 · 2p−1 ; then 0 ≤ kn,p ≤ 2p − 1, kn,ln = an,ln = 1 and, for every p > ln , kn,p = 0.

For any n ∈ N, let un : ]0, 1[→ R, un = ∞ p=1 p · ½ kn,p kn,p +1 . 2p

,

2p

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For every t ∈]0, 1[, un (t) =

ln

p · ½ kn,p 2p

p=1

,

kn,p +1 2p

(t)

+

every n ∈ N, un is well deﬁned.

∞

p · ½

p=ln +1

0,

1 2p

(t)

∈ R; thus, for

Proposition 2.5. (un )n ⊆ M(R) is a tight sequence but it is not a finite-tight sequence. Proof. For every n ∈ N,

0

1

|un (t)|dt =

∞ p=1

1

p·

1 = 2; thus (un )n is a bounded sequence 2p

in L (]0, 1[, R) and then it is tight (see deﬁnition 1.2). Let ε = 12 ; to show that (un )n is not a ﬁnite-tight sequence it is enough to notice that, for every k > 0, every q ∈ N, and all A1 , . . . , Aq ∈ A with μ(Ai ) < ε, ∀i = 1, . . . , q, there exists n ∈ N such that (|un | > k) Ai , ∀i = 1, . . . , q. Let k > 0, q ∈ N and A1 , . . . , Aq ∈ A be arbitrary such that, for any i = 1, . . . , q,

μ(Ai ) < 12 ; ﬁx p ∈ N, p > k.

kn,p kn,p +1 , 2p 2p

⊆ (|un | > k), for all n ∈ N. i1 i1 + 1 1 p A1 . As μ(A1 ) < 2 , there exists i1 ∈ {0, . . . , 2 − 1} such that p 2 2p Let n1 = 2p−1 · i1 ; then kn1 ,p = i1 and from i1 i1 + 1 kn1 ,p kn1 ,p + 1 = p, , (|un1 | > k) ⊇ 2p 2p 2 2p

we deduce that (|un1 | > k) A1 .

i2 i2 + 1 A2 . As μ(A2 ) < there exists i2 ∈ {0, . . . , 2 − 1} such that 2p 2p 2 2 Let n2 = n1 + 22p−1 · i2 ; then kn2 ,p = i1 , kn2 ,2p = i2 and from

1 , 2

2p

i2 i2 + 1 i1 i1 + 1 ∪ 2p , 2p (|un2 | > k) ⊇ p , 2 2p 2 2

we conclude that (|un2 | > k) A1 and (|un2 | > k) A2 . 1 Continuing this reasoning, as μ(A that there exists q ) < 2 , we show iq + 1 iq q−1 iq ∈ {0, . . . , 22 ·p − 1} such that 2q−1 ·p , 2q−1 ·p Aq . 2 2 2q−1 ·p−1 Let nq = n1 + · · · + nq−1 + 2 · iq ; then knq ,p = i1 , knq ,2p = i2 , . . . , knq ,2q−1 ·p = iq and we have iq + 1 i2 i2 + 1 iq i1 i1 + 1 ∪ 2p , 2p ∪ · · · ∪ 2q−1 ·p , 2q−1 ·p |unq | > k ⊇ p , 2 2p 2 2 2 2 Thus, for all i = 1, . . . , q, (|unq | > k) Ai . In the sequel we present some suﬃcient conditions for a set to be ﬁnite-tight. Proposition 2.6. Let X be a separable Banach space. (i) Any finite set H ⊆ M(X) is finite-tight.

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(ii) Let H = {un : n ∈ N} ⊆ M(X) be a tight set; if H satisfies one of the following two conditions ∞ μ(u−1 (S) ∃K0 ∈ KX such that n (X \ K0 )) < +∞, ∀K ∈ KX , ∃nK ∈ N such that

n=0 u−1 n+1 (X

\ K) ⊆ u−1 n (X \ K), ∀n ≥ nK ,

(M)

then H is a finite-tight set. Proof. (i) For every u ∈ M(X), μ ◦ u−1 : BX → R+ , (μ ◦ u−1 )(C) = μ(u−1(C)), for every C ∈ BX , is a σ-additive measure; as X is a Polish space, μ ◦ u−1 is a Radon measure and then for every ε > 0 there exists K ∈ KX such that μ(u−1(X \ K)) < ε. Thus every ﬁnite subset H ⊆ M(X) is ﬁnite-tight. (ii) Let H be a tight set which satisﬁes condition (S); for every ε > 0 there exists K ∈ KX such that μ(u−1 n (X \ K)) < ε for all n ∈ N; obviously, we can assume that −1 K ⊇ K0 . Let An = un (X \ K) for n ∈ N. ∞ According to condition (S) there exists p ∈ N such that μ(An ) < ε. n=p Thus the family Af = {A0 , A1 , . . . , Ap−1 , ∞ n=p An } satisﬁes the conditions of deﬁnition 2.1. If H satisﬁes condition (M) then the family Af = {A0 , A1 , . . . , AnK } satisﬁes the conditions of deﬁnition 2.1. In the following examples we show that both conditions of the previous proposition are not necessary for a set H to be ﬁnite-tight. Example 2.7. (1) Let Ω = [0, 1], A1 = 13 , 23 , A2 = 312 , 322 ∪ 372 , 382 , 20 25 26 A3 = 313 , 323 ∪ 373 , 383 ∪ 19 ∪ 33 , 33 . . . be the sets removed from [0, 1] to obtain , 33 33 the Cantor set; let us remind that An is a union of 2n−1 intervals, each of length 31n . As μ(An ) → 0, the set H = {un = 3n · ½An : n ∈ N∗ } ⊆ M(R) is tight. We notice that H is not bounded in L1 ([0, 1], R). Let K0 = [−1, 1] ∈ KR ; then ∞

μ(u−1 n (R

n=1

\ K0 )) =

∞ n=1

μ(|3 · ½An | > 1) = n

∞

μ(An ) = 1 < +∞

n=1

and hence, according to condition (S) of the previous proposition, H is a ﬁnite-tight set. Nevertheless, H does not satisfy condition (M) of (ii). Indeed, for K = [−1, 1] and for every n ∈ N∗ , u−1 n (R \ K) = An , although the sets An are pairwise disjoint. (2) For every n ∈ N∗ , let un : [0, 1] → R, un = n2 · ½ 1 . Then (un )n ⊆ M(R); the ∗

[0, n ] 1

set H = {un : n ∈ N } is tight but H is not bounded in L ([0, 1], R). / K, for every n ≥ nK . Then for For all K ∈ KR there exists nK ∈ N such that n2 ∈ every n ≥ nK we have ⎧ ⎪ ⎨ [0, 1 ] , 0 ∈ K n −1 un (R \ K) = ; ⎪ ⎩ [0, 1] , 0 ∈ /K

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thus u−1 n (R \ K) n≥nK is a decreasing sequence of sets, that is H satisﬁes the condition (M) and therefore H is ﬁnite-tight. Nevertheless, condition (S) is not satisﬁed; indeed, ∞ ∞ −1 1 = +∞. μ un (R \ {0}) = n n=1 n=1

Definition 2.8. Let X be a separable Banach space; a sequence (un )n∈N ⊆ M(X) is w 2 -convergent to a measurable function u : Ω → X if there exist a decreasing sequence (Bp )p∈N ⊆ A withμ(Bp ) ↓ 0 such that, for every p ∈ N, (un )n ⊆ L1 (Ω \ Bp , X), u ∈

L1 (Ω \ Bp , X) and

w2

½Ω \ B

p

· un

is weakly convergent to n∈N

½Ω \ B

p

· u; let us denote this

situation by un −→ u. If (un )n∈N is w 2-convergent to u then every subsequence of (un )n∈N is w 2-convergent to u. Remark 2.9. (i) This deﬁnition generalizes the deﬁnition of [1] where it was assumed that u ∈ L1 (Ω, X) and un ∈ L1 (Ω, X) for every n ∈ N. (ii) If X is a reﬂexive Banach space then one can reformulate Biting Lemma (see 1.4): every bounded sequence of L1 (Ω, X) has a subsequence w 2 -convergent to a mapping u ∈ L1 (Ω, X). The following result is an alternative of Biting Lemma for ﬁnite-tight sets. Theorem 2.10 (see Thm. 4 of [8]). Let X be a Banach space such that X and its dual space X ∗ have the Radon-Nikodym property (particularly let X be a reflexive space) and let (un )n∈N ⊆ M(X) be a finite-tight sequence; then there exist a subsequence (ukn )n∈N w2

and a measurable mapping u ∈ M(X) such that ukn −→ u. Proof. I. As (un )n is a ﬁnite-tight sequence, for every ε > 0 there exist K ∈ KX and a ﬁnite sub-family Af of A with μ(A) < ε for all A ∈ Af , such that for every n ∈ N there exists An ∈ Af with u−1 n (X \ K) ⊆ An . Then there exist a set A ∈ Af and an inﬁnite subset N ⊆ N such that u−1 n (X \ K) ⊆ A, for any n ∈ N. Therefore, for every ε > 0 there exist K ∈ KX , A ∈ A with μ(A) < ε and an inﬁnite subset N of N such that u−1 n (X \ K) ⊆ A for every n ∈ N. For ε = 1 choose K1 ∈ KX , B1 ∈ A with μ(B1 ) < 1 and let N1 be an inﬁnite subset of N such that u−1 n (X \ K1 ) ⊆ B1 , for all n ∈ N1 . 1 For ε = 2 choose K2 ∈ KX , B2 ∈ A with μ(B2 ) ≤ 12 and let N2 be an inﬁnite subset of N1 such that u−1 n (X \ K2 ) ⊆ B2 , for all n ∈ N2 ; obviously we can choose K2 ⊇ K1 and B2 ⊆ B1 . Generally, for p ∈ N∗ and ε = 1p , choose Kp ∈ KX with Kp ⊇ Kp−1 , Bp ∈ A with Bp ⊆ Bp−1 and μ(Bp ) < 1p and let Np be an inﬁnite subset of Np−1 such that u−1 n (X \ Kp ) ⊆ Bp , for every n ∈ Np .

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Let n1 ∈ N1 , n2 ∈ N2 , . . . , np ∈ Np , . . . be such that n1 < n2 < · · · < np < . . . . Then (uni )i∈N is a subsequence of (un )n∈N and for each p ∈ N and i ≥ p we have that ni ∈ Np ; therefore u−1 ni (X \ Kp ) ⊆ Bp . Thus, for every p ∈ N and every i ≥ p

Hence uni |Ω\Bp

i≥p

uni (Ω \ Bp ) ⊆ Kp ⊆ X. is uniformly bounded, so uni |Ω\Bp

i∈N

is bounded in L1 (Ω \ Bp , X)

and absolutely continuous. As for every A ∈ A, A ⊆ Ω \ Bp with μ(A) > 0 and for every i ≥ p, uni dμ ∈ μ(A) · co(uni (A)) ⊆ μ(A) · co(Kp ),

A

uni dμ : i ∈ N is relatively weakly compact. From Dunford’s theo rem (see [6], thm. 1, pp. 101), uni |Ω\Bp is relatively weakly compact in the space i∈N ˇ L1 (Ω \ Bp , X), for every p ∈ N. From Eberlein-Smulyan theorem, uni |Ω\Bp is rela-

we deduce that

A

i∈N

tively sequentially weakly compact in L1 (Ω \ Bp , X), for every p ∈ N. II. Let M1 be an inﬁnite subset of N and let v1 ∈ L1 (Ω\B1 , X) be such that (uni )i∈M1 is weakly convergent to v1 . Let M2 be an inﬁnite subset of M1 and let v2 ∈ L1 (Ω \ B2 , X) be such that (uni )i∈M2 is weakly convergent to v2 in L1 (Ω\B2 , X). As Ω\B1 ⊆ Ω\B2 , v2 = v1 almost everywhere on Ω \ B1 . Generally, let Mp be an inﬁnite subset of Mp−1 and let vp ∈ L1 (Ω\Bp , X) be such that (uni )i∈Mp is weakly convergent to vp in L1 (Ω \ Bp , X); then vp = vp−1 almost everywhere on Ω \ Bp−1 . is still a Choose i1 ∈ M1 , i2 ∈ M2 , . . . so that i1 < i2 < . . . . Then uniq q∈N is weakly convergent to subsequence of (un )n and for every p ∈ N the sequence uniq q∈N

vp in L1 (Ω \ Bp , X). Let u : Ω → X be deﬁned by u = vp on Ω\Bp and u = 0 on ∞ p=1 Bp . Then u ∈ M(X) w2

and uniq −→ u. Now we give a substitute of the theorem of Saadoune and Valadier (see [10]).

Theorem 2.11 (see Thm. 5 of [8]). Let X be a separable Banach space such that X and X ∗ have the Radon-Nikodym property (particularly, let X be a separable reflexive space) and let (un )n ⊆ M(X) be a finite-tight sequence; then there exist a subsequence (ukn )n∈N S of (un )n∈N and a Young measure τ ∈ Y(X) such that ukn −→ τ . If τ : Ω → PX is the disintegration of τ then, for almost every t ∈ Ω, τt has a barycenter xdτt (x) ∈ X.

u(t) = bar(τt ) = X

w2

The mapping u : Ω → X is measurable (u ∈ M(X)), ukn −→ u and u(t) ∈ co(Lsn (ukn (t))), for almost every t ∈ Ω (for every t ∈ Ω, Lsn (ukn (t)) = ∞ p=1 {uki (t) : i ≥ p}).

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Proof. According to theorem 2.10 there exist a subsequence (un )n∈N of (un )n∈N and a w2

mapping v ∈ M(X) such that un −→ v. Let (Bp )p∈N ⊆ A be a decreasing sequence with μ(Bp ) ↓0 such that, for every p ∈ N, is weakly convergent to (un )n ⊆ L1 (Ω \ Bp , X), v ∈ L1 (Ω \ Bp , X) and ½Ω \ B · un

½Ω \ B

p

p

n∈N

· v. From Prohorov’s theorem (theorem 1.3) there exist a subsequence (ukn )n∈N of w2

S

and a Young measure τ such that ukn −→ τ ; obviously, ukn −→ v. For every p ∈ N, the mapping Ψp : Ω × X → R, deﬁned by Ψp (t, x) = = ½Ω \ B (t) · x for every (t, x) ∈ Ω × X, is a positive l.s.c. integrand. Thus from p theorem 2.1.3 (2) of [4], Ψ(t, x)dτ (t, x) ≤ lim inf ukn (t)dμ(t) < +∞ (un )n

n

Ω×X

((ukn )n is bounded in L1 (Ω \ Bp , X)). Therefore Ω\Bp

Ω\Bp

xdτt (x) dμ(t) < +∞, X

so for almost every t ∈ Ω \ Bp there exists a measurable mapping up : Ω \ Bp → X, up (t) = bar(τt ) = X xdτt (x). As Ω \ Bp ⊆ Ω \ Bp+1 , up+1|Ω\Bp = up almost everywhere. Thus one can deﬁne u : Ω → X, letting ⎧ ⎪ ⎨ bar(τt ), if t ∈ ∞ (Ω \ Bp ) and there exists bar(τt ), p=1 u(t) = ⎪ ⎩ 0X , otherwise. Then u ∈ M(X) and u ∈ L1 (Ω \ Bp , X) for every p ∈ N. Let us show that u = v. As X ∗ has the Radon-Nikodym property, the dual of L1 (Ω, X) is L∞ (Ω, X ∗ ) (see theorem 1, pp. 98 of [6]). Let g be an arbitrary element of L∞ (Ω, X ∗ ) and p ∈ N. As ½Ω \ B · ukn −−1−w−−→ ½Ω \ B · v, p

L (Ω,X)

p

Ω\Bp

g(t), ukn (t) dμ(t) −−−→ n→∞

Ω\Bp

g(t), v(t) dμ(t).

(1)

Let Φp : Ω × X → R be deﬁned by Φ(t, x) = ½Ω \ B · g(t), x for t ∈ Ω and x ∈ X; then p Φp is separately measurable in t and continuous in x and so, by Lemma III.14 of [5], it is jointly measurable. Thus Φp is a Carath´eodory integrand. Furthermore, for every t ∈ Ω and every n ∈ N, Φ t, ½ (t) · u (t) p ≤ g∞ · ½Ω \ Bp (t) · ukn (t)X kn Ω \ Bp and since ½Ω \ B · ukn is uniformly integrable, p n∈N Φp ·, ½Ω \ B (·) · ukn (·) is uniformly integrable. p

n∈N

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According to theorem 6 of [11], Φp t, ½Ω \ B (t) · ukn (t) dμ(t) →

p

Ω

Ω\Bp

g(t), ukn (t) dμ(t) →

From (1) and (2), Ω\Bp

= Ω\Bp

Ω×X

Ω\Bp

629

Φp (t, x)dτ (t, x) or

g(t), x dτt (x) dμ(t).

(2)

X

g(t), v(t) dμ(t) =

g(t), xdτt (x) dμ(t) = X

Ω\Bp

g(t), u(t) dμ(t).

As g is an arbitrary element of the dual, ½Ω \ B (t) · v(t) = ½Ω \ B (t) · u(t) for almost p p every t ∈ Ω and for every p ∈ N, therefore v = u almost everywhere and consequently w2 ukn −→ u. S As ukn −→ τ , suppτt ⊆ Ls(ukn (t)) = ∞ n=1 {ukp (t) : p ≥ n} (see theorem 4.3.12 of [4]). Thus u(t) = X xdτt (x) = suppτt xdτt (x) ∈ co(suppτt ) ⊆ coLs(ukn (t)) for almost every t ∈ Ω.

3

Jordan finite-tight sets

First we explain the terms used in this section. A d-dimensional interval I ⊆ Rd is a product of bounded closed intervals of R: I = d i=1 [ai , bi ]. An elementary setE is the union of a ﬁnite family of nonoverlapping ddimensional intervals, i.e. E = pk=1 Ik and μ(Ik ∩ Il ) = 0 for every k, l ∈ {1, . . . , p} with k = l. Let us denote by E the family of all elementary sets; E generates the σ-algebra of Borel sets on Rd . Let Ω ⊆ Rd be a Lebesgue measurable bounded set, let A be the family of all Lebesgue measurable subsets of Ω and let μ be the Lebesgue measure in Ω. Definition 3.1. Let X be a separable Banach space; a set H ⊆ M(X) is a Jordan finite-tight set if for every ε > 0 there exist K ∈ KX and a ﬁnite subfamily Ef ⊆ E satisfying μ(E) < ε for every E ∈ Ef and such that for any u ∈ H there exists Eu ∈ Ef with u−1 (X \ K) ⊆ Eu . A sequence (un )n ⊆ M(X) is a Jordan finite-tight sequence if the set H = {un : n ∈ N} is a Jordan ﬁnite-tight set. Remark 3.2. (i) As E ⊆ A, every Jordan ﬁnite-tight set is a ﬁnite-tight set and therefore a tight set. (ii) If X is a Euclidean space then H ⊆ M(X) is a Jordan ﬁnite-tight set if and only if for every ε > 0 there exist k > 0 and a ﬁnite subfamily Ef ⊆ E with μ(E) < ε for every E ∈ Ef and such that for any u ∈ H there exists Eu ∈ Ef with {t ∈ Ω : u(t) > k} ⊆ Eu .

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The following proposition gives a justiﬁcation for the naming of Jordan finite-tight set. For every A ⊆ Ω let μ∗J (A) = inf{μ(E) : E ∈ E, A ⊆ E} be the Jordan outer measure of A; obviously, μ∗J (A) = 0 if and only if A is a Jordan-negligible set. Proposition 3.3. H ⊆ M(X) is a Jordan finite-tight set if and only if for every ε > 0 there exist K ∈ KX and a finite cover {H1 , . . . , Hp } of H such that μ∗J

u−1 (X \ K)

<ε

u∈Hi

for all i = 1, . . . , p. The proof is rather the same as that of proposition 2.2. Theorem 3.4. Let X be a Euclidean space; for every H ⊆ M(X) let ¯ AH (∞) = t ∈ Ω : lim sup sup u(s) = +∞ . s→t

u∈H

A set H ⊆ M(X) is a Jordan finite-tight set if and only if for every ε > 0 there exists a finite cover {H1 , . . . , Hp } of H such that, for any i = 1, . . . , p, μ∗J (AHi (∞)) < ε. Proof. (=⇒): At ﬁrst we remark that lim sups→t supu∈H u(s) = +∞ if and only if there exist a sequence (un )n∈N ⊆ H and a sequence (sn )n∈N ⊆ Ω with sn → t such that un (sn ) → +∞. Let us suppose that H ⊆ M(X) is a Jordan ﬁnite tight set; according to (ii) of remark 3.2, for every ε > 0 there exist k > 0 and a ﬁnite subfamily Ef ⊆ E with μ(E) < ε for every E ∈ Ef , such that for all u ∈ H there exists Eu ∈ Ef with (u > k) ⊆ Eu . For every E ∈ Ef let HE = {u ∈ H : (u > k) ⊆ E}; then {HE : E ∈ Ef } is a ﬁnite cover of H. Let us show that AHE (∞) ⊆ E for all E ∈ Ef . Indeed, for every t ∈ AHE (∞), lim sups→t supu∈HE u(s) = +∞ and thus there exist a sequence (sn )n∈N ⊆ Ω with sn → t and a sequence (un )n∈N ⊆ HE such that un (sn ) → +∞; we can assume that, for any n ∈ N, un (sn ) > k and therefore (sn )n∈N ⊆ (un > k) ⊆ E. As E is closed, t ∈ E. Hence μ∗J (AHE (∞)) < ε for every E ∈ Ef . (⇐=): For every ε > 0, let {H1 , . . . , Hp } be a ﬁnite cover of H such that, for any i = 1, . . . , p, μ∗J (AHi (∞)) < ε and let Ei ∈ E be such that AHi (∞) ⊆ Ei and μ(Ei ) < ε. ˚i of Ei . Then, Obviously, we can assume that every AHi (∞) is contained in the interior E ˚i and for for any i = 1, . . . , p, there exists ki > 0 such that u(t) ≤ ki for every t ∈ Ω \ E ˚i every u ∈ Hi (if we suppose on the contrary that there exist a sequence (tn )n∈N ⊆ Ω \ E

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and a sequence (un )n∈N ⊆ Hi with un (tn ) > n, for any n ∈ N, then (tn )n∈N has a ¯ \E ˚i , therefore t ∈ AH (∞); but AH (∞) ⊆ E ˚i ). subsequence (tkn )n∈N convergent to t ∈ Ω i i Let k = max{k1 , . . . , kp }. For every u ∈ H there exists i ∈ {1, . . . , p} such that u ∈ Hi ; thus (u > k) ⊆ (u > ki) ⊆ Ei . Remark 3.5. (i) Let X be a Euclidean space and let u ∈ M(X); then, from the ¯ : previous theorem, {u} is a Jordan ﬁnite-tight set if and only if Au (∞) = {t ∈ Ω lim sups→t u(s) = +∞} is a Jordan-negligible set. We notice that Au (∞) is the set of points which have no neighborhood on which u is bounded. (ii) Let Q ∩ ]0, 1[= {q0 , q1 , . . . , qn , . . . } be all the rational points of ]0, 1[ and let u :

]0, 1[→ R, u = ∞ n=0 n · ½{qn } . Then, according to (i) of proposition 2.6, H = {u} is a ﬁnite-tight set. Observe that Au (∞) = [0, 1]. Indeed, for every t ∈ [0, 1] and every n ∈ N∗ , there exists kn ≥ n such that |qkn − t| < n1 ; so, qkn → t and u(qkn ) = kn → +∞. Thus, from the previous remark, H is not a Jordan ﬁnite-tight set. We notice that u = 0 almost everywhere and that H1 = {0} is a Jordan ﬁnite-tight set. Thus being Jordan ﬁnite-tight is a property of sets of measurable functions and not a property of their equivalence classes with respect to the relation of being equal almost everywhere. (iii) Let X be a Euclidean space; for every bounded sequence (un )n in ∞ (L (Ω, X), · ∞ ) there exists a Jordan ﬁnite-tight sequence (vn )n such that un = vn almost everywhere for any n ∈⎧N. Indeed, let k > 0 be such that un ∞ ≤ k for every ⎪ ⎨ un (t), un X ≤ k n ∈ N. If we deﬁne vn (t) = , then (vn )n is a uniformly bounded ⎪ ⎩ 0X , un X > k sequence and thus (vn )n is a Jordan ﬁnite-tight sequence and vn = un almost everywhere for every n ∈ N. We present now a suﬃcient condition for being a Jordan ﬁnite-tight set. Proposition 3.6. Let Ω ⊆ Rd be a bounded measurable set; for every A ⊆ Ω let δ(A) = sup{t − s : t, s ∈ A} be the diameter of A. Every H ⊆ M(X) which satisfies the condition ∀ε > 0, ∃K ∈ KX such that δ u−1 (X \ K) < ε, ∀u ∈ H

(δ)

is a Jordan finite-tight set. Let a, b ∈ Rd , a = (a1 , . . . , ad ), b = (b1 , . . . , bd ), a < b be such that d ! Ω ⊆ [a, b] = [ai , bi ]; μ([a, b]) = (b1 − a1 ) . . . (bd − ad ). Proof.

i=1

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For every ε > 0, let n ∈ N be such that 3d · μ([a, b]) < ε. nd

(1)

Let Γ = {0, 1, . . . , n − 1}d . For every i ∈ {1, . . . , d} and every j ∈ {0, 1, . . . , n} let us d ! ji ji +1 j bi −ai ai , ai . denote ai = ai + j · n and, if γ = (j1 , . . . , jd ) ∈ Γ, let Iγ = i=1 Then {Iγ : γ ∈ Γ} is a partition of [a, b]; thus Ω ⊆ γ∈Γ Iγ . Furthermore μ(Iγ ∩Iγ ) = 1 d 0 if γ = γ and μ(Iγ ) = b1 −a · · · · · bd −a = μ([a,b]) . n n nd According to condition (δ) there exists K ∈ KX such that −1 bi − ai : i = 1, . . . , d . (2) δ u (X \ K) < ε1 = min n " d # ! j −1 j +2 Let Ef = ai i , ai i : (j1 , . . . , jd ) ∈ Γ ; then Ef is a ﬁnite subfamily of E and, aci=1

d d ! ! ji −1 ji +2 ji −1 ji +2 3d · μ([a, b]) < ε, for every interval ai , ai = ai , ai ∈ cording to (1), μ d n i=1 i=1 Ef . For any function u ∈ H for which u−1 (X \ K) = ∅ there exists γ = (j1 , . . . , jd ) ∈ Γ such that u−1 (X \ K) ∩ Iγ = ∅. d ! ji −1 ji+2 −1 Thus, according to (2), u (X \ K) ⊆ ai , ai ∈ Ef . i=1

Indeed, if t0 = (t01 , . . . , t0d ) ∈ u−1(X \ K) ∩ Iγ , then, for every t = (t1 , . . . , td ) ∈ u−1 (X \ K) and every i = 1, . . . , d, ti < t0i + ε1 ≤ aji i +1 + ε1 ≤ aji i +1 + ti > t0i − ε1 ≥ aji i − ε1 ≥ aji i −

bi − ai = aji i +2 and n bi − ai = aji i −1 . n

Example 3.7. (1) Let Q∩]0, 1[= {q0 , q1 , . . . , qn , . . . } and, for any n ∈ N∗ , let un = . Obviously H = {un : n ∈ N∗ } is a tight set but it is not bounded in n2 · ½ 1 ]qn , qn +

n

[

L1 (]0, 1[, R). For every k > 0 and every n ∈ N∗ , ⎧ ⎪ ⎨ ∅ , n2 ≤ k, {t ∈]0, 1[: |un (t)| > k} = ⎪ ⎩ qn , qn + 1 , n2 > k. n 1 Then δ (|un | > k) ≤ √ for every n ∈ N∗ , thus H = {un : n ∈ N∗ } satisﬁes condition (δ) k of the previous proposition and hence H is a Jordan ﬁnite-tight set; consequently H is a ﬁnite-tight set.

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√ 1 Nevertheless ∞ n=1 μ(|un | > k) = n≥ k n = +∞, thus H satisﬁes neither condition (S) nor condition (M) of proposition 2.6. (2) For every n ∈ N∗ , let un : [0, 1] → R, un = n·½ 1 . For a ﬁxed ε > 0 there 1 [0, n ] ∪ [1 − n , 1] 1 1 2 ∪ 1 − , 1 ∈ E. exists n0 ∈ N such that n0 < ε; let K = [0, n0 ] ∈ KR and E = 0, n0 n0 Then μ(E) < ε. For any n ∈ N∗ , ⎧ ⎪ ⎨ ∅, n ≤ n0 , −1 −1 ⊆ E. un (R \ K) = un (n0 , +∞) = ⎪ ⎩ [0, 1 ] ∪ [1 − 1 , 1], n > n0 n n Thus H = {un : n ∈ N∗ } is a Jordan ﬁnite-tight set. Nevertheless, for every K ∈ KR and n1 ∈ N with K ⊆ [−n1 , n1 ], δ u−1 n (R \ K) = 1, for every n > n1 . The main result of this section asserts that a tight set H ⊆ W 1,1 (Ω, Rm ), for which there exists a Jordan ﬁnite-tight set of gradients ∇H, is necessarily relatively compact in measure. We recall and clarify some notions and results. Definition 3.8. Let Ω ⊆ Rd be a bounded open set; v = (vji ) 1≤i≤m ∈ L1 (Ω, Rmd ) is the 1≤j≤d

gradient of u = (ui )1≤i≤m ∈ L1 (Ω, Rm ) if for every i ∈ {1, . . . , m} and j ∈ {1, . . . , d} the equality ∂φ ui (t) · (t)dμ(t) = − vji (t) · φ(t)dμ(t) ∂tj Ω Ω holds true, for every mapping φ ∈ Cc∞ (Ω) (φ is a smooth function with compact support in Ω). If v and v are two gradients of u then v = v almost everywhere. We denote the gradient of u by ∇u and, for every i ∈ {1, . . . , m} and j ∈ {1, . . . , d}, we write ∇j ui = vji . We remark that, for every i ∈ {1, . . . , m}, ∇ui = (vji )1≤j≤d so that ∇u = (∇ui)1≤i≤m = (∇j ui) 1≤i≤m . 1≤j≤d

For every p ≥ 1, the Sobolev space W 1,p (Ω, Rm ) consists of all mappings u ∈ Lp (Ω, Rm ) whose gradient ∇u belongs to Lp (Ω, Rmd ). For every H ⊆ W 1,p (Ω, Rm ) we write ∇H = {∇u : u ∈ H} and we say that ∇H is a gradient of H. Definition 3.9. For every mapping u : Ω → Rm , let us denote u(t) − u(s) : t, s ∈ Ω, t = s ; L(u, Ω) = sup t − s if L(u, Ω) < +∞, u is a Lipschitz function on Ω.

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We recall that if Ω ⊆ Rd is a bounded open convex set and if u ∈ L1 (Ω , Rm ) is a continuous mapping with ∇u = (∇j ui) 1≤i≤m ∈ L∞ (Ω , Rmd ) then u is a Lipschitz 1≤j≤d

function on Ω and L(u, Ω ) ≤

m d

12 ∇j ui 2L∞ (Ω ,R)

< +∞.

i=1 j=1

Proposition 3.10 (see Thm. 6 of [8]). Let Ω ⊆ Rd be a bounded open convex set and let H ⊆ W 1,1 (Ω, Rm ) ∩ C(Ω, Rm ) be a tight set. If for every u ∈ H there exists a gradient ∇u such that ∇H = {∇u : u ∈ H} ⊆ L1 (Ω, Rmd ) is a Jordan finite-tight set, then H is a Jordan finite-tight set too. Proof. Assume that for every u ∈ H there exists a gradient ∇u such that ∇H = {∇u : u ∈ H} is a Jordan ﬁnite-tight set; we remark that, for any other gradient vu of u, vu = ∇u almost everywhere but, according to (ii) of remark 3.5, we cannot assert that the set {vu : u ∈ H} is Jordan ﬁnite-tight. For every ε > 0 let k > 0 and Ef = {E1 , . . . , Ep } ⊆ E be a ﬁnite family of elementary sets, with μ(Ei ) < ε for every i ∈ {1, . . . , p}, such that for every u ∈ H there exists i ∈ {1, . . . , p} with {t ∈ Ω : ∇u(t) > k} ⊆ Ei . For any i ∈ {1, . . . , p} we can ﬁnd a ﬁnite family of open convex sets {Ωij : j = i 1, . . . , pi} such that Ω \ Ei = pj=1 Ωij ; let δ = min{μ(Ωij ) : 1 ≤ i ≤ p, 1 ≤ j ≤ pi } > 0. As H is a tight set, there exists k1 > 0 such that for every u ∈ H μ(u > k1 ) < δ.

(1)

Fix an arbitrary u ∈ H; then u ∈ L1 (Ω, Rm ) and u is continuous on Ω. Let i ∈ {1, . . . , p} such that (∇u > k) ⊆ Ei ; then, for every j ∈ {1, . . . , pi } and every t ∈ Ωij , ∇u(t) ≤ k and hence ∇u ∈ L∞ (Ωij , Rmd ). Thus u is a Lipschitz function on Ωij and

12 m d √ ∇b ua 2L∞ (Ωi ,R) ≤ md · k. (2) L(u, Ωij ) ≤ a=1 b=1

j

As μ(Ωij ) ≥ δ, according to (1), Ωij (u > k1 ); thus there exists t0 ∈ Ωij with u(t0 ) ≤ k1 .

(3)

According to (2) and (3), for every t ∈ Ωij ,

√ u(t) ≤ u(t) − u(t0 ) + u(t0 ) ≤ md · k · t − t0 + k1 ≤ √ ≤ md · k · diam(Ω) + k1 ≡ k2 .

As j is arbitrary in {1, . . . , pi }, u(t) ≤ k2 , for every t ∈

pi j=1

Ωij .

(4)

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According to (4), (u > k2 ) ⊆ Ei and hence H is a Jordan ﬁnite-tight set.

635

Remark 3.11. (i) According to the proof of the previous proposition, if H ⊆ W 1,1 (Ω, Rm ) is a tight set and if ∇H is a Jordan ﬁnite-tight set then for every ε > 0 there exist k > 0 and a ﬁnite family of elementary sets Ef = {E1 , . . . , Ep } with μ(Ei ) < ε for any i ∈ {1, . . . , p}, such that for every u ∈ H there exists i ∈ {1, . . . , p} with {t ∈ Ω : u(t) > k} ⊆ Ei and {t ∈ Ω : ∇u(t) > k} ⊆ Ei . (ii) We cannot remove the condition of continuity for the mappings of H in the

previous proposition. Indeed, if H = {u} ⊆ W 1,1 (]0, 1[, R), where u = ∞ n=0 n · ½{qn } is the function described in remark 3.5(ii), then H is a tight set and ∇H = {0} is a Jordan ﬁnite-tight set, but H is not a Jordan ﬁnite-tight set. (iii) The tightness condition for H is necessary in the previous proposition. Indeed, if un = n is the constant function on ]0, 1[ for every n ∈ N, then H = {un : n ∈ N} ⊆ W 1,1 (]0, 1[, R) ∩ C(]0, 1[, R) and ∇H = {0} is a Jordan ﬁnite tight set, but H is not even a tight set. Theorem 3.12 (see Thm. 7 of [8]). Let Ω ⊆ Rd be a bounded open convex set and let H ⊆ W 1,1 (Ω, Rm ) ∩ C(Ω, Rm ) be a tight set such that there exists a Jordan finite-tight gradient ∇H. Then H is relatively compact in the topology of convergence in measure on M(Rm ). Proof. According to the previous proposition and to remark 3.11(i), for every ε > 0 there exist k > 0 and a ﬁnite family of d-dimensional intervals I = {I1 , . . . , Ip } such that for any u ∈ H we can ﬁnd a subfamily Iu ⊆ I with μ( Iu ) < ε and (u > k) ⊆ Iu , (1)

(∇u > k) ⊆

Iu ,

(2)

Iu is the union of the intervals of the sub-family Iu ⊆ I. d ! i i aj , bj and let q ∈ N be such that For every i ∈ {1, . . . , p}, let Ii =

where

j=1

d·k·

√ m < q · ε.

(3)

Let us show that H satisﬁes the conditions of Fr´echet theorem of compactness in measure (see theorem IV.11.1 in [7]). d ! aj bj , . Let a1 , . . . , ad , b1 , . . . , bd ∈ Z be such that Ω ⊆ q q j=1 r

For each j ∈ {1, . . . , d} we enumerate by t1j , . . . , tj j the elements of the set bj − 1 bj 1 aj aj + 1 d 1 d , ,..., , , aj , . . . , aj , bj , . . . , bj in a such way that q q q q rj 1 2 tj < tj < · · · < tj .

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Let N = {1, . . . , r1 −1}×· · ·×{1, . . . , rd −1} ⊆ Nd and, for every n = (n1 , . . . , nd ) ∈ N, let En = tn1 1 , tn1 1 +1 × · · · × tnd d , tnd d +1 ; then (4) {Ω ∩ En : n ∈ N} is a partition of Ω.

Fr(En ) , where Fr(En ) is the boundary of For every u ∈ H let Fu = ( Iu ) ∪ n∈N

En . Then μ(Fu ) = μ

Iu < ε for every u ∈ H

(5)

and, according to (1), sup u(t) ≤ k

for every u ∈ H.

(6)

t∈Ω\Fu

Notice that for every n ∈ N and for every u ∈ H, the set (Ω ∩ En ) \ Fu is either empty or equal to the convex open set Ω ∩ E˚n . As, according to (2), ∇u ∈ L∞ ((Ω ∩ En ) \ Fu , Rmd ), u is a Lipschitz function on (Ω ∩ En ) \ Fu and √ (7) L(u, (Ω ∩ En ) \ Fu ) ≤ md · k for every n ∈ N and every u ∈ H. Thus for every n ∈ N, for every u ∈ H and every t, s ∈ (Ω ∩ En ) \ Fu we have √ √ √ √ √ d m · dk = u(t) − u(s) ≤ md · k · t − s ≤ md · k · diam(En ) ≤ md · k · q q and, according to (3), sup t,s∈(Ω∩En )\Fu

u(t) − u(s) < ε.

(8)

Now the result follows from theorem IV.11.1 of [7] in virtue of (4), (5), (6) and (8). In some situations we can avoid the continuity assumption in the previous theorem. Corollary 3.13. Let Ω ⊆ Rd be a bounded open convex set, let p ≥ 1 and let H be a tight set in W 1,p (Ω, Rm ) such that ∇H is a Jordan finite-tight set. If either p > d or d = 1, then H is relatively compact in measure. Proof. In both cases, for every u ∈ H there exists u¯ ∈ W 1,p (Ω, Rm ) ∩ C(Ω, Rm ) such ¯ = {¯ that u = u¯ almost everywhere. Then H u : u ∈ H} ⊆ W 1,1 (Ω, Rm ) ∩ C(Ω, Rm ) is a ¯ = ∇H. Therefore H, ¯ tight set for which there exists a Jordan ﬁnite-tight gradient ∇H and consequently also H, is relatively compact in measure. Remark 3.14. Let Ω ⊆ Rd be a bounded open convex set, let H be a tight set in W 1,1 (Ω, Rm ) ∩ C(Ω, Rm ) such that ∇H is a bounded set in (L∞ (Ω, Rm ), · ∞ ). According to remark 3.5(iii), for every sequence (un )n ⊆ H there exists (vn )n ⊆ ∞ L (Ω, Rm ) such that ∇un = vn almost everywhere for n ∈ N and {vn : n ∈ N} is a Jordan ﬁnite-tight set. Then, according to the previous theorem, H is relatively compact

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with respect to the topology of convergence in measure. But in this case we can obtain a stronger result. As a consequence of Arzel`a-Ascoli theorem, H is relatively compact in the topology of uniform convergence on C(Ω, Rm ). Indeed, H is uniformly Lipschitz and hence it is uniformly bounded and uniformly equicontinuous. In some relaxed conditions we can obtain an alternative to Rellich-Kondrachov theorem. Corollary 3.15. Let Ω ⊆ Rd be a bounded open convex set and let (un )n∈N ⊆ W 1,1(Ω, Rm )∩ C(Ω, Rm ) be a uniformly integrable sequence for which (∇un )n∈N is a Jordan finite-tight sequence. Then (un )n∈N , up to a subsequence, converges in L1 (Ω, Rm ). Proof. Because (un )n∈N is uniformly integrable, it is bounded in L1 (Ω, Rm ) and so (un )n∈N is a tight sequence for which (∇un )n∈N is a Jordan ﬁnite-tight sequence. Thus (un )n∈N has a subsequence (ukn )n∈N convergent in measure to a map u. As (un )n∈N is L1

uniformly integrable, u ∈ L1 (Ω, Rm ) and ukn −→ u. The Jordan ﬁnite-tightness condition oﬀers a very good frame for applying the ﬁber product lemma; we illustrate this in the following corollary. Corollary 3.16. Let Ω ⊆ Rd be a bounded open convex set and let H ⊆ W 1,1 (Ω, Rm ) ∩ C(Ω, Rm ) be a tight set such that ∇H is a Jordan finite-tight set. Then for every (un )n ⊆ M(Rm ) there exist a subsequence (ukn )n of (un )n , a mapping u ∈ M(Rm ) and a Young measure τ ≡ τ. ∈ Y(Rmd ) such that: i) (ukn )n is convergent in measure to u, S ii) ∇ukn −→ τ , w2

iii) ∇ukn −→ bar τ. iv) For every bounded below l.s.c. integrand Ψ : Ω × Rm × Rmd → R, Ψ(t, u(t), y)dτ (t, y) ≤ lim inf Ψ (t, ukn (t), ∇ukn (t)) dμ(t). Ω×Rmd

n

Ω

Proof. By theorem 3.12, H is relatively compact in measure in M(Rm ) and, by 2.11, ∇H is sequentially stable compact in Y(Rmd ). Thus for every (un )n ⊆ M(Rm ) there exist a subsequence (ukn )n of (un )n , a mapping u ∈ M(Rm ) and a Young measure τ ≡ τ. ∈ Y(Rmd ) such that i), ii) and iii) are accomplished. From the ﬁber product lemma (theorem 1.1), S (ukn , ∇ukn ) −→ δu(·) ⊗ τ. which proves iv).

4

Acknowledgements

1. I warmly thank the anonymous referees for their careful reading which improved the form of this paper.

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2. This work has been supported by ANCS and CNCSIS through grants 2-CEx06-1110/25.07.2006, 2-CEx06-11-56/25.07.2006, CEx05-D11-23/05.10.2005, GR 214/20.09.2006.

References [1] J.K. Brooks and R.V. Chacon: “Continuity and compactness of measures”, Adv. in Math., Vol. 37, (1980), pp. 16–26. [2] Ch. Castaing and P. Raynaud de Fitte: “On the ﬁber product of Young measures with application to a control problem with measures”, Adv. Math. Econ., Vol. 6, (2004), pp. 1–38. [3] Ch. Castaing, P. Raynaud de Fitte and A. Salvadori: “Some variational convergence results for a class of evolution inclusions of second order using Young measures”, Adv. Math. Econ., Vol. 7, (2005), pp. 1–32. [4] Ch. Castaing, P. Raynaud de Fitte and M. Valadier: Young measures on topological spaces. With applications in control theory and probability theory, Kluwer Academic Publishers, Dordrecht, 2004. [5] Ch. Castaing and M. Valadier: Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. [6] J. Diestel and J.J. Uhl: Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977. [7] N. Dunford and J.T. Schwartz: Linear Operators. Part I, Reprint of the 1958 original, Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988. [8] L.C. Florescu and C. Godet–Thobie: “A Version of Biting Lemma for Unbounded Sequences in L1E with Applications”, AIP Conference Proceedings, no. 835, (2006), pp. 58–73. [9] J. Hoﬀmann-Jørgensen: “Convergence in law of random elements and random sets”, High dimensional probability (Oberwolfach, 1996), Progress in Probability, no. 43, Birkh¨auser, Basel, 1998, pp. 151–189. [10] M. Saadoune and M. Valadier: “Extraction of a good subsequence from a bounded sequence of integrable functions”, J. Convex Anal., Vol. 2, (1995), pp. 345–357. [11] M. Valadier: “A course on Young measures”, Rend. Istit. Mat. Univ. Trieste, Vol. 26, (1994), suppl., pp. 349–394.

DOI: 10.2478/s11533-007-0025-1 Research article CEJM 5(4) 2007 639–653

Metrics in the sphere of a C∗-module Esteban Andruchow∗ and Alejandro Varela† Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez 1150, (1613) Los Polvorines, Argentina

Received 25 January 2007; accepted 16 July 2007 Abstract: Given a unital C ∗ -algebra A and a right C ∗ -module X over A, we consider the problem of finding short smooth curves in the sphere SX = {x ∈ X : x, x = 1}. Curves in SX are measured considering the Finsler metric which consists of the norm of X at each tangent space of SX . The initial value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any element x0 ∈ SX and any tangent vector v at x0 , there exists a curve γ(t) = etZ (x0 ), Z ∈ LA (X ), Z ∗ = −Z and Z ≤ π, such that γ(0) = x0 and γ(0) ˙ = v, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x0 , x1 ∈ SX , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f0 the selfadjoint projection I − x0 ⊗ x0 , if the algebra f0 LA (X )f0 is finite dimensional, then there exists a curve γ joining x0 and x1 , which is minimizing along its path. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: C∗ -modules, spheres, geodesics MSC (2000): 46L08, 53C22, 58B20

1

Introduction

The sphere SX of a right Hilbert C ∗ -module X over a unital C ∗ -algebra A, which consists of the elements x ∈ X such that x, x = 1, is a C∞ submanifold of the (Banach space) X . Its basic topological and diﬀerentiable aspects were considered in [2]. In this paper we consider the geometric problem of ﬁnding short smooth curves in SX . To measure the length of a smooth curve we endow each tangent space (which we describe below, and is a complemented real Banach subspace of X ), with the norm of X . Therefore the length ∗ †

E-mail: [email protected] E-mail: [email protected]

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of a curve γ(t) ∈ SX , t ∈ [a, b] is measured by b length(γ) = γ(t) ˙ dt, a

where denotes the norm of X . We refer the reader to [11] for basic facts on C ∗ -modules. As is usual notation, let LA (X ) be the C ∗ -algebra of adjoinable linear operators acting on X . If y, z ∈ X , let y ⊗ z ∈ LA (X ) be the operator y ⊗ z(x) = yz, x. For example, it is easy to see that if x ∈ SX , then x ⊗ x is a self-adjoint projection, which we shall denote by ex . Let U(X ) be the unitary group of LA (X ). Perhaps the main feature in the geometry of SX (as with classical spheres) is the natural action of U(X ) on SX : U · x = U(x), U ∈ U(X ), x ∈ SX . In [2] it was shown that if x0 , x1 ∈ SX verify x0 − x1 < 1/2, then x0 and x1 are conjugate by this action, moreover, one can ﬁnd a unitary operator U(x0 ,x1 ) , which is a C∞ function in (x0 , x1 ) such that U(x0 ,x1 ) (x0 ) = x1 . In particular the action is locally transitive. It is globally transitive in some cases (e.g. if X is self-dual [15] and A is a ﬁnite von Neumann algebra). In general, SX has many components: take for instance X = B(H) with the inner product X, Y = X ∗ Y , then the sphere is the set of isometries of H, whose connected components are parameterized by the codimension of the range. The existence of local cross sections for the action (namely, the unitaries U(x0 ,x1 ) ), implies that for any ﬁxed x0 ∈ SX , the map πx0 : U(X ) → SX , πx0 (U) = U(x0 ) is a locally trivial ﬁbre bundle and a C∞ submersion. It follows that any smooth curve γ(t) ∈ SX can be lifted to a smooth curve μ(t) ∈ U(X ), and therefore represented γ(t) = μ(t) · x0 for some x0 ∈ SX . This enables one to compute the tangent spaces of SX : (T SX )x0 = {A(x0 ) : A ∈ LA (X ), A∗ = −A}. Clearly these elements are also characterized by the condition (T SX )x0 = {v ∈ X : v, x0 + x0 , v = 0}. It is natural to ask whether one can ﬁnd curves of the form γ(t) = etZ (x0 ), t ∈ [0, 1], Z ∗ = −Z, which have minimal length joining their endpoints, or more strictly, which have minimal length along their paths. There are two main problems. (1) The initial value problem: for any tangent vector v ∈ (T SX )x0 ﬁnd a curve γ as above (in particular γ(0) = x0 ), with γ(0) ˙ = v, such that γ has minimal length. (2) The boundary value problem: given x0 , x1 in the same component of SX , ﬁnd a minimal curve γ as above, which joins x0 and x1 .

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In this paper we solve the initial value problem: we show that if A is a von Neumann algebra and X is a right C ∗ -module, which is self-dual [15], then for any x0 ∈ SX and any tangent vector v ∈ (T SX )x0 with v ≤ π there exists a curve γ(t) = etZ (x0 ) with γ(0) = x0 and γ(0) ˙ = v, which has minimal length along its path for t ∈ [0, 1]. The antihermitic operator Z implementing this geodesic is the solution of the extension problem by M.G. Krein [10], in the context of von Neumann algebras (see [6]), as it will be shown in the next section. We call such Z minimal lifts, following [7]. We also consider the boundary value problem. We prove that if x0 , x1 is a scalar multiple of the identity, then x0 and x1 can be joined by a minimizing geodesic (Proposition (4.1)). Another case in which there exists a short geodesic joining x0 and x1 occurs when the (non empty) set {Z : Z ∗ = −Z, eZ (x0 ) = x1 } has a minimum (Theorem (4.3)). As a consequence, we obtain that if f0 (X ) is ﬁnite dimensional (f0 = I −ex0 ), then there exists such a geodesic. In section 5 we introduce a metric in SX , by means of the states of A, which induce Hilbert space representations of the sphere SX . We compare this metric with the Finsler metric. For example, it is shown that they coincide whenever there exist minimal lifts (Theorem (5.4)).

2

Extension problem in von Neumann algebras

A simpliﬁed version of the extension problem ([10], [14], [6]) could be stated as follows: given a closed subspace L of a Hilbert space H and a bounded symmetric operator A0 : L → H, ﬁnd a selfadjoint extension A : H → H with A = A0 . This problem was solved, and all solutions parameterized. We remark that extensions can, but in general need not, be unique. See for example [6] or [14] for explicit parameterizations. M.G. Krein [10] showed that there exist a minimal and a maximal solution (in terms of the usual order of self-adjoint operators), and that all solutions lie in between. For our purposes, we need the additional requirement that if P = PL (=the orthogonal projection onto L) and A0 lie in a von Neumann algebra B, then there exists a solution of the extension problem in B. By this we mean the following result, which is a consequence of the parametrization of solutions given by Davis, Kahan and Weinberger in [6], or the results by Parrott [14]. It is certainly well known, we state it here with proof. Lemma 2.1. Let A be a selfadjoint element and P a selfadjoint projection in a von Neumann algebra B. Then there exists a selfadjoint element Z in B such that ZP = AP and Z = AP . Proof. Let A and P ∈ B be as above. Choose a representation of the von Neumann algebra B in B(H) with H a Hilbert space. Let us consider the following selfadjoint 2 × 2 block operators in terms of P and (I − P ): ⎛ ⎞ (I − P )AP ⎟ ⎜ P AP ZX = ⎝ ⎠ P A(I − P ) X

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where X is a selfadjoint operator in B ((I − P )H). These ZX ∈ B(H) satisfy ZX P = AP and ZX ≥ AP . As it was mentioned at the begining of this section, several authors dealt with the problem of minimizing the norm of ZX . Theorem 1 in [14], for example, proves that in our context there exists an X0 ∈ B((I − P )H) such that ZX0 = P A and X0 is the weak limit of the following elements of B: −cn (I − P )(I − dn P AP AP )−1P AP A(I − P ) (where {cn } and {dn } are sequences of real numbers). Therefore this X0 belongs to B and then ZX0 belongs to B, and veriﬁes ZX0 = P A. We now state a consequence of the result above, in the context of the modular spheres. Let x0 ∈ SX , and v ∈ (T SX )x0 . We call an antihermitic operator Z ∈ La (X ) a minimal lift of v if Z(x0 ) = v and Z = v. Corollary 2.2. Let x0 ∈ SX , with X a selfdual module over the von Neumann algebra A, and v ∈ (T Sx )x0 . Then there exists a minimal lift Z of v. Proof. In this case, LA (X ) is a von Neumann algebra [15]. Since v ∈ (T Sx )x0 , there exists A ∈ LA (X ) such that −A = A∗ and A(x0 ) = v. Note that this implies that A(x0 ⊗x0 ) = v⊗x0 . Moreover, the operator v⊗x0 has norm equal to the norm of v. Indeed, clearly v ⊗ x0 ≤ vx0 = v because x0 = 1, and v ⊗ x0 ≥ v ⊗ x0 (x0 ) = v. Since ex0 = x0 ⊗ x0 is a selfadjoint projection in LA (X ), by the above lemma there exists Z ∈ LA (X ) such that Z ∗ = −Z, Zex0 = Aex0 and Z = Aex0 . In other words, Z(x0 ) = Zex0 (x0 ) = Aex0 (x0 ) = A(x0 ) = v, and Z = v.

3

The initial value problem

We shall now state our main result. Theorem 3.1. Let x0 ∈ SX and v ∈ (T SX )x0 with v ≤ π. Let Z be a minimal lift of v, i.e. Z ∗ = −Z, Z(x0 ) = v and Z = v. ˙ = v, has Then the curve ν(t) = etZ (x0 ), t ∈ [0, 1] which veriﬁes ν(0) = x0 and ν(0) minimal length along its path among smooth curves in SX . Proof. Given a positive element A of a C ∗ -algebra, there exists a faithful representation of the algebra (for instance, the universal representation) and a unit vector ξ in the Hilbert space H of this representation, such that Aξ = Aξ (here we identify A with its image under the representation). Let us call such a vector ξ a norming eigenvector for A. Let us apply this folklore fact to the positive operator −eZ 2 e, where e = ex0 . Let ξ be a (unit) norming eigenvector for −eZ 2 e. Again we identify the operators with their images under this representation, and regard them as operators in this Hilbert space. Clearly ξ lies in the range of e. We claim that ξ is a norming eigenvector for −Z 2 as well. Indeed, −Z 2 ξ = −Z 2 eξ = −eZ 2 P ξ − (I − e)Z 2 P ξ = eZ 2 eξ + ξ1 ,

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where ξ1 = −(I − e)Z 2 eξ is orthogonal to ξ. Note that eZ 2 e = Ze2 = Z2 = Z 2 . Then Z 2 2 ≥ Z 2 ξ2 = eZ 2 e2 + ξ1 2 = Z 2 2 + ξ1 2 . It follows that ξ1 = 0 and our claim is proved. Consider the curve νˆ(t) = etZ (ξ). Clearly ˆ = 1, i.e. νˆ(t) is a curve in the unit sphere SH of the Hilbert space H. Let us prove ν(t) that it is a minimizing geodesic of this Riemann-Hilbert manifold. Indeed, ν¨ˆ(t) = etZ Z 2 ξ = −Z2 etZ ξ = −Z2 νˆ(t). That is, νˆ satisﬁes the diﬀerential equation of the geodesics of the sphere SH . Moreover, the length of νˆ is 1 νˆ˙ (t) dt = Zξ ≤ π. length(ˆ ν) = 0

It follows that νˆ is a minimizing geodesic of the unit sphere. Note also that Zξ2 = Zξ, Zξ = −Z 2 ξ, ξ = Z 2 = Z2 . Clearly, if [t0 , t1 ] ⊂ [0, 1], the length of νˆ restricted to [t0 , t1 ] (or shortly νˆ|[t0 ,t1 ] ) is (t1 − t0 )Z. On the other hand, t1 length(ν|[t0 ,t1 ] ) = ν ˙ dt = (t1 − t0 )Z(x0 ) = (t1 − t0 )Z. t0

It follows that length(ˆ ν ) = length(ν) on any subinterval of [0, 1]. Suppose now that γ : [a, b] → SX is a smooth curve joining ν(t0 ) and ν(t1 ). Consider the curve γˆ (t) := γ(t) ⊗ x0 (ξ). Note that γˆ is also a curve in the unit sphere of H: ˆ γ (t), γˆ (t)H = (γ(t) ⊗ x0 )∗ (γ(t) ⊗ x0 )ξ, ξH = (x0 ⊗ γ(t))(γ(t)) ⊗ x0 ξ, ξH = eξ, ξH = 1. Moreover, γˆ˙ (t) = (γ(t) ˙ ⊗ x0 )ξ ≥ γ(t) ˙ ⊗ x0 = γ(t). ˙ This implies that length(γ) ≤ length(ˆ γ ). Finally, let us show that νˆ|[t0 ,t1 ] and γˆ join the same endpoints of SH : νˆ(t0 ) = et0 Z ξ = et0 Z eξ = et0 Z (x0 ⊗ x0 )ξ = (et0 Z (x0 ) ⊗ x0 )ξ = (ν(t0 ) ⊗ x0 )ξ = (γ(t0 ) ⊗ x0 )ξ = γˆ (t0 ), and similarly for t1 . By the minimality of νˆ, it follows that length(ˆ ν |[t0 ,t1 ] ) ≤ length(ˆ γ ). Therefore length(ν|[t0 ,t1 ] ) = length(ν|[t0 ,t1 ] ) ≤ length(ˆ γ ) ≤ length(γ), which completes the proof.

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Corollary 3.2. If A is a von Neumann algebra and X is a selfdual module, then for any element x0 ∈ SX and tangent vector v ∈ (T SX )x0 with v ≤ π, there exists a geodesic δ ˙ = v, such that δ is minimizing along its path for t ∈ [0, 1]. with δ(0) = x0 , δ(0) Proof. In this case, minimal lifts exist for any tangent vector v.

4

Geodesics joining given endpoints

The problem of ﬁnding minimizing geodesics given any pair of points (in the same component) of the sphere SX is more diﬃcult. It is related to the analogous problem for abstract homogeneous spaces [8]. In this section we ﬁnd solutions in certain cases. These results work for arbitrary C ∗ -algebras and modules. Proposition 4.1. Let x0 , x1 ∈ SX with x0 , x1 = α.1, for α ∈ C. Then there exists a smooth curve in SX with minimal length along its path, which joins x0 and x1 . Proof. Note that since x0 , x1 ≤ x0 x1 = 1, it follows that |α| ≤ 1. If |α| = 1, then α = eir with |r| ≤ π. In this case clearly x1 = αx0 . Indeed, x1 − αx0 , x1 − αx0 = x1 , x1 − x1 , αx0 − αx0 , x1 + αx0 , αx0 = 0. Put γ(t) = eirt x0 . Apparently γ is minimizing along its path (for instance, re = r, i.e. the operator rI is a minimal lift). If |α| < 1, let β ∈ C be such that |α|2 + |β|2 = 1 (note that β = 0), and consider y = αβ −1 x0 − β −1 x1 . Then clearly x0 , y = αβ −1 1 − β −1 x0 , x1 = 0, and y, y =

|α|2 α ¯ α 1 − x0 , x1 − x1 , x0 + 2 = 1. 2 2 2 |β| |β| |β| |β|

in other words, x1 = αx0 +βy with y ∈ SX . That is, x1 lies in the complex plane generated by two orthogonal elements x0 and y of SX . The situation resembles what happens in a classic ﬁnite dimensional sphere, and the proof follows as in that case. Namely, let (α(t), β(t)) be a minimal geodesic of the sphere SC2 of C2 , joining (1, 0) (at t = 0) and (α, β) (at t = 1). Consider the curve γ(t) = α(t)x0 + β(t)y. Clearly it is a smooth curve with γ(0) = x0 and γ(1) = x1 , which lies in SX : γ(t), γ(t) = |α(t)|2 + |β(t)|2 = 1. Moreover, it has constant speed equal to 2 2 ˙ ˙ = α(t)x ˙ ˙ γ(t) ˙ 0 + β(t)x1 , α(t)x 0 + β(t)x1 2 2 2 2 ˙ 2 = |α(0)| ˙ = |α(t)| ˙ + |β(t)| ˙ + |β(0)| = γ(0) ˙ .

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We claim that it is minimizing along its path. Let ϕ be a state in A. Then the form [x, y]ϕ := ϕ(x, y), x, y ∈ X is positive semideﬁnite in X . Let Hϕ be the completion of (X /Z, [ , ]ϕ ), where Z = {z ∈ X : [z, z]ϕ = 0}. Denote by x¯ be the class of x ∈ X in X /Z ⊂ Hϕ . In other words, x¯ is the element x regarded as a vector in the Hilbert space Hϕ . Note that the elements of SX induce elements in the unit sphere of Hϕ : clearly [¯ x, x¯]ϕ = ϕ(x, x) = 1 The geodesic (α(t), β(t)) of SC2 satisﬁes the Euler equation of the sphere: ¨ (α(t), ¨ β(t)) = −κ2 (α(t), β(t)). It follows that γ¯ satisﬁes the diﬀerential equation γ¨¯ (t) = −κ2 γ¯ (t), in the sphere SHϕ of Hϕ . Moreover, the length of γ¯ restricted to the interval [t1 , t2 ] ⊂ [0, 1], is given by t1 t1 1/2 1/2 ˙ ˙ [γ¯˙ (t), γ¯˙ (t)] dt = ϕ(α(t)x ˙ ˙ dt 0 + β(t)y, α(t)x 0 + β(t)y) t0

t0 t1

=

2 ˙ 2 .1)1/2 dt = (t1 − t0 )γ(0). ϕ(|α(t)| ˙ .1 + |β(t)| ˙

t0

It follows that γ¯ is minimizing along its path in SHϕ , and length(¯ γ ) = length(γ). Let ν(t), t ∈ [0, 1] be another smooth curve in SX joining ν(0) = γ(t0 ) and ν(1) = γ(t1 ). Then ν¯ is a smooth curve in SHϕ , and the inequality [ν¯˙ , ν¯˙ ]ϕ = ϕ(ν, ˙ ν) ˙ ≤ ν, ˙ ν ˙ implies that length(ν) ≥ length(¯ ν ). It follows that ν is not shorter than γ|[t0 ,t1 ] . If x0 , x1 ∈ SX satisfy that x0 − x1 < 1/2, then they are conjugate by the action of U(X ) (see [2]). Let us state the following result, estimating the distance between the identity and the unitary operator performing this conjugacy. Lemma 4.2. Let x0 , x1 ∈ SX with x0 −x1 < 1/2. Then there exists a unitary U ∈ U(X ) such that U(x0 ) = x1 with U − I < 3/2. Proof. First we transcribe the construction of the unitary U given in [2]. Let e0 = ex0 and e1 = ex1 . Since x0 − x1 < 1/2, it follows that e0 − e1 ≤ e0 − x0 ⊗ x1 + x0 ⊗ x1 − e1 = x0 ⊗ (x0 − x1 ) + (x1 − x0 ) ⊗ x1 .

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Note that x0 ⊗ (x0 − x1 ) ≤ x0 − x1 (in fact equality holds because x0 ∈ SX ), and analogously for the other term. Therefore e0 − e1 < 1. It is a standard fact that two such projections are unitarily equivalent, moreover, the unitary V such that V e0 V ∗ = e1 can be chosen V = eY with Y ∈ LA (X ) such that Y ∗ = −Y and Y < π/2 (moreover, Y is codiagonal in terms of e0 and sin Y = e0 − e1 , see for instance [3], page 151). Therefore √ I − V = r(I − V ) = supp{|1 − eω | : ω ∈ sp(Y )} < 2, because |ω| ≤ Y < π/2 (here sp and r stand for the spectrum and the spectral radius, respectively). Consider U = x1 ⊗ x0 + V (I − e0 ). This unitary veriﬁes that U(x0 ) = x1 , and moreover, I − U = e1 − x1 ⊗ x0 + (I − e1 ) − V (I − e0 ). Since V (I − e0 )V ∗ = e1 , it follows that the operators e1 − x1 ⊗ x0 and (I − e1 ) − V (I − e0 ) have orthogonal ranges (in any Hilbert space representation for LA (X )). Indeed, the range of e1 − x1 ⊗ x0 = e1 (I − x1 ⊗ x0 ) is contained in the range of e1 , and the range of (I − e1 ) − V (I − e0 ) = (I − e1 ) − (I − e1 )V ∗ is contained in its orthogonal complement. Thus I − U ≤ e1 − x1 ⊗ x0 2 + I − e0 − V (I − e0 )2 . Note that e1 − x1 ⊗ x0 = x1 ⊗ (x1 − x0 ) = x1 − x0 < 1/2 and I − e0 − V (I − e0 ) = (I − e0 )(I − V ) ≤ I − V ≤

√ 2.

Then I − U < 3/2. In particular, by a standard argument involving the continuous functional calculus in the C ∗ -algebra LA (X ), the unitary U of the lemma above is of the form U = eZ for Z ∈ LA (X ), with Z ∗ = −Z and Z < π/3 (using the same computation as in the norm of I − V above). Denote by Lx0 ,x1 = {Z ∈ LA (X ) : Z ∗ = −Z, eZ (x0 ) = x1 }. If x0 − x1 < 1/2, then Lx0 ,x1 is non empty. If x0 , x1 are not that close, but they lie in the same component of SX , the algebra A is a von Neumann algebra, and the module X is selfdual, one also has that Lx0 ,x1 is non empty, with the unitary chosen such that Z ≤ π. If moreover A is ﬁnite, then S(X ) is connected, and any pair of elements in the sphere are conjugate by an exponential. The following result is an adaptation of Theorem 3.2 in [8] to our particular context, where the Finsler metric is given by the norm of X (in [8] quotient norms are considered).

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Theorem 4.3. Let x0 , x1 ∈ SX , with x0 − x1 < 1/2. Suppose that there exists Z0 ∈ Lx0 ,x1 such that Z0 = inf{Z : Z ∈ Lx0 ,x1 }. Then Z0 is a minimal lift and therefore ν(t) = etZ0 (x0 ) is minimizing along its path. In particular, it is shorter than any other piecewise smooth curve joining x0 and x1 in SX . Proof. The proof, as in 3.2 of [8], proceeds in three steps: • a) Let Z0 ∈ Lx0 ,x1 with Z0 = inf{Z : Z ∈ Lx0 ,x1 }, ﬁx s ∈ (0, 1) and denote xs = esZ0 (x0 ). Then sZ0 ∈ Lx0 ,xs and sZ0 = inf{Z : Z ∈ Lx0 ,xs }. • b) Suppose that X, Y are antihermitic operators of small norms in order that eX eY lies in the domain of the power series of the logarithm log deﬁned on a neighborhood of I with antihermitic values. (for instance, eX eY − I < 1). Then log(eX eY ) = X + Y + R2 (X, Y ), where lim

s→0

R2 (sX, sY ) = 0. s

• c) Let e = ex0 . For any Y ∗ = −Y such that Y = (I − e)Y (I − e), one has that Z0 ≤ Z0 + Y . Let us prove these steps, and show how they prove our result. Step a): For an element X ∗ = −X, denote by γX (t) = etX . We claim that the condition Z0 = inf{Z : Z ∈ Lx0 ,x1 } implies that the curve γZ0 is the shortest among piecewise smooth curves of unitaries joining I to the set {U ∈ U(H) : U(x0 ) = x1 }. Indeed, by the remark above, since x0 − x1 < 1/2, there exists X ∈ Lx0 ,x1 such that X ≤ π/3. It follows that Z0 ≤ π/3. Suppose that μ(t) is another smooth curve of unitaries with μ(0) = I and μ(1)(x0 ) = x1 , which is shorter than γZ0 . Let LA (X )∗∗ be the von Neumann enveloping algebra of LA (X ). Then there is a curve of the form etΩ , Ω∗ = −Ω ∈ LA (X )∗∗ and Ω < π/3, with eΩ = μ(1), which is shorter than μ. This follows from the folklore fact that exponentials are short curves in the unitary group of a von Neumann algebra, when the length is measured with the Finsler metric given by the usual norm (see for instance [5]). It follows that I − μ(1) < 3/2. Therefore the unitary μ(1) which lies in the C ∗ -algebra LA (X ), is also of the form μ(1) = eW , for W ∗ = −W ∈ LA (X ). That is, W lies in Lx0 ,x1 , and therefore W ≥ Z0 . Then length(μ) ≥ W ≥ length(γZ0 ). Let us show that sZ0 = inf{Z : Z ∈ Lx0 ,xs }. Suppose that there exists X ∈ Lx0 ,xs such that X < sZ0 . Consider the curve δ(t) = e(1−t)sZ0 +tZ0 which joins esZ0 with eZ0 in U(X ), and σ(t) = δ(t)e−sZ0 eX , joining eX and e(1−s)Z0 eX (in both cases t ∈ [0, 1]). Note that they have the same length, for they diﬀer on an element of U(X ): length(δ) = length(σ) = (1 − s)Z0 . Note also that the endpoint of σ satisﬁes σ(1)x0 = x1 . Let γ˜ be the piecewise smooth curve which consists of the curve γX followed by σ. Then γ˜ joins

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I to the ﬁber {U ∈ U(X ) : U(x0 ) = x1 } in U(X ), and therefore, by the fact remarked above, length(˜ γ ) ≥ Z0 . On the other hand, length(˜ γ ) = length(γX ) + length(σ) = X + (1 − s)Z0 < sZ0 + (1 − s)Z0 = Z0 . Step b): The linear part of the series of log(eX eY ) is X + Y . So that log(eX eY ) = X + Y + R2 (X, Y ) Where the remainder term R2 (X, Y ) is an inﬁnitesimal of the order X+Y . Therefore R2 (sX, sY ) = 0. s→0 s lim

Step c): By step a), for any s ∈ (0, 1), sZ0 = inf{Z : Z ∈ Lx0 ,xs }. Let Y ∗ = −Y such that Y = (I − e)Y (I − e). Then clearly eY (x0 ) = x0 . Therefore log(eZ0 eY ) ∈ Lx0 ,x1 . Analogously, log(esZ0 esY ) ∈ Lx0 ,xs . Then sZ0 ≤ log(esZ0 esY ) = sZ0 + sY + R2 (sZ0 , sY ) ≤ sZ0 + Y + R2 (sZ0 , sY ). Then

R2 (sZ0 , sY ) . s Taking limits, Z0 ≤ Z0 + Y , which proves step c). The theorem follows. The set {Z0 + Y : Y ∗ = −Y, (I − e)Y (I − e) = Y } parameterizes the set of all Z such that Ze = Z0 e. This means that Z0 is a minimal lift, and therefore ν(t) = etZ0 (x0 ) is a minimizing geodesic, joining x0 and x1 . Note that if x0 , x1 are conjugate by the action of U(X ), then the projections ex0 and ex1 are unitarily equivalent: if U(x0 ) = x1 , ex1 = U(x0 ) ⊗ U(x0 ) = U(x0 ⊗ x0 )U ∗ = Uex0 U ∗ . Z0 ≤ Z0 + Y +

Corollary 4.4. Let x0 , x1 ∈ SX , with x0 − x1 < 1/2. Denote f0 = 1 − ex0 . If the algebra f0 LA (X )f0 is ﬁnite dimensional, then there exists a geodesic ν(t) = etZ (x0 ) with ν(1) = x1 , which is minimizing along its path. Proof. Note that if U, U ∈ U(X ) with U(x0 ) = U (x0 ) it follows that U ∗ U (x0 ) = x0 . Let e0 = ex0 . This last statement is equivalent to U ∗ U e0 = e0 . The group Ge0 = {V ∈ U(X ) : V e0 = e0 } when written as 2 × 2 matrices in terms of e0 , consists of matrices of the form ⎞ ⎛ ⎜ e0 0 ⎟ ⎠, ⎝ 0 f0 V f0

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where f0 V f0 is a unitary operator in U(f0 (X )), which identiﬁes with the unitary group of the reduced C ∗ -algebra f0 LA (X )f0 . It follows that Ge0 is compact in the norm topology. Therefore the set {U ∈ U(X ) : U (x0 ) = x1 } is compact, which implies that the set {Z : Z ∈ Lx0 ,x1 } has a minimum, and the theorem above applies.

Remark 4.5. If A is a von Neumann algebra and X is selfdual, then the hypothesis x0 − x1 < 1/2 of the above results can be replaced by the requirement that x0 , x1 lie in the same connected component, or by no requirements at all if A is ﬁnite.

5

Hilbert space spheres

Denote by d the metric in SX determined by the Finsler metric given by the norm of X at every tangent space of SX : d(x0 , x1 ) = inf{length(γ) : γ joins x0 and x1 }, with length(γ) measured as before. As in the proof of the proposition (4.1) at the beginning of the preceding section, one may endow X with a semideﬁnite scalar product by means of a state ψ of A. Namely, put [x, y]ψ = ψ(x, y), x, y ∈ X . If the state ψ is non faithful this inner product degenerates. Let Z = {z ∈ X : [z, z]ψ = 0} be the subspace of degenerate vectors, and Hψ the completion of X /Z. Denote by x¯ the class of x ∈ X in Hψ . Note that the quotient map maps SX into SHψ . If x0 , x1 ∈ SX , denote by dψ (x0 , x1 ) = inf{length(α) : α a smooth curve in SHψ joining x¯0 and x¯1 }, i.e. the geodesic distance of x¯0 and x¯1 as elements in the unit sphere SHψ . Let ds (x0 , x1 ) = sup {dψ (x0 , x1 ) : ψ a state in A}. If x0 , x1 < 1, a fact which implies that [x0 , x1 ]ψ < 1, then it is a standard fact from the geometry of spheres (ﬁnite or inﬁnite dimensional), that the distance equals dψ (x0 , x1 ) = arccos(Re([x¯0 , x¯1 ]ψ )) = arccos(Re(ψ(x0 , x1 ))). Note that, for ﬁxed elements x0 , x1 ∈ SX , the map ψ → arccos(Re(ψ(x0 , x1 ))) is continuous for the w ∗ -topology of the state space of A. Therefore the supremum at the deﬁnition of ds is attained at a certain state. Note also that dψ is in fact a pseudometric in SX , if ψ is not faithful.

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Proposition 5.1. ds is a metric in SX . Moreover ds (x0 , x1 ) ≤ d(x0 , x1 ). Proof. The metric ds is the supremum of a family of pseudometrics in SX , therefore it is a pseudometric. Let us show that if ds (x0 , x1 ) = 0 then x0 = x1 . Clearly this implies that x¯0 = x¯1 in every Hilbert space Hψ , that is, ψ(x0 − x1 , x0 − x1 ) = 0 for all states ψ. This implies that x0 − x1 , x0 − x1 = 0 and therefore x0 = x1 . If γ is a smooth curve in SX with γ(0) = x0 and γ(1) = x1 , then ˙ γ) ˙ ≤ γ ˙ 2. [γ¯˙ , γ¯˙ ]ψ = ψ(γ, Next we show that these two metrics coincide if there exists a minimizing geodesic giving by a minimal lift as in the ﬁrst section (Theorem 3.1). To prove this fact we need the following elementary results concerning states and operators in LA (X ). Lemma 5.2. Let x0 ∈ SX and e = ex0 . Then A is isomorphic to the reduced algebra eLA (X )e, via the mapping a → x0 a ⊗ x0 . Proof. The map a → x0 a⊗x0 is clearly linear, and takes values in eLX (A)e: e(x0 a⊗x0 )e = x0 a ⊗ x0 . It is multiplicative: (x0 a ⊗ x0 )(x0 b ⊗ x0 ) = x0 ax0 , x0 b ⊗ x0 = x0 ab ⊗ x0 . It preserves the adjoint: (x0 a ⊗ x0 )∗ = x0 ⊗ x0 a = x0 a∗ ⊗ x0 . It is isometric: as remarked before, x0 a ⊗ x0 = x0 a x0 = a. Finally, it is onto: if T ∈ eLA (X )e, then T = (x0 ⊗ x0 )T (x0 ⊗ x0 ) = (x0 ⊗ x0 )(T (x0 ) ⊗ x0 ) = x0 x0 , T (x0 ) ⊗ x0 , i.e. T is the image of x0 , T (x0 ) ∈ A. A straightforward consequence of this result is the following (see [4]).

Lemma 5.3. If Φ is a state of LA (X ) with support less or equal than e = x0 ⊗ x0 (i.e. Φ(e) = 1), then there exists a state ψ of A such that Φ(T ) = ψ(x0 , T (x0 )), T ∈ LA (X ). Theorem 5.4. Let x0 , x1 ∈ SX with x0 , x1 < 1, and suppose that there exists a minimal lift Z at x0 (i.e. Z ∈ LA (X ), Z ∗ = −Z, with Z = Ze = Z(x0 ) ≤ π) such that eZ (x0 ) = x1 . Then the length of the geodesic ν(t) = etZ (x0 ) equals the distance ds (x0 , x1 ). In other words, d(x0 , x1 ) = ds (x0 , x1 ) = Z. In particular, ν is a minimizing geodesic in SX .

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Proof. As in the proof of theorem 3.1, let ξ be a norming (unit) eigenvector for eZ 2 e in a faithful representation of LA (X ): (as before we identify operators with their images under this representation) eZ 2 eξ = −Ze2 ξ = −Z2 ξ. Recall that ξ lies in the range of e, and is also a norming eigenvector for Z 2 . Consider the state Φ of LA (X ) given by ξ: Φ(T ) = [T ξ, ξ]H (here [ , ]H denotes the inner product of H). Then Φ(e) = 1, and therefore there exists a state ϕ of A such that ϕ(a) = Φ(x0 a ⊗ x0 ). We claim that the state ϕ realizes the maximum above: ds (x0 , x1 ) = max{arccos(Re(ψ(x0 , x1 ))) : ψ a state of A}. To prove our claim, let us show that arccos(Re(ϕ(x0 , x1 ))) = Z = d(x0 , x1 ), which ends the proof. Note that Φ(eZ ) = Φ((x0 ⊗ x0 )eZ (x0 ⊗ x0 )) = Φ((x0 x0 , eZ (x0 ) ⊗ x0 ) = ϕ(x0 , x1 ). On the other hand, Φ(eZ ) = [eZ ξ, ξ]H. Since Z 2 ξ = −Z2 ξ, it follows that 1 1 1 1 eZ ξ = (1 − Z2 + Z4 + . . . )ξ + (1 − Z2 + Z4 + . . . )Zξ. 2 4! 3! 5! Note that since Z is antihermitic, it follows that Re([eZ ξ, ξ]H) = cos Z. Therefore Re(ϕ(x0 , x1 )) = Re(Φ(eZ )) = cos Z. It is a standard fact that given a state ψ of A, the algebra LA (X ) can be represented in Hψ . Let us denote by ρψ this representation. Namely, if x, y ∈ X and A ∈ LA (X ), then A(x − y), A(x − y) = A∗ A(x − y), x − y ≤ A2 x − y, x − y, therefore [A(x − y), A(x − y)]ψ = ψ(A(x − y), A(x − y)) ≤ A2 ψ(x − y, x − y) = A2 [x − y, x − y]ψ . This implies that if x and y are equivalent in X /Z, then A(x) and A(y) are also equivalent, ¯ extends to a bounded operator ρψ (A) on Hψ . and the linear map x¯ → A(x) Remark 5.5. Let x0 ∈ SX and v ∈ (T SX )x0 with v ≤ π. Suppose that there exists a minimal lift Z ∈ LA (X ) for v. Let ϕ be a state in A constructed as in the proof of the previous result. Then x¯0 ∈ Hϕ is an eigenvector for ρϕ (Z 2 ), with eigenvalue −Z2 = −v2 . Let Z be a minimal lift for v, i.e. Z ∗ = −Z, Z(x0 ) = v and Z = v. By Theorem (3.1), the curve ν(t) = etZ x0 has minimal length along its path in SX . Then ν¯ is a

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minimizing geodesic in the Hilbert space sphere SHϕ . Then ν¨¯ = −k 2 ν¯ for some real constant k. Therefore −k 2 ν¯(t) = ν¨¯(t) = ρϕ (Z 2 )¯ ν (t), i.e. etρϕ (Z) (−k 2 x¯0 ) = etρϕ (Z) (ρϕ (Z 2 )(x¯0 )), which implies that ρϕ (Z 2 )(x¯0 ) = −k 2 x¯0 . On the other hand [ρϕ (Z 2 )(x¯0 ), x¯0 ]ϕ = ϕ(Z 2 (x0 ), x0 ) = Φ(eZ 2 e) = −Z2 . It follows that k 2 = Z2 . Combining the previous theorem with (4.4) one obtains the following: Corollary 5.6. If the algebra f0 LX (A)f0 is ﬁnite dimensional, and x0 , x1 lie in the same connected component of SX , then d(x0 , x1 ) = ds (x0 , x1 ). Proof. Note that x0 , x1 ≤ 1. Suppose that x0 , x1 < 1. By (4.4), there exists a minimal lift Z ∈ LA (X ), Z ∗ = −Z, Z ≤ π, such that eZ (x0 ) = x1 . Then the above theorem (5.4) applies and ds (x0 , x1 ) = d(x0 , x1 ). If x0 , x1 = 1, then x1 can be approximated by xn ∈ SX (in the norm of X ), with x0 , xn < 1. It follows that ds (x0 , xn ) = d(x0 , xn ). Next note that if xn − x1 → 0, then [x¯n − x¯1 , x¯n − x¯1 ]ψ → 0 for every state ψ. On the other hand also it is clear that d(xn , x1 ) → 0. Therefore the result follows.

References [1] E. Andruchow, G. Corach and M. Mbekhta: “On the geometry of generalized inverses”, Math. Nachr., Vol. 278, (2005), no. 7-8, pp. 756–770. [2] E. Andruchow, G. Corach and D. Stojanoﬀ: “Geometry of the sphere of a Hilbert module”, Math. Proc. Cambridge Philos. Soc., Vol. 127, (1999), no. 2, pp. 295–315. [3] E. Andruchow, G. Corach and D. Stojanoﬀ: “Projective spaces of a C ∗ -algebra”, Integral Equations Operator Theory, Vol. 37, (2000), no. 2, pp. 143–168. [4] E. Andruchow and A. Varela: “C ∗ -modular vector states”, Integral Equations Operator Theory, Vol. 52, (2005), pp. 149–163. [5] C.J. Atkin: “The Finsler geometry of groups of isometries of Hilbert space”, J. Austral. Math. Soc. Ser. A, Vol. 42, (1987), pp. 196–222. [6] C. Davis, W.M. Kahan and H.F. Weinberger: “Norm preserving dilations and their applications to optimal error bounds”, SIAM J. Numer. Anal., Vol. 19, (1982), pp. 445–469.

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[7] C.E. Dur´an, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C∗ -algebra. Part I. Minimal curves”, Adv. Math., Vol. 184, (2004), no. 2, pp. 342–366. [8] C.E. Dur´an, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C∗ -algebra. Part II. Geodesics joining ﬁxed endpoints”, Integral Equations Operator Theory, Vol. 53, (2005), no. 1, pp. 33–50. [9] S. Kobayashi and K. Nomizu: Foundations of diﬀerential geometry, Vol. II. Reprint of the 1969 original, Wiley Classics Library, John Wiley & Sons, New York, 1996. [10] M.G. Krein: “The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications”, Mat. Sb., Vol. 20, (1947), pp. 431–495, Vol. 21, (1947), pp. 365–404 (in Russian). [11] E.C. Lance: “Hilbert C∗ -modules, A toolkit for operator algebraists”, London Math. Soc. Lecture Note Ser., Vol. 210, Cambridge University Press, Cambridge, 1995. [12] P.R. Halmos and J.E. McLaughlin: “Partial isometries”, Paciﬁc J. Math., Vol. 13, (1963), pp. 585–596. [13] L.E. Mata-Lorenzo and L. Recht: “Inﬁnite-dimensional homogeneous reductive spaces”, Acta Cient. Venezolana, Vol. 43, (1992), pp. 76–90. [14] S. Parrott: “On a quotient norm and the Sz.-Nagy–Foias lifting theorem”, J. Funct. Anal., Vol. 30, (1978), no. 3, pp. 311–328. [15] W.L. Paschke: “Inner product modules over B ∗ -algebras”, Trans. Amer. Math. Soc., Vol. 182, (1973), pp. 443–468. [16] F. Riesz and B. Sz.-Nagy: Functional Analysis, Frederick Ungar Publishing Co., New York, 1955.

DOI: 10.2478/s11533-007-0023-3 Research article CEJM 5(4) 2007 654–664

The Cauchy Harish-Chandra Integral, for the pair up,q , u1 Andrzej Daszkiewicz1 and Tomasz Przebinda2∗ 1

Faculty of Mathematics, N. Copernicus University, 87-100 Toru´ n, Poland

2

Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A.

Received 29 May 2007; accepted 13 June 2007 Abstract: For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Orbital integrals, dual pairs MSC (2000): primary 22E45, secondary 22E46

1

Introduction

This article was completed in summer 1998, while the work [5] was still in progress. We publish the original version without any essential changes. One of the main problems in the theory of dual pairs is the description of the correspondence of characters of representations in Howe duality, [3]. In [2] a formula describing this correspondence was obtained under some very strong assumptions. In [5] the second author has developed a notion of a Cauchy Harish–Chandra integral for any real reductive pair, in order to describe this correspondence of characters. In this paper a special case of this integral will be studied. The results obtained here are crucial for the estimates needed in [5]. (See the proof of Theorem 10.19, page 343, in [5].) ∗

E-mail: [email protected]

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In this paper we consider the Lie algebra g = up,q = {z ∈ glp+q (C); z Ip,q +Ip,q z t = 0}. We assume, ⎛ for convenience, that p ≤ q. Let h ⊆ g be the diagonal Cartan subalgebra. ⎞ ⎜Ip 0 ⎟ Here Ip,q = ⎝ ⎠, as usual. Let Hj = Ejj ∈ hC , 1 ≤ j ≤ p + q, be the diagonal 0 −Iq matrix with 1 in the jth row and jth column and zeros elsewhere. Then H1 , H2 , ..., Hp+q is a basis of the vector space hC . Let e1 , e2 , ..., ep+q ∈ h∗C denote the dual basis. We ﬁx the following system of positive roots of hC in gC , Φ(h) = {ej − ek ; 1 ≤ j < k ≤ p + q}. Let k ⊆ g be the centralizer of Ip,q . Then k is the Lie algebra of a maximal compact subgroup of G = Up,q = {g ∈ GL(C); g Ip,q g t = Ip,q }, and h ⊆ k. The set of non-compact roots in Φ(h) is Φn (h) = {ej − ek ; 1 ≤ j ≤ p < k ≤ p + q}. Let π denote the product of all the roots in Φ(h): π= (ej − ek ) . 1≤j
For a root α ∈ Φ (h), let cα ∈ End(gC ) be the Cayley transform and let Hα ∈ i h be the corresponding element, as in [1, 3.1], (Hα = Hj − Hk , if α = ej − ek ). For a strongly orthogonal set S ⊆ Φn (h) let cS = α∈S cα , and let hS = g ∩ cS (hC ) be the corresponding Cartan subalgebra, as in [6, sec. 2]. Denote by HS ⊆ G the corresponding Cartan r subgroup. For any α ∈ S, the root −α ◦ c−1 α ◦ c−1 S = S of hS,C in gC , is real. For x ∈ hS , the set of regular elements in hS , set S (x) = α∈S sgn(α ◦ c−1 S (x)). The formula, n

(x, y ∈ g)

P (x, y) = tr(xy t )

(1.0)

deﬁnes a real valued, positive deﬁnite scalar product on g, viewed as a real vector space. This scalar product determines a Lebesgue measure dx on g, such that for any basis e1 , e2 , ..., en of g the volume of the parallelepiped Ie1 + Ie2 + ... + Ien , where I = (0, 1) is the unit interval, is equal to det(P (ei, ej ))1/2 . Similarly P determines an Lebesgue measure on each subspace of g, an invariant measure on the group G, on each closed unimodular Lie subgroup and on each quotient of two such subgroups. Recall the Harish–Chandra integral deﬁned with respect to the negative roots:

−1 ψ(gxg −1) dg˙ (x ∈ hrS , ψ ∈ S(g)). (1.1) ψS (x) = π ◦ cS (x) S (x) G/HS

Here h ⊆ h is the subset of regular elements, (see [8, 0.2.1]). Set αj = ej − ep+j , 1 ≤ j ≤ p, and for m = 1, 2, 3, ..., p, deﬁne Sm = {α1 , α2 , ..., αm }. Let S0 = ∅. Then each Sm , 0 ≤ m ≤ p, is a strongly orthogonal set and, in terms of the Cartan subalgebras corresponding to these sets, the Weyl integration formula may be written as follows

p 1 ψ(x) dx = π ◦ c−1 (1.2) Sm (x) ψSm (x) dx, + (p − m)!(q − m)! hS g m=0 r

m

= h , and for m ≥ 1, where ψ ∈ S(g), −1 1 α ˜ j = − 2 αj ◦ cSm . Let h+ S0

Y =

p j=1

r

i Hj −

p+q j=p+1

h+ Sm

i H j , Y Sm =

= {x ∈ hS ; α ˜ 1 (x) > ... > α ˜ m (x) > 0}, and

ej ⊥Sm

ej (Y ) Hj

(0 ≤ m ≤ p).

(1.3)

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Here, “ej ⊥ Sm ” means “α(Hj ) = 0, for all α ∈ Sm ”. If this condition is empty then YSm = 0. Let g+ = {y ∈ g; −iyIp,q > 0}. This is an open convex cone in g. In terms of limits of holomorphic functions, [4, 3.1.15], deﬁne the following temperate distribution: 1 1 = +lim det(x + i 0) g y→0 det(x + i y)

(x ∈ g).

(1.4)

Clearly, this distribution is Ad(G)-invariant. The goal of this paper is to prove the following, seemingly obvious theorem, which expresses the distribution (1.4) in terms of integrals over various Cartan subgroups. Theorem 1.5. For any ψ ∈ S(g),

1 ψ(x) dx g det(x + i 0)

p 1 = lim →0+ (p − m)!(q − m)! h+S m=0

m ,

π ◦ c−1 Sm (x + i YSm ) ψSm (x) dx det ◦c−1 Sm (x + i YSm )

+ r where h+ ˜ 1 (x) > ... > α ˜ m (x) > } for m ≥ 1, and the integrals S0 , = h , hSm , = {x ∈ hS ; α on the right hand side are absolutely convergent. Let

t G+ C = {g ∈ GLn (C); the hermitian matrix Ip,q − g Ip,q g is positive deﬁnite}. + + Clearly, G+ C is a sub-semigroup of GLn (C), and G · GC ⊆ GC . In terms of limits of holomorphic functions, deﬁne the following distribution on G:

1 1 = lim + det(1 − g · 1) {p→1, p∈GC } det(1 − g · p)

(g ∈ G).

(1.6)

For a strongly orthogonal set S ⊆ Φn (h), let HS = exp(hS ) ⊆ G be the corresponding Cartan subgroup, and let CS : HC → HS,C , be the Cayley transform. Let Δ(h) = (h(ej −ek )/2 − h(ek −ej )/2 ) (h ∈ HC ). j
Recall the Harish–Chandra integral, deﬁned with respect to the negative roots:

−1 S (h) Ψ(ghg −1)dg, ˙ ΨS (h) = Δ ◦ CS (h)˜ G/HS −1 sgn(1 − h−α◦cS ) (h ∈ HSr ), ˜(h) =

(1.7)

α◦c−1 real S

where HSr ⊆ HS is the subset of regular elements. Set HS+m = exp(h+ Sm ). Then the Weyl integration formula for G says

p 1 Ψ(g) dg = Δ ◦ CS−1 (h)ΨSm (h) dh (1.8) m + (p − m)!(q − m)! G H S m=0 m

A. Daszkiewicz and T. Przebinda / Central European Journal of Mathematics 5(4) 2007 654–664 657

where Ψ ∈ Cc (G). With the hSm , as in (1.5), set HS+m , = exp(hSm , ). Theorem 1.9. For Ψ ∈ Cc∞ (G),

1 Ψ(g) dg G det(1 − g · 1)

p 1 = lim →0 (p − m)!(q − m)! HS+ m=0

m ,

Δ ◦ CS−1 (h exp(iYSm )) m ΨSm (h) dh, −1 det(1 − CSm (h exp(iYSm )))

where the integrals on the right hand side are absolutely convergent.

2

Integration by parts

Let V be a ﬁnite dimensional space over the reals. Let V ∗ denote the linear dual to V . Fix elements e ∈ V and e∗ ∈ V ∗ such that e∗ (e) = 1. Let f be a smooth function on V and let φ be a bounded, smooth function on V \ ker e∗ , the complement of ker e∗ in V . Recall the directional derivative: ∂(e)f (x) =

d f (x + t e)|t=0 dt

(x ∈ V ).

(2.1)

Assume that f and all derivatives of f are of at most polynomial growth at inﬁnity, and that φ and all derivatives of φ are rapidly decreasing at inﬁnity. Suppose we have an Euclidean norm on V . Assume that e has norm 1 and that it is orthogonal to ker(e∗ ). Then every subspace of V is equipped with a Lebesgue measure dx, normalized so that the volume of the unit cube is 1. Integration by parts veriﬁes the following formula

(f (x)(∂(en )t φ(x)) − (∂(en )f (x))φ(x))dx e∗ (x)>

n−1

=

(2.2) n−1−k

∂(e

k t

)f (x + e)∂(e ) φ(x + e) dx

( > 0, n = 1, 2, ....).

ker e∗ k=0

Here ∂(en )t = (−1)n ∂(en ) stands for the adjoint of the diﬀerential operator ∂(en ). Let us assume that the following limits exist ∂(ek )φ(x ± 0 e) = lim ∂(ek )φ(x ± t e) t→0+

(x ∈ ker e∗ ; k = 0, 1, 2, ....).

(2.3)

Then (2.2) implies

e∗ (x) =0

(f (x)(∂(en )t φ(x)) − (∂(en )f (x))φ(x))dx

=

n−1

ker e∗ k=0

where n = 1, 2, ... .

(2.4) n−1−k

∂(e

)f (x)(∂(e ) φ(x + 0 e) − ∂(e ) φ(x − 0 e)) dx, k t

k t

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Recall that the map ∂, deﬁned in (2.1), extends to an isomorphism of the symmetric algebra Sym(VC ) and the algebra of constant coeﬃcient diﬀerential operators on V . For any w ∈ Sym(VC ) there are uniquely determined elements wn ∈ Sym(ker e∗ ) such that w= wn en . (2.5) n≥0

By combining (2.2), (2.4) and (2.5) we deduce that for any w ∈ Sym(VC ) and any > 0, the following two formulas hold

(f (x)(∂(w)t φ(x)) − (∂(w)f (x))φ(x))dx e∗ (x)> n−1

(2.6) n−1−k

∂(e

=

n≥1 k=0

ker e∗

k t

)∂(wn )f (x + e)∂(e ) φ(x + e) dx,

(f (x)(∂(w)t φ(x)) − (∂(w)f (x))φ(x))dx

e∗ (x) =0 n−1

(2.7) n−1−k

∂(e

=

n≥1 k=0

3

ker e∗

)∂(wn )f (x)(∂(e ) φ(x + 0 e) − ∂(e ) φ(x − 0 e)) dx, k t

k t

Proof of the Theorem 1.5

We identify g with g∗ via the bilinear form B(x, y) = tr(xy)

(x, y ∈ g).

(Notice that B takes only real values.) Given a polynomial function P on gC , let P # be the corresponding element of the symmetric algebra Sym(gC ). Lemma 3.1. In terms of germs of holomorphic functions, the following formula holds: ∂(det # ) log(det(z)) =

(n − 1)! det(z)

(z ∈ gC , det(z) = 0),

where n = p + q, and log is the natural logarithm. Proof. Notice that, by the deﬁnition (2.1), the map ∂ depends on the real form g ⊆ gC . We shall write ∂g in order to indicate this dependence. For a function f deﬁned on g and for an element g ∈ GLn (C), let λ(g)f (x) = f (g −1x) be a function deﬁned on the set gg. Then for a polynomial P on gC , λ(g)∂g(P # )f = ∂gg((ρ(g)P )# )λ(g)f,

(3.2)

where ρ(g)P (x) = P (x g). Let g = Ip,q . Then gg = un . Hence, (3.2) implies that, with f (z) = log(det(z)), ∂up,q (det # )f = λ(g −1)∂un (det(g) · det # )λ(g)f = det(g)λ(g −1)∂un (det # )λ(g)f = ∂un (det # )f,

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where the last equation holds because (locally) f is a holomorphic function on un,C = un + i un = gln (C). Thus, in order to prove the lemma, we may assume that g = un . Harish–Chandra’s theorem on the radial component of an invariant diﬀerential operator, [8, 7.A.2.9], implies that our lemma will follow as soon as we show that (n − 1)! 1 ∂((det |h)# )π(x) log(det(x)) = π(x) det(x)

(x ∈ hr ).

(3.3)

The equation (3.3) is equivalent to det(z) ∂z1 ∂z2 ... ∂zn (π(z) log(det(z))) = (n − 1)!π(z)

(z ∈ hC ),

(3.4)

where zj = ej (z), 1 ≤ j ≤ n. Let Sn denote the group of permutations of elements of the set {0, 1, 2, ... , (n − 1)}. Recall (Vandermonde), that σ(n−1) σ(n−2) π(z) = sgn(σ)z1 z2 ... znσ(0) (z ∈ hC ). σ∈Sn

A straightforward calculation shows that for γk = 0, 1, 2, ... , z1 z2 ... zn ∂z1 ∂z2 ... ∂zn (z1γ1 z2γ2 ... znγn log(z1 z2 ... zn )) n γ1 γ2 γn γ1 γ2 ... γk ... γn z1γ1 z2γ2 ... znγn , = γ1 γ2 ... γn z1 z2 ... zn log(z1 z2 ... zn ) + k=1

where the hat, γk , indicates that γk is missing in the product. Hence, the left hand side of (3.4) coincides with σ(n−1) σ(n−2) sgn(σ) σ(n − 1)σ(n − 2) ... σ(0)z1 z2 ... znσ(0) log(z1 z2 ...zn ) σ∈Sn n

+

σ(n − 1)σ(n − 2) ... σ (n −

σ(n−1) σ(n−2) k) ... σ(0)z1 z2 ... znσ(0)

k=1

=

σ(n−1) σ(n−2) z2 ... znσ(0)

sgn(σ)(n − 1)! z1

= (n − 1)!π(z),

σ∈Sn

which coincides with the right hand side of (3.4).

Lemma 3.5. Let u = (deth)# ∈ Sym(hC ), and let F (z) = log ◦ det(z) · π(z), z ∈ hC . Then for ψ ∈ S(g)

(p + q − 1)! ψ(x) dx g det(x + i0)

p 1 t = lim F (c−1 Sm (x + iYSm ))∂(cSm u) ψSm (x) dx. →0+ + (p − m)!(q − m)! h S , m=0 m

Proof. The limit log(det(x + i 0)) =

lim

{y→0, y∈g+ }

log(det(x + i y))

(x ∈ g)

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exists in the sense of distributions, [4, 3.1.15], and coincides with the indicated locally integrable function. Also, by (3.1),

(p + q − 1)! ψ(x) dx = g det(x + i0)

log(det(x + i 0))∂(det # )t ψ(x) dx. g

Thus the lemma follows from the Weyl integration formula and Harish-Chandra’s theorem on the radial component of an invariant diﬀerential operator, [8, 7.A.2.9]. Set

1 t ISm , = (F ◦ c−1 Sm (x + iYSm )∂(cSm u) ψSm (x)− (p − m)!(q − m)! h+S , (3.6) m −1 ∂(cSm u)(F ◦ cSm )(x + iYSm )ψSm (x)) dx, where u = (det |h)# , ψ ∈ S(gC ), and h+ Sm , is as in (1.5). In order to prove Theorem 1.5 it will suﬃce to show that p ISm , = 0. (3.7) lim →0

m=0

For a root α ∈ Φn (h) let (as in (2.5)) u=

un,α (iHα )n

(un,α ∈ Sym(ker α)).

(3.8)

n≥0

Lemma 3.9. Let us multiply the form (1.0) by a positive constant such that the norm of each iHα is 1. Then, with the above notation we have n−1

1 ISm , = ∂((iHαm )n−1−k )∂(un,αm ) + (p − m)!(q − m)! m=0 m=1 n≥1 k=0 hSm ,,0 −1 F (cSm (x + iYSm−1 ))i ∂((cαm iHαm )k )t ψSm (x)

−∂((cαm iHαm )k )t ψSm (x − cαm Hαm ) dx, p

p

˜ 1 (x) > ... > α ˜ m−1 (x) > , α ˜ m (x) = 0}. where h+ Sm ,,0 = {x ∈ hSm ; α Proof. Consider the integral (3.6). Suppose ﬁrst that supp ψSm ∩

ker(β ◦ c−1 Sm ) = ∅.

⊥ β∈Φn (h)∩Sm

Notice that cSm u =

n≥0

=

n≥0

cSm un,αm (cSm iHαm )n =

cαm un,αm (cαm iHαm )n

n≥0

(−i)n cαm un,αm (−cαm Hαm )n .

A. Daszkiewicz and T. Przebinda / Central European Journal of Mathematics 5(4) 2007 654–664 661

By applying (2.6) with V = hS , e = −cαm Hαm and the w replaced by cSm u, we see that ISm , 1 = (p − m)!(q − m)! n≥1 n−1

k=0

h+ Sm ,,0

∂((−cαm Hαm )n−1−k )(−i)n ∂(cSm un,αm )

c−1 Sm )(x

− cαm Hαm + iYSm )∂((−cαm Hαm )k )t ψSm (x − cαm Hαm ) dx n−1

1 =− ∂((iHαm )n−1−k )∂(un,αm ) (p − m)!(q − m)! n≥1 k=0 h+S ,,0

(F ◦

(3.10)

m

F (c−1 Sm (x

− cαm Hαm + iYSm ))i∂((cαm iHαm )k )t ψSm (x − cαm Hαm ) dx

Recall that, with S = Sm , for β ∈ Φn (h) ∩ S ⊥ we have the Harish-Chandra’s matching condition, [1, 3.1], ∂(v)ψS (x + 0iHβ ) − ∂(v)ψS (x − 0iHβ ) = i∂(cβ v)ψS∪{β} (x),

(3.11)

where β ◦ c−1 S (x) = 0, and x is not annihilated by any other non-compact imaginary root of hS . Suppose now that the support of ψS is disjoint with the set where α ˜ (x) = for all n α ∈ S. The we apply (2.7) with V = hS and e = iHβ for β ∈ Φ (h) ∩ S ⊥ , and the matching condition (3.11), to see that ISm , 1 = (p − m)!(q − m)! β n≥1 k=0 n−1

h+ Sm , ,

β◦c−1 Sm (x)=0

∂((iHβ )n−1−k )∂(cSm un,β )

(3.12)

k t (F ◦ c−1 Sm )(x + iYSm )i∂((cSm ∪{β} iHβ ) ) ψSm ∪{β} (x) dx.

Notice also that, by deﬁnition (1.3), −Hαm+1 + iYSm+1 = i(iHαm+1 + YSm+1 ) = iYSm .

(3.13)

From the deﬁnition (3.6) we deduce that the summands corresponding to various β in ⊥ has (p − m)(q − m) elements, (3.12) are all equal to each other. Since the set Φn (h) ∩ Sm (3.12) is equal to ISm , 1 = (p − m − 1)!(q − m − 1)! n≥1 k=0 n−1

h+ S

m+1

∂((iHαm+1 )n−1−k )∂(un,αm+1 )

(3.14)

, ,0

k t F (c−1 Sm+1 (x − cαm+1 Hαm+1 + iYSm+1 ))i∂((cSm+1 iHαm+1 ) ) ψSm+1 (x) dx.

The integral (3.14) is non-zero only if m < p. (Otherwise there are no non-compact roots β.) Hence, the lemma follows (via partition of unity) by adding (3.10) and (3.14) and grouping the terms with the same ψSm .

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Proof of Theorem 1.5 Notice that (det |h) = #

p+q

Hj .

(3.15)

j=1

Also, for 1 ≤ m ≤ p, Hm Hp+m = 14 ((Hm + Hp+m )2 − (Hm − Hp+m )2 ). Since Hm − Hp+m = Hαm , the decomposition (3.8) can be rewritten as 1 1 # 2 (det |h) = (Hm + Hp+m) Hj + Hj (iHαm )2 . (3.16) 4 4 j =m,p+m j =m,p+m Therefore (3.9) shows that p

ISm ,

m=0 p

1 = 4(p − m)!(q − m)! k=0 m=1 1

∂((iHαm )1−k )∂(

Hj )

(3.17)

−1 In the formula (3.17), c−1 Sm (x + iYSm−1 ) = cSm−1 (x) + iYSm−1 . Hence, m−1 p p+q c−1 (zj Hj + z j Hp+j ) + zj Hj + zj Hj , Sm (x + iYSm−1 ) = i

(3.18)

F (c−1 Sm (x

h+ Sm ,,0

j =m,p+m

k t

+ iYSm−1 ))i(∂((cαm iHαm ) ) ψSm (x)

− ∂((cαm iHαm )k )t ψSm (x − cαm Hαm )) dx.

j=1

where

j=m

j=p+m

Im zj > for 1 ≤ j ≤ m − 1 Im zj = for m ≤ j ≤ p

(3.19)

Im zj = − for p + m ≤ j ≤ p + q. and Re zm = Re zp+m .

(3.20)

Consider the functions Hj )F (z) = ∂z1 ∂z2 ... ∂zm ... ∂zm+p ... ∂zp+q log(z1 z2 ... zp+q ), ∂(

(3.21)

j =m,p+m

and

∂(Hm − Hp+m )∂(

Hj )F (z)

j =m,p+m

= (∂zm − ∂zp+m )∂z1 ∂z2 ... ∂zm ... ∂zm+p ... ∂zp+q log(z1 z2 ... zp+q ).

(3.22)

The function (3.21) is a linear combination of terms log(z1 z2 ... zp+q ) · polynomial(z1 , z2 , ... , zp+q ), polynomial(z1 , z2 , ... , zp+q ), zm zp+m · (zj − zk ), zl 1≤j
(l = m, l = p + m)

(3.23)

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The function (3.22) is a linear combination of terms log(z1 z2 ... zp+q ) · polynomial(z1 , z2 , ... , zp+q ), polynomial(z1 , z2 , ... , zp+q ), zp+m · (zj − zk ), zm 1≤j
(3.24)

where a = m, or a = p + m; l = m, and l = p + m. By combining (3.18) - (3.24) we see that each of the functions (3.21), (3.22) when evaluated at c−1 Sm (x + iYSm−1 ), M can be dominated by a constant multiple of | log(|x|)| + |x| , for M ≥ 0 large enough, independently of 0 < < 1. Hence, by dominated convergence, (3.7) holds, and we are done.

4

A sketch of a proof of the Theorem 1.9

If the support of the test function Ψ (see the statement of Theorem 1.9) is disjoint with the singular support of the distribution (1.6), then the limit formula (1.9) holds, for trivial reasons. Consider a semisimple point h in the singular support of the distribution (1.6). We may, and shall, assume that h belongs to one of the Cartan subgroups HS , (see (1.7)). Let Z = Gh denote the centralizer of h in G. Let U ⊆ Z be a connected, completely invariant open neighborhood of h, contained in the set of regular elements of Z, (see [7]). Since the sets of the form G · U = {gug −1; g ∈ G, u ∈ U} cover the singular support of the distribution (1.6), we may assume that Ψ ∈ Cc∞ (G · U). The group G acts on the space V = Cp+q as the group of isometries of the hermitian form (u, v) = v t Ip,q u, (u, v ∈ V ). Let V = V1 ⊕ V2 ⊕ ... be a decomposition of V into the direct sum of eigenspaces for h. It is easy to see that the restriction of the form ( , ) to a Vj is either non-degenerate or zero. Hence, the group Z is isomorphic to a Cartesian product GLn1 (C) × ...GLns (C) × Up1 ,q1 × ...Upc ,qc , by restriction. Moreover, there is only one factor (necessarily Upj ,qj ) in this product, such that the restriction of the distribution (1.6) to U is singular on it. Thus, by descent, we may assume that h = 1. Let U0 ⊆ g be a completely invariant open neighborhood of 0 ∈ g, such that x → exp(x) is an analytic diﬀeomorphism of U0 onto U = exp(U0 ). Then, with the standard normalization of the Haar measure on G, we have

exp(−ad(x) − 1 Ψ(g) dg = Ψ(exp(x)) det dx. −ad(x) G g

The function j(x) = det

exp(−ad(x) − 1 −ad(x)

(x ∈ U0 )

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is invariant, analytic and positive. Hence the positive square root j 1/2 (x), x ∈ U0 , is well deﬁned and extends to a holomorphic function in a neighborhood of U0 in gC . Thus

1 1 Ψ(g) dg = j 1/2 (x) · j 1/2 (x)Ψ(exp(x)) dx det(1 − g · 1) det(1 − exp(x) · 1) G g

1 x 1/2 det j (x) j 1/2 (x)Ψ(exp(x)) dx, = 1 − exp(x) g det(x + i0) where the function in brackets is invariant and holomorphic in a neighborhood of the support of the test function Ψ(exp(x)). A slight modiﬁcation of the proof of Theorem 1.5 shows that this theorem holds with f (x) 1 replaced by det(x+i0) , where f is any invariant, holomorphic function in a the det(x+i0) neighborhood of the support of the test function ψ. A straightforward application of the limit formula 1.5 completes the proof.

Acknowledgements Research of the ﬁrst author was partially supported by KBN grant 2P03A00611. Research of the second author was partially supported by NSF grant DMS 9622610 and NSA grant DMA 904-96-1-0023. Both authors were partially supported by NATO Linkage Grant OUTR.LG 951526

References [1] A. Bouaziz: “Int´egrales orbitales sur les alg`ebres de Lie r´eductives”, Invent. Math., Vol. 115, (1994), pp. 163-–207. [2] A. Daszkiewicz and T. Przebinda: “The oscillator character formula, for isometry groups of split forms in deep stable range”, Invent. Math., Vol. 123, (1996), pp. 349—376. [3] R. Howe: “Transcending classical invariant theory”, J. Amer. Math. Soc., Vol. 2, (1989), pp. 535-–552. [4] L. H¨ormander: The analysis of linear partial diﬀerential operators, I, Springer Verlag, Berlin, 1983. [5] T. Przebinda: “A Cauchy Harish–Chandra integral, for a real reductive dual pair”, Invent. Math., Vol. 141, (2000), pp. 299—363. [6] W. Schmid: “On the characters of the discrete series. The Hermitian symmetric case”, Invent. Math., Vol. 30, (1975), pp. 47—144. [7] V.S. Varadarajan: Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer Verlag, Berlin-New York, 1977. [8] N. Wallach: Real Reductive Groups, I, Pure and Applied Mathematics, 132, Academic Press, Inc., Boston, MA, 1988.

DOI: 10.2478/s11533-007-0033-1 Research article CEJM 5(4) 2007 665–685

On components of the Auslander-Reiten quiver of trivial extensions of 2-fundamental algebras which contain projective modules Alicja Jaworska∗ Faculty of Mathematics and Computer Science Nicholaus Copernicus University 87-100 Toru´ n, Poland

Received 10 May 2007; accepted 21 August 2007 Abstract: Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Trivial extension, Minimal 2-fundamental algebra, Auslander-Reiten quiver, Generalized standard components. MSC (2000): 16G20, 16G70

1

Introduction

Let K be an algebraically closed ﬁeld and A be a ﬁnite -dimensional associative K-algebra with an identity. We shall also assume that A is basic and connected. For an algebra A we shall denote by mod(A) the category of the left ﬁnite-dimensional A-modules. In order to characterize mod(A) the Auslander-Reiten quiver ΓA is usually examined. But the description of its structure can be problematic even for tame representation type algebras. One of the classes of algebras whose Auslander-Reiten quiver has not been known yet is a class of tame algebras which are not of polynomial growth. Several authors have studied connected components of string algebras (special biserial algebras) [3, 5–7]. The whole structure of the Auslander-Reiten quiver of such algebras is unknown in the case of ∗

E-mail: [email protected]

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A. Jaworska / Central European Journal of Mathematics 5(4) 2007 665–685

nonpolynomial growth algebras. In particular, we do not know all components containing projective vertices. A class of minimal 2-fundamental algebras was introduced in [7] . The AuslanderReiten quiver of their trivial extension was considered in [6]. Some results on starting and ending components of nonpolynomial growth algebras were given. We continue these examinations. Let A be a minimal 2-fundamental algebra of the certain form, as it was in [6]. Our objective is to describe the connected components of ΓT (A) which contain indecomposable projective T (A)-modules where T (A) is the trivial extension algebra of A. The algebra T (A) is special biserial which is not of polynomial growth (see [7]). Using unsophisticated but powerful methods we aim at providing the most important information. We shall study the structure of such components, their number and relations betweeen them. The question whether they are generalized standard or not is also examined. All obtained properties allow us to construct a certain full subgraph of the components graph GA . This is the ﬁrst step in understanding the structure of GA . The main result of the paper is given in Theorem, which states that all indecomposable projective T (A)-modules, where A is an algebra from one of the families under discussion, lie in nonstable tubes or in components of pseudotype ZA∞ ∞. The article is organized in the following way. In Section 1 we present all deﬁnitions and facts applied in our examinations. Section 2 is devoted to the proof of Theorem. In the ﬁnal Section 3 the more precise description of components introduced in Theorem is given. Because of some combinatorial complexity we will often give detailed arguments in selected cases only, as long as other cases are analogous. Throughout the paper, we shall use freely the basic results of the representation theory which can be found in [1].

2

Preliminaries

Let A be a ﬁnite-dimensional basic K-algebra. It comes from Gabriel that A can be associated to a bound quiver (QA , IA ) in such a way that A ∼ = KQA /IA , where KQA is a path algebra of the quiver QA and IA is an admissible two-sided ideal in the sense of Gabriel [4]. If QA does not have any oriented cycles we call A triangular. An algebra A is said to be special biserial [10] if there exists a bound quiver (QA , IA ) with A ∼ = KQA /IA such that: (1) Every vertex of QA is a source of at most two arrows. (2) Every vertex of QA is a target of at most two arrows. (3) For every arrow α in QA there exists at most one arrow β (resp. γ) such that αβ ∈ IA (resp. γα ∈ IA ). To every vertex x of Q we attach a trivial walk denoted by e(x). Recall that a nontrivial walk in a quiver Q is a formal composition of arrows and their formal inverses. A nontrivial walk ω in the bound quiver (Q, I) is a walk in the quiver Q which does not contain any subpath υ such that υ or υ −1 belongs to I.

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Let A ∼ = KQA /IA be a special biserial algebra, where the bound quiver (QA , IA ) satisﬁes the conditions from the deﬁnition. Then A is said to be a string algebra if IA is generated only by paths. There is a full classiﬁcation of ﬁnite dimensional indecomposable left modules over a string algebra. For every such a module X we have two possibilities. The ﬁrst one refers to X induced by a walk ω satisfying: ω = ω1 αα−1 ω2 , ω = ω1 β −1 βω2 and ω does not contain a subwalk of the form υ or υ −1 with υ ∈ IA . In this case we denote X by X(ω). The second possibility is that there is a small closed walk ν, an integer n ≥ 1 and an element λ ∈ K ∗ such that X is uniquely determined (up to isomorphism) by these data. In this case we use the notation X(ν, n, λ). Recall that a closed walk ν in a bound quiver (Q, I) is called small if it is not of the form un for any integer n ≥ 2, and for any positive integer m the walk ν m does not contain αα−1 or α−1 α, and it is not of the form ν m = ν1 uν2 , where u is a path (respectively its inverse) such that either u (respectively u−1) lies in I, or u − v (respectively u−1 − v) belongs to I for some path v. For the ﬁrst type of modules we have the following algorithm for computing the Auslander-Reiten sequences. It was given by Skowro´ nski and Waschb¨ usch in [10]. −1 −1 −1 −1 If ω = δ1,s1 . . . δ1,1 δ2,1 . . . δ2,s2 . . . δr−1,sr−1 . . . δr−1,1 δr,1 . . . δr,sr is a walk in the bound quiver −1 −1 (QA , IA ), where δj,t is an arrow in QA and δ1,s1 . . . δ1,1 or δr,1 . . . δr,s may be trivial, then r −1 −1 −1 −1 −1 ωR = δ1,s1 . . . δ1,1 δ2,1 . . . δ2,s2 . . . δr−1,sr−1 . . . δr−1,1 δr,1 . . . δr,sr δr,sr +1 δr+1,sr+1 . . . δr+1,1 , where δr+1,sr+1 . . . δr+1,1 ∈ IA is maximal nonzero path, provided that such a walk exists. If the −1 −1 −1 walk δr,s δ . . . δr+1,1 does not exist then ωR = δ1,s1 . . . δ1,1 δ2,1 . . . δ2,s . . . δr−1,sr−1 . . . r +1 r+1,sr+1 2 δr−1,2 . Using the same rules on the other end of the walk ω we obtain ωL . The composition of these constructions gives us the walk ωRL . Moreover, for X(ω) being a noninjective module we have the following Auslander-Reiten sequence in mod(A): 0 → X(w) → X(wR ) ⊕ X(wL ) → X(wRL ) → 0. m -separated if for any two subquivers We say that a triangular string algebra A is A m such that KQ ∩ IA = 0 = KQ ∩ IA we have Q ∩ Q = ∅, Q , Q in QA of type A 0 0 where Q0 , Q0 denotes the set of vertices of Q , Q , respectively. m -separated algebra A is said to be 2-fundamental According to [7] a triangular string A if A ∼ = KQA /IA is connected and the following conditions are satisﬁed: m in (QA , IA ) such that (1) There exist exactly two full subquivers Q1 , Q2 of type A ¯ A obtained from QA by they are disjoint, KQj ∩ IA = 0, j = 1, 2, and the quiver Q removing the arrows from Qj , j = 1, 2, and identifying the vertices of Qj with vertex 0j , j = 1, 2, is a tree. ¯ A , j = 1, 2, there exists either a maximal path v in Q ¯A (2) For any vertex 0j in Q ¯ A ending at 0j such that u ∈ IA . starting at 0j such that v ∈ IA , or a maximal path u in Q If v (treated as a path in QA ) starts at some vertex x in Qj that is the ending point of two maximal paths v1 , v2 in Qj then v1 v ∈ IA or v2 v ∈ IA . If u (treated as a path in QA ) ends at some vertex y in Qj that is the starting point of two maximal paths u1 , u2 in Qj then uu1 ∈ IA or uu2 ∈ IA . A 2-fundamental algebra A is said to be minimal if the graph obtained from the quiver

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¯ A by forgetting orientations of the arrows is of the form 01 ··· 02 . Q For any ﬁnite dimensional K-algebra A we may consider its Auslander-Reiten quiver denoted by ΓA . Recall that the vertices of it are the isoclasses [X] of the indecomposable ﬁnite dimensional left A-modules X. The number of arrows from [X] to [Y ] is equal to dimK (Irr(X, Y )/ Irr2 (X, Y )), where Irr(mod(A)) is the two-sided ideal in mod(A) generated by the irreducible morphisms. In this article we will not distinguish between indecomposable A-modules and their isoclasses. A component of ΓA will always mean a connected component. For a given connected and acyclic quiver Q = (Q0 , Q1 ) we can deﬁne a translation quiver (ZQ, τ ) in the following way. The set of vertices of ZQ is equal to Z × Q0 . The set of arrows is completely determined by Q1 : for each arrow α : x −→ y from Q1 and z ∈ Z there exist two arrows in (ZQ)1 : (z, α) : (z, x) −→ (z, y) and (z, α ) : (z + 1, y) −→ (z, x). The translation is given as follows: τ (z, x) = (z + 1, x) for all (z, x) ∈ (ZQ)0 . A component C is said to be of ZQ -type for a certain quiver Q if there is an isomorphism of translation quiver (ZQ, τ ) and (C, τA ), where τA is the Auslander-Reiten translation. To introduce a deﬁnition of a tube we need to consider a structure of 2-dimensional complex on ΓA . The vertices are 0-dimensional simplices, the arrows and pairs of the form (τA x, x) are 1-dimensional simplices whereas 2-dimensional are given for each α : x −→ y by a triple (τ y, x, y). A component T of ΓA is called a tube if its geometric realization is homeomorphic with + 1 1 S × R+ 0 , where S is the one-dimensional sphere and R0 is the set of the nonnegative real numbers. We shall denote by A∞ an inﬁnite quiver of the form • −→ • −→ • −→ . . .. We call a component T a stable tube of rank r if it is isomorphic as a translation quiver to ZA∞ /(τ r ). A component C of ΓA is said to be a pseudotube [9] if after removing all projectiveinjective vertices and all arrows which have sources or targets in these vertices we obtain a tube. We call a component to be of ZQ-pseudotype if after removing all projective and injective vertices together with their τA -orbits and arrows which have sources or targets in these orbits we obtain a component of type ZQ. In other words, a component is of ZQ-pseudotype if its stable quiver is isomorphic as a translation quiver to ZQ. For the purpose of this article we shall assume that pseudotypes components contain at least one indecomposable projective A-module. By rad∞ (mod(A)) we denote the intersection of all positive powers of the Jacobson radical rad(mod(A)) of the category mod(A). For C and D being components of the Auslander-Reiten quiver of an algebra A we shall claim that HomA (C, D) = 0 whenever there exist modules X ∈ C and Y ∈ D such that there is a nonzero homomorphism f : X −→ Y . Of course f ∈ rad∞ (X, Y ). With ΓA we may associate an oriented graph GA which shows the relation among its components. The vertices of GA are the components. An arrow from C to D belongs to

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GA if and only if Hom(C, D) = 0. We call GA the components graph. A component C of Auslander-Reiten quiver is said to be generalized standard if ∞ rad (X, Y ) = 0 for all A-modules X, Y whose isoclasses belong to C. If a component is not generalized standard then its graphic representation in GA has a loop. The above deﬁnition is due to Skowro´ nski [8].

3

Main results

Let A be a ﬁnite-dimensional K-algebra. Recall that the trivial extension T (A) of A by its minimal injective cogenerator bimodule D(A) = HomK (A, K) is the algebra which has an additive structure of the group A ⊕ D(A), and multiplication deﬁned by the formula: (a, f )(b, g) = (ab, ag+f b) for a, b ∈ A and f, g ∈ D(A). Notice that T (A) is a self-injective ﬁnite-dimensional K-algebra. According to [6] we consider certain families of minimal 2-fundamental algebras. They are of the form Ai = KQ(i) /I(i) for i ∈ {1, 2, 3}, where QA(i) are given in the following way: Q(1)

•@ ~~ @@@ ~ @ ~~ β1 @@ ~ ~

• α2

•

.

.. .

αn−1

δt−1

• @@

~ @@ δt ~~ αn @@@ ~~~ ~

•

n, m, t, s, l, r, q, p ≥ 1

•

.

•

• 99

.

α2

βr

99 9 δ1 ε1 999 9

•

.. .

.. . βm

• β2

β2

.

•@ ~~ @@@ ~ @ ~~ α1 @@ ~ ~ β1

α1

...

99 · 99 εl 9 εs 99 99

•

·

·

ε1 δ1

.

.. .

.

δp−1

αq−1

• @@

@@ ~~ @@ ~~ ~ δp @ ~ ~ αq

•

•

670

A. Jaworska / Central European Journal of Mathematics 5(4) 2007 665–685

Q(2)

•C ~~ CCCC ~ C ~~ β1 CC ~ ~ !

•@ ~~ @@@ ~ @ ~~ α1 @@ ~ ~ β1

α1

• α2

•

β2

β2

.. .

.. .

αn−1

• ;;

;; ;; βm αn ; ; • 77 77 77 ε1 77 77

·

.9

· 99 εl 99 εs 99 99

α2

• ~ CCC CC ~~ ~ C ~ β1 CC ~ ~ !

•

.. .

βm−1

• @@

·

αq−1

•

66 66 αq δp 66 66

•@ xx @@ @@ xx x xx α1 @@ x| x

• β2

•

α2

.. . βr−1

αq−1

• FF

..

·

.9

· 99 εl 99 εs 99 99

.. .

•

{ @@ βm {{{ @ { αn @@ {{ }{ • 99 99 99 ε1 99 9

• 66

•

.. .

αn−1

·

β1

β2

..

•

α1

•

·

δp−1

n, m, s, l, r, p, q ≥ 1 Q(3)

•

δ1 . ε1

·

.. .

•

..

•

α2

βr

βm−1

•

·

~ FF ~~ FF ~ F ~ βr FF ~ " ~ • ε1 αq

•

•

and I(i) is the two-sided ideal generated by βm ε1 and βr ε1 , n, m, s, l, r, q ≥ 1. Moreover, we know (see [6]; Lemma 2.1.) that trivial extensions of these minimal T 2-fundamental algebras are special biserial. They are of the form T (Ai ) ∼ , = KQT(i) /I(i) where:

A. Jaworska / Central European Journal of Mathematics 5(4) 2007 665–685

QT(1)

•L R @ ~~ @@@ ~ @ ~~ β1 @@ ~ ~

α2

•

.

• β2

β2

c

d

αn−1

99 9 δ1 ε1 999 9

..

a

.9

δt−1

• @@

• ~

· 99 εl 99 εs 99 99

·

·

H ε1 δ1 b

. e

z

.. .

.

δp−1

αq−1

•

@@ ~ @@ δ~t ~~ @ ~~

αn

.

•

• 9e 9 .. .

.

α2

βr

βm

• @@

• ~

αq ~~ @@ @@ ~ @ ~~~

δp

•

•

n, m, t, s, l, r, q, p ≥ 1 QT(2)

•J eC ~~ CCCC ~ C ~~ β1 CC ~ ~ !

α2

•

.. .

βm−1

• ;;

.. .

d

αn−1

• β2

β2

;; ;; βm αn ; ; • 77 77 77 ε1 77 77

•

α2

·

.. . βr

6• ε1 δ1 .

a

•

..

· e

z

·

..

b

.9

· 99 εl 99 εs 99 99 •

n, m, s, l, r, p, q ≥ 1

•L R @ ~~ @@@ ~ @ ~~ α1 @@ ~ ~ β1

α1

•

•

.. .

.. .

.

•L R @ ~~ @@@ ~ @ ~~ α1 @@ ~ ~ β1

α1

•

671

·

δp−1

·

αq−1

• 66

•

66 66 αq δp 66 66

•

672

A. Jaworska / Central European Journal of Mathematics 5(4) 2007 665–685

QT(3)

< •J C ~~ CCCC ~ C ~~ β1 CC ~~ !

• α2

•

β2

.. .

β2

d βm−1

• { @@ βm {{{ αn @@@ {{{ }{ • 99 99 99 ε1 99 9

• @@

a

•

.. .

αn−1

x6 •J @@@ xx @@ x x xx α1 @@ |xx β1

α1

...

α2

.. . βr−1

b

αq−1

·

·

.. .

e

• FF

99 · 99 εl 9 εs 99 99

•

FF αq ~~~ FF F ~~ βr FF ~ " ~ • ε1

•

•

n, m, s, l, r, q ≥ 1, where T is generated by the listed relations: I(1) αj . . . αn cα1 . . . αj ,

α1 . . . αn c − β1 . . . βm δ1 . . . δt d,

αn d,

dα1 ,

βx . . . βm δ1 . . . δt dβ1 . . . βx ,

cα1 . . . αn − dβ1 . . . βm δ1 . . . δt ,

βm ε1 ,

cβ1 ,

δy . . . δt dβ1 . . . βm δ1 . . . δy ,

ε1 . . . εl b − δ1 . . . δp eβ1 . . . βr ,

εs b,

εl a,

εu . . . εs aε1 . . . εu ,

β1 . . . βr δ1 . . . δp e − α1 . . . αq z,

aδ1 ,

bδ1 ,

εv . . . εl bε1 . . . εv ,

aε1 . . . εs − bε1 . . . εl ,

δt c,

δp z,

βw . . . βr δ1 . . . δp eβ1 . . . βw ,

δ1 . . . δt dβ1 . . . βm − ε1 . . . εs a,

αq e,

eα1 ,

δj 1 . . . δp eβ1 . . . βr δ1 . . . δj 1 ,

, eβ1 . . . βr δ1 . . . δp − zα1 . . . αq , βr ε1 ,

zβ1 ,

αj 2 . . . αq zα1 . . . αj 2 ; T I(2) has generators of the form:

βj1 . . . βm dβ1 . . . βj1 ,

α1 . . . αn ε1 . . . εs a − β1 . . . βm d,

αn d,

dα1 ,

αj2 . . . αn ε1 . . . εs aα1 . . . αj2 ,

ε1 . . . εs aα1 . . . αn − dβ1 . . . βm ,

aβ1 ,

zβ1 ,

εj3 . . . εs aα1 . . . αn ε1 . . . εj3 ,

aα1 . . . αn ε1 . . . εs − bε1 . . . εl ,

εs b,

εl a,

βj 4 . . . βr δ1 . . . δp eβ1 . . . βj 4 ,

eβ1 . . . βr δ1 . . . δp − zα1 . . . αq ,

eα1 ,

αq e,

δj 5 . . . δp eβ1 . . . βr δ1 . . . δj 5 ,

ε1 . . . εl b − δ1 . . . δp eβ1 . . . βr ,

βr ε1 , βm ε1 ,

αj 6 . . . αq zα1 . . . αj 6 ,

β1 . . . βr δ1 . . . δp e − α1 . . . αq z,

bδ1 ,

εj . . . εl bε1 . . . εj ;

δp z,

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T and I(3) is generated by:

αj1 . . . αn ε1 . . . εs aα1 . . . αj1 ,

α1 . . . αn ε1 . . . εs a − β1 . . . βm d,

αn d,

αq e,

εj2 . . . εs aα1 . . . αn ε1 . . . εj2 ,

aα1 . . . αn ε1 . . . εs − bα1 . . . αq ε1 . . . εl ,

aβ1 ,

dα1 ,

dβ1 . . . βm − ε1 . . . εs aα1 . . . αn ,

βm ε1 ,

βr ε1 ,

εj5 . . . εl bα1 . . . αq ε1 . . . εj5 ,

eβ1 . . . βr − ε1 . . . εl bα1 . . . αq ,

eα1 ,

bβ1 ,

βj 6 . . . βr eβ1 . . . βj 6 ,

α1 . . . αq ε1 . . . εl b − β1 . . . βr e,

εl a,

εs b,

βj3 . . . βm dβ1 . . . βj3 ,

αj 4 . . . αq ε1 . . . εl bα1 . . . αj 4 . In [2] the reduced algebras are studied. In the case of self-injective special biserial algebra A the reduced algebra Ard is simply equal to A/ soc(A), where soc(A) denotes the socle of the A-module A. We will base our examination of the structure of the components of ΓT (Ai ) which contain a projective module on the following lemma. Lemma 3.1. Let A be a self-injective special biserial algebra and Ard = A/ soc(A). Then all indecomposable A-modules which are not Ard -modules are non-serial projectiveinjective ones. This comes from Auslander-Reiten (see its generalization in [2]). Note that Ard is a string algebra. Moreover, each projective-injective A-module P occurs in the AuslanderReiten sequence of the form: / rad(P ) / P ⊕ rad(P )/ soc(P ) / P/ soc(P ) /0 0 where rad denotes the Jacobson radical of a module. It allows us to use the Skowro´ nskiWaschb¨ usch algorithm for building ΓT (Ai ) for i = 1, 2, 3. Definition 3.2. Let A be a string algebra and X = X(ω) an indecomposable module from the component C of Auslander-Reiten quiver of the algebra A. We call a sequence of indecomposable modules {X(ωLη )}η∈Z to be a left diagonal in the component C. A sequence {X(ωRη )}η∈Z is called a right diagonal in C. A diagonal is said to be inﬁnite iﬀ all its terms are diﬀerent from the zero module. Notice that in a component of ZA∞ ∞ -type each pair ({X(ωLη )}η∈Z , {X(νRη )}η∈Z ) of left and respectively right diagonal has a common term. Moreover, if there exists an inﬁnite left or right diagonal in a component C of ΓA which does not contain any projective or injective modules then C is of ZA∞ ∞ -type, which follows from [3]. Let us now introduce some useful notation. By P (z) we denote an indecomposable projective T (A)i -module in a vertex z. Here z stands for an arbitrary symbol. We shall denote the following vertices: s(α1 ) by 1, s(α1 ) by 1 , t(αn ) by 2, t(αq ) by 2 , t(εs ) = t(εl ) by 3, 4 := s(ε1 ) if s(ε1 ) = t(αn ), and 4 := s(ε1 ) if s(ε1 ) = t(αq ). We shall denote a component of ΓT (Ai ) which contains a projective-injective T (Ai )module P (j) as Cji for a well deﬁned j from {1, 1, 2, 2, 3, 4, 4} .

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Lemma 3.3. For i = 1, 2, 3 the indecomposable projective T (Ai )-modules P (1), P (1), P (2), P (2), P (3), and P (4), P (4) lie in components of pseudotype ZA∞ ∞ of the AuslanderReiten quiver of T (Ai ). Proof. We will examine only the case of i = 1 since other cases are similar. (a) The component C11 : The projective module P (1) occurs in mod T (A1 ) in the following Auslander-Reiten sequence: / X(α ...α cd−1 δ −1 ...δ −1 β −1 ...β −1 ) / X(δ −1 ...δ −1 β −1 ...β −1 ) ⊕ P (1) ⊕ X(α ...α ) 0 2 n 2 n t t m m 1 2 1 1 / X(δ −1 ...δ −1 β −1 ...β −1 α ...α ) / 0 where rad P (1) 1 n t 1 1 m −1 ...β2−1 ) is an injective T (A1 )rd -module and P (1)/ soc P (1) = = X(α2 ...αn cd−1 δt−1 ...δ1−1 βm −1 X(δt−1 ...δ1−1 βm ...β1−1 α1 ...αn ) is a projective T (A1 )rd -module. The simple T (A1 )-module S(2) lies in the same component as P (1) because there is a chain of indecomposable modules and irreducible morphisms: / X(αn ) / X(αn−1 αn ) / ... / X(α2 ...αn ) / P (1)/ soc P (1) / S(2) / ... / X(α ...α δ −1 ...δ −1 β −1 ...β −1 α ...α ) / ... X(αn δt−1 ...δ −1 β −1 ...β −1 α1 ...αn ) 2 n t 1 n 1

1

m

1

m

1

in which the ηth term is simply of the form X(e(2)Lη ). It is not hard to see that paths are changing periodically, thus the above sequence is indeed well deﬁned for all η ∈ N. On the other hand, applying the Skowro´ nski-Waschb¨ usch algorithm we get a chain of irreducible morphisms X(e(2)L−μ )μ∈N . It is easy to see that this sequence is inﬁnite by examining the paths corresponding to modules in the sequence: / X(αn cd−1 cd−1 ) / ... / X(α2 ...αn cd−1 cd−1 ) / X(cd−1 ) / X(αn cd−1 ) ... / ... / X(α ...α cd−1 ) / S(2) . 2 n In that way we obtain a sequence D1L = {X(e(2)Lη )}η∈Z in C11 , where e(2)L0 = e(2). (b) The component C21 : We have the following Auslander-Reiten sequence: / X(α ...α δ −1 ...δ −1 β −1 ...β −1 ) / P (2) ⊕ X(δ −1 ...δ −1 β −1 ...β −1 ) ⊕ X(α1 ...αn−1 ) 0 1 n t t−1 1 1 m m 1 1 / X(δ −1 ...δ −1 β −1 ...β −1 dc−1 α1 ...αn−1 ) t−1

1

m

1

/0

−1 ...β1−1 ) where X(α1 ...αn δt−1 ...δ1−1 βm

= rad P (2) is an injective Trd -module and −1 −1 P (2)/ soc P (2) = X(δt−1 ...δ1−1 βm ...β1−1 dc−1 α1 ...αn−1 ) is a projective Trd -module. We also have a chain D2R := {X(e(1)Rμ )}μ∈Z of irreducible morphisms in C21 : / X(α ...α δ −1 ...δ −1 β −1 ...β −1 α ...α / ... / X(α ...α δ −1 ...δ −1 β −1 ...β −1 ) ... 1 n t 1 n−1 ) 1 n t m m 1 1 1 1 / X(α1 ...αn−1 ) / ... / X(α1 ) / S(1) / X(d−1 cα ...α / ... 1 n−1 ) −1 −1 −1 −1 / X(d cα ) / X(d c) / X(d cd cα ...α / ... 1 1 n−1 ) which is inﬁnite because of its obvious periodicity in both directions. Note that walks −1 ...β1−1 α1 are not critical in the sense of Geiss, d−1 cd−1 cα1 ...αn−1 and α1 ...αn δt−1 ...δ1−1 βm thus we could use [5, Prop. 2] to state that above sequence D2R is well deﬁned for all integers and that some periodicity must occur in this case.

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(c) The case of the component C11 and C21 : This follows immediately from (a) and (b). We denote by D1L the left diagonal in C11 which passes through S(2 ). The right diagonal in C21 which contains S(1 ) will be denoted by D2R . (d) The component C31 : Consider the following Auslander-Reiten sequence: / X(ε ...ε ε −1 ...ε −1 ) / X(ε −1 ...ε −1 ) ⊕ P (3) ⊕ X(ε ...ε / 0 1 s−1 ) 1 s l 1 l−1 1 /0. X(εl−1 −1 ...ε1 −1 ba−1 ε1 ...εs−1 ) There are also chains of irreducible morphisms of the form: / ... / X(ε −1 ) / S(4 ) and X(ε1 ...εs−1 ) / ...X(ε1 ) X(εl−1 −1 ...ε1 −1 ) 1 Like in the above cases it can be proved that there are inﬁnite sequences D3R := {X(e(4)Rη )}η∈Z and D3L := {X(e(4 )Lμ )}μ∈Z of modules in C31 . (e)The component C41 : The module P (4) occurs in the following Auslander-Reiten sequence:

0

/ X(δ

−1 −1 −1 / X(ε−1 ...ε−1 ) ⊕ P (4) 2 ...δt dβ1 ...βm a εs ...ε2 ) s 2 / X(ε−1 ...ε−1 δ ...δ dβ ...β 1 t 1 m−1 ) 1 s

/ S(4) .

⊕ X(δ2 ...δt dβ1 ...βm−1 ) /0

/ X(ε ) / ...) , There are also chains of irreducible morphisms of the form: S(3) l / X(ε ...ε / X(ε−1 ) / ... / X(ε−1 ...ε−1 ) . S(3) 2 l s s 2 Note that X(ε2 ...εl ) is a middle term of the Auslander-Reiten sequence given below: / X(ε ...ε bβ −1 ...β −1 e−1 δ −1 ...δ −1 ) 0 2 r 1 p 2 l −1 −1 −1 −1 e δp ...δ2 −1 ) r−1 ...β1 / X(β −1 ...β −1 e−1 δ −1 ...δ −1 ε ...ε ) /0. r−1 1 p 1 1 l

/ X(ε ...ε ) ⊕ P (4 ) ⊕ X(β 2

l

It means that C41 = C41 . Let us consider a chain of irreducible morphisms and modules X((δ2 ...δt d)Rμ ) where μ ∈ N: / X(δ ...δ dc−1 dβ ...β δ ...δ ) / ... / X(δ ...δ dc−1 dβ ...β ) / ... . X(δ2 ...δt d) 2 t 1 m 1 t−1 2 t 1 m It is not hard to see that it is indeed inﬁnite because it is cyclic for the sake of adding paths: X((δ2 ...δt d)Rt+l+r+p+s+m ) = X(δ2 ...δt dω) for an adequate path ω in QT (A1 ) . On the other hand X(δ2 ...δt dβ1 ...βm−1 ) = X((δ2 ...δt d)R−(m−1) ) and once more X((δ2 ...δt d)Rμ ) is deﬁned for all μ ∈ (−N) - one can ﬁnd a regularity appearing here. We denote {X((δ2 ...δt d)Rμ )}μ∈N by D4R . Now we are able to conclude that all Cj1 for j ∈ {1, 1, 2, 2, 3, 4} are of pseudotype ∞ ZA∞ . They cannot be tubes because they have inﬁnite diagonals and the only drawback in the structure of type ZA∞ ∞ is when a projective T (A1 )-module P occurs (see [3]) . Since they arise only in the following Auslander-Reiten sequences / rad P / rad P/ soc P ⊕ P / soc P /0 0 where rad P is injective, soc P a projective T (A1 )rd -module, it is obvious that the stable

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quiver of Cj1 over algebra T (A1 ) is indeed of type ZA∞ ∞.

Lemma 3.4. The components Cji for j ∈ {1, 1, 2, 2, 3, 4 }, j ∈ {1, 1, 2, 2, 3}, respectively, of the Auslander-Reiten quiver of algebra T (Ai ) for i = 1, 2, and i = 3, respectively, are pairwise distinct. Proof. As before, we will show the required conditions only for i = 1. We shall use the names of diagonals from the above proof. • The components C11 and C21 are distinct: Notice that C11 = C21 would mean that S(1), S(2) lie in the same component. For that reason the left diagonal D1L from C11 which contains S(2) would be intersected either by D2R = {X(e(1)Rη )}η∈Z or D2L := {X(e(1)Lη )}η∈Z . But this is impossible because of the form of modules from diagonals: Suppose there exist μ ∈ Z and η ∈ N such that X(e(2)Lμ ) ∼ = X(e(1)Rη ). This means −1 that e(2)Lμ contains a subpath of the form d c or equivalently c−1 d. It is easily seen that there is no μ ∈ Z fulﬁlling the above condition. Of course X(e(2)Lμ ) X(e(1)Rη ) for any η, μ ∈ (−N). Assume there exists μ ∈ N such that X(e(2)Lμ ) ∼ = X(e(1)Rη ) for some η ∈ (−N). But n + 1), where n is a for any μ ∈ N X(e(2)Lμ ) has in vertex 2 a k-space of dimension ( dimension of a k-linear space in vertex 1. For any η ∈ (−N) we have a reverse situation with X(e(1)Rη ). In the sequence D2L each nonsimple module lives on the path which contains an arrow βi (or its formal inverse) for at least one i from {1, ..., m}. Thus it is easily seen that there is no intersection of D2L and X(e(2)Lμ ) where μ is negative. For μ being positive a subpath neither of the form d−1 c nor a occurs in e(2)Lμ . Obviously S(1) X(e(2)Lμ ) and S(2) X(e(1)Rη ) for all integers μ, η . This implies that no module from D2L can appear in the sequence X(e(2)Lμ ) for μ ∈ Z. In that way we obtain that components C11 , C21 are diﬀerent. • The same arguments will show that C11 , C21 are distinct in ΓT (A1 ) . • The component C11 is diﬀerent both from C11 and C21 : Imagine that we present QT (A1 ) as the union of two quivers Q∪Q which are determined by arrows with their sources and targets in such a way that α1 , ..., αn , β1 , ..., βm , δ1 , ..., δt , ε1 , ..., εs , c, d, a are the only ones which belong to Q, whereas α1 , ..., αq , δ1 , ..., δp , β1 , ..., βr , ε1 , ..., εl , e, z, b belong to Q . Moreover, (Q ∩ Q )0 = {3}. Using the Skowro´ nski-Wachb¨ usch algorithm we build D1R - right diagonal in C11 which passes through S(2). In C21 we consider a left diagonal D2L containing a simple module S(1). Observe that all modules both from diagonal D1L and D2R lie on the paths containing only the arrows from Q, whereas D1L , D2R are completely determined by arrows from Q . Moreover, each module from D1R and D2L satisﬁes the condition supp X ∩ Q = ∅ where supp denotes the support of a module. In this situation it is by no means possible for D1L to intersect any of given diagonals from C11 and C21 .

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Thereby we can conclude that C11 = C11 and C11 = C21 . • The analogous consideration shows that C21 = C21 and C21 = C11 . • The components C31 and C11 are distinct: −1 Notice that in C31 all e(4)Rη for η ∈ N start with βm (because there is a nonzero ho/ X(e(4)Rη ) which is a composition of irreducible morphisms) momorphism f : S(4) and all e(4)Rη for η ∈ (−N) start with ε1 . Similarly for μ ∈ N all e(4 )Lμ end with βr and for μ ∈ (−N) they end with ε−1 1 . In this situation there is no intersection of the R L diagonals D3 and D3 with the left diagonal from C11 because all its nonsimple modules start with c, αn , ..., α2 , δt−1 and end in d−1 or αn ( equivalently start with d, αn−1 and end in c−1 , αn−1 , ..., α2−1 , δt ). Thus we obtain that C31 and C11 are distinct. • The components C31 and C21 are not the same: It is easy to see that there is no intersection of the left diagonal D3L from C31 with the right diagonal D2R from C21 because its modules except S(1), start with α1 or d ( end in α1−1 or −1 d−1 ) and end in β1−1 , αn−1 , ..., α1 , c ( start with β1 , αn−1 , ..., α1−1 , c−1 ). Taking into account that also D3R will not cross it, we conclude that C31 and C21 are diﬀerent. • Analogous arguments show that C31 is diﬀerent both from C21 and C11 . • The component C11 is diﬀerent from C41 : Let us now examine the case of C41 . All modules from the sequence D4R = {X((δ2 ...δt d)Rμ )}μ∈Z have the form X(δ2 ...δt dσ) for some walk σ in QT (A1 ) . All modules from the sequence D4L := {X((δ2 ...δt dβ1 ...βm−1 )Lη )}η∈Z lie on walks which end in βm−1 (respectively start in −1 βm−1 ). Such representations do not appear in the diagonal D1L . Thus C11 = C41 . • The component C41 is diﬀerent from C21 : We remind that all modules from the right diagonal D2R in C21 live on walks consisting only of arrows (and their formal inverse) from Q. Both considered sequences from C41 have ﬁnitely many terms which also satisfy this condition. One can easily check that none of them belong to D2R . It shows that C21 = C41 . • The components C41 and C11 are diﬀerent: To simplify the proof we shall introduce a new diagonal in C11 . We study the right diagonal in C11 which contains X(α2 ...αq ). All its modules lie on walks starting with α2 ...αq , whereas in already deﬁned diagonals in C41 no walk starts with α2 ...αq or ends with αq−1 ...α2−1 . We obtain that C41 = C11 . • The case of C21 and C41 : The right diagonal D2R in C21 satisﬁes the condition that supp Y ∩ Q = ∅ for any module Y belonging to it. This does not hold for diagonals from C41 . We conclude that these components are diﬀerent. • The components C41 and C31 are distinct: All modules in the sequence D4L lie on walks which end in dβ1 ...βm−1 (start in −1 ...β1−1 d−1 ). Such modules do not appear in D3R . The underlying walks of modules βm−1 from D4R start with δ2 ...δt d (end in d−1 δt−1 ...δ2−1 ). This condition is also satisﬁed by

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modules from D3R of the form X(e(4)R ) where = −(s+p+r+l)−k(t+m+s+p+r+l−1) for some k ∈ N. One can easily verify that these modules do not belong to D4R . Thus C41 = C31 . To summarize the above considerations, we can state that components C11 , C21 , C11 , C21 , C31 and C41 are pairwise distinct. Remark 3.5. Note that in the case of stable ZA∞ ∞ -components, veriﬁcation whether they are diﬀerent is equivalent to comparing the unique modules of minimal dimension from each component [5, Prop. 3]. Compare with our case of C3 . Lemma 3.6. For i = 1, 2, 3 the indecomposable projective T (Ai )-modules of the form P (s(ω)), where ω is an arrow from the set {α2 , ..., αn , α2 , ..., αq , β2 , ..., βm , β2 , ..., βr , δ2 , ..., δt , δ2 , ..., δp , ε2 , ..., εs , ε2 , ..., εl }, lie in nonstable tubes of the Auslander-Reiten quiver of the algebra T (Ai). Proof. We will prove only the case of i = 1 since the other are similar. nski - Waschb¨ usch Let us start with P (s(αj )) for j ∈ {2, ..., n}. Using the Skowro´ algorithm we get the following chain of irreducible morphisms: fj / X(αj+1 ...αn cα1 ...αj−2 ) / ... / X(αj+1 ...αn c) / (∗) X(αj+1 ...αn cα1 ...αj−1 ) / ... / X(αj+1 ...αn cd−1 c) / X(αj+1 ...αn cd−1 cα1 ...αn−1 ) / ... X(αj+1...αn cd−1 cd−1 cα1 ...αn−1 ) for all j ∈ {1, ...n}, where for j = 1 we adopt αj+1 ...αn dα1 ...αj−1 := α2 ...αn d and αj+1...αn dα1 ...αj−1 := dα1 ...αn−1 for j = n. Moreover, there is no other irreducible morphism starting at X(αj+1 ...αn dα1 ...αj−1 ) than fj in mod(T (A1 )/ soc T (A1 )). But for j > 1 we have the following Auslander-Reiten sequence: / X(αj+1 ...αn cα1 ...αj−1 ) / P (s(αj )) ⊕ X(αj+1 ...αn cα1 ...αj−2 ) / (∗∗) 0 /0 X(αj ...αn cα1 ...αj−2 ) in mod T (A1 ). Because there is also the Auslander-Reiten sequence of the form: / X(α2 ...αn c) / X(α ...α cd−1 cα ...α / X(cα1 ...αn−1 ) /0 (∗ ∗ ∗) 0 2 n 1 n−1 ) we obtain a circle of irreducible morphisms consisting of sequences (∗∗) considered for various j and (∗ ∗ ∗). They build a mouth of a tube T11 of rank n and inﬁnite rays of the form (∗). In the same way we show that indecomposable projective modules P (s(α2 )), ..., P (s(αq )) are situated on a mouth of a tube T21 of rank q. Now we can notice that there are the Auslander-Reiten sequences: / X(βk+1 ...βm δ1 ...δt dβ1 ...βk−1 ) / P (s(βk )) ⊕ X(βk+1 ...βm δ1 ...δt dβ1 ...βk−2 ) / 0 /0 X(βk ...βm δ1 ...δt dβ1 ...βk−2 ) in mod T (A1 ) for all k ∈ {2, ..., m}. For 0 in index we put β1 β0 := es(β1 ) and we adopt βm+1 βm to be et(βm ) . The indecomposable projective modules P (s(δi )) for i ∈ {2, ..., t} are the summands of the middle terms of the following Auslander-Reiten sequences: / X(δi+1 ...δt dβ1 ...βm δ1 ...δi−1 ) / P (s(δi )) ⊕ X(δi+1 ...δt dβ1 ...βm δ1 ...δi−2 ) / 0

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/0 X(δi ...δt dβ1 ...βm δ1 ...δi−2 ) where for i = t, i = 2 we deﬁne δi+1 ...δt dβ1 ...βm δ1 ...δi−1 , δi ...δt dβ1 ...βm δ1 ...δi−2 as dβ1 ...βm δ1 ...δt−1 and δ2 ...δt dβ1 ...βm respectively. The modules P (s(εj )) for j ∈ {2, ..., l} occur in the Auslander-Reiten sequences: / X(ε ...ε bε ...ε ) / P (s(ε )) ⊕ X(ε ...ε bε ...ε ) / X(ε ...ε bε ...ε ) 0 j+1 j−1 j j+1 j−2 j j−2 l 1 l 1 l 1 /0 where εj+1...εl bε1 ...εj−1 := bε1 ...εl−1 for j = l and εj ...εl bε1 ...εj−2 := ε2 ...εl b for j = 2. Let η := t + m + s + r + p + l − 1. Consider a chain {Bg }g∈{1,2,...η} of indecomposable modules: Bk = X(βk+1 ...βm δ1 ...δt dβ1 ...βk−1 ), Bm+i = X(δi+1 ...δt dβ1 ...βm δ1 ...δi−1 ), Bm+t+j = X(εj+1...εl bε1 ...εj−1 ), Bm+t+l = S(s(βr )), ..., S(s(β2 )), X(α1 ...αq ), S(s(δp )), ..., S(s(δ2 )), X(ε1 ...εl ), S(s(εs )), ..., S(s(ε2 )) = Bη , where 1 k m, 1 i t,1 j l. It is easy to see that in mod T (A1 ) the Auslander-Reiten sequences starting at B1 , Bm+1 , Bm+t+1 , Bm+t+l+1 , ..., Bη have only one middle term and that for all g ∈ {1, ..., η} the Auslander-Reiten sequence is: / Bg / Yg / Bg+1 / 0 (for g = η : 0 / Bη /Y /B /0) 0 1 where Yg is either indecomposable or has two direct summands one of which is a projectiveinjective indecomposable T (A1 )-module from the set B = {P (s(β2 )), ..., P (s(βm )), P (s(δ2 )), ..., P (s(δt )), P (s(ε2)), ..., P (s(εl ))}. Thus we obtain a cycle of indecomposable modules in the component T31 of ΓT (A1 ) such that each nonzero map f : Bg −→ Z for Z lying in the same component of ΓT (A1 ) and f = f1 f2 ...fh where h ∈ N, f1 , ..., fh are irreducible maps, needs to factorize through Yg or its direct summand. The argument that projective-injective T (A1 )-modules from the set B = {P (s(β2 )), ..., P (s(βr )),P (s(δ2 )), ..., P (s(δp )), P (s(ε2)), ...,P (s(εs ))} lie on a cycle in the component T41 is similar. Moreover, because of the structure of Auslander-Reiten sequences in which they occur it is obvious that T41 = T31 . Note that in this way we end a process of placing projective-injective indecomposable modules in connected components of ΓT (A1 ) (which has started in Lemma 3.3). Thus the set B contains all projective and injective indecomposable T (A1 )-modules which belong to this component and there are no more projective and injective indecomposable T (A1 )rd modules in T31 . Starting from any ω for which there exists g ∈ {1, ..., η} such that X(ω) = Bg one can build a sequence of diﬀerent paths {ω, ωR, ωR2 := (ωR )R , ...} in QT (A1 )rd . Observe that from some place a periodicity in the above sequence appears. For example: consider X(ω) = Bm+t . Then constructing new paths from ωRt+l+r+q we get endings which occur cyclically in the following forms: ez −1 α1 ...αq−1 ,ez −1 α1 ...αq−2 , ...,ez −1 α1 ,ez −1 . Thus an inﬁnite ray in this component emerges. Note that ωRt+l+r+2q+1 is a critical string in the sense of [5] and we could use [5, Prop. 2] to deduce that {ωRn }n∈N is inﬁnite. The above characterization of component T31 in ΓT (A1 ) is suﬃcient to determine it as the tube whose rank equals (t + m + s + r + p + l) − 1. Similarly we obtain that T41 is the tube of rank (t + m + s + r + p + l+) − 1 with modules from the set B lying on its mouth.

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Remark 3.7. For i = 1, 2, 3 there are four various tubes in the Auslander-Reiten quiver of the trivial extension of the algebra Ai which contain projective modules: • the tube T1i of rank n with modules P (s(α2 )), ..., P (s(αn )) on a mouth • the tube T2i of rank q with modules P (s(α2 )), ..., P (s(αq )) on a mouth • the tube T3i of rank (t + m + s + r + p + l − 1) with modules from the set B on a mouth • the tube T4i of rank (t + m + s + r + p + l − 1) with modules from the set B on a mouth In the case of T3i , T4i for i = 2 we have t = 0 and for i = 3 we have p = 0, t = 0. Now we are ready to formulate the main statement of this article:

Theorem 3.8. Let Ai = kQi /Ii where Qi , Ii have been previously deﬁned for i = 1, 2, 3. Then all projective indecomposable T (Ai )-modules lie in nonstable tubes or in components of pseudotype ZA∞ ∞. Proof. Naturally C1i , C1i , C2i , C2i , C3i and, respectively C4i , together with tubes T1i , T2i , T3i , T4i form a complete family of those components of the quiver ΓT (Ai ) which contain projective T (Ai )-modules. Let us emphasize that also every projective and injective T (Ai )rd module is lying in these components as the starting vertex and the ending vertex of the Auslander-Reiten sequence containing a projective T (Ai )-module in the middle. The proof follows immediately from Lemma 3.3 and Lemma 3.6.

4

Further characterization of the components of ZA∞ ∞ -pseudotype

The knowledge about the structure of components C1i , C1i , C2i , C2i , C3i and C4i encourages us to examine whether a complete characterization of homomorphisms between modules from these components can be obtained. The answer is given in the following lemma. Lemma 4.1. The components of ΓT (Ai ) (i = 1, 2, 3) which are of pseudotype ZA∞ ∞ and contain projective T (Ai )-modules are not generalized standard. Proof. The proof reduces to considering pairs of modules (X, Y ) from a given component C such that there is a nonzero homomorphism f : X −→ Y which cannot be represented as a ﬁnite sum of ﬁnite compositions of irreducible morphisms. Thus, we will examine a set consisting of only those modules U in C such that there is a ﬁnite path from U to Y in this component (from X to U respectively). For C1i where i = 1, 2, 3 the assertion has been proved in [6]. The cases of C1i are analogous. −1 Let us start with C21 . We shall consider two modules X((α1 ...αn δt−1 ...δ1−1 βm ...β1−1 )Lt ) =

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−1 −1 X(βm ...β1−1 ) and X((α1 ...αn δt−1 ...δ1−1 βm ...β1−1 )Rn ) = S(1). There is a nonzero homomor−1 ...β1−1 ) −→ S(1). Let phism f : X(βm

W = {U ∈ C21 : there is a ﬁnite path in C21 from U to S(1) whose composition is nonzero}. −1 Assume that X(βm ...β1−1 ) ∈ W. This means that f is a composition of irreducible f f / ... f1 / morphisms f0 , f1 , ...fk for some natural k : X(β −1...β −1 ) k / Yk k−1 / Yk−1 m

f0

1

/ S(1) .

Of course f cannot factorize through P (2). Thus Yk is either Y1 −1 −1 −1 −1 −1 X(βm ...β1 d cα1 ...αn−1 ) or X(εl−1 ...ε1−1 b−1 aβm ...β1−1 ). But there is no nonzero mor−1 phism g : X(βm ...β1−1 d−1 cα1 ...αn−1 ) −→ S(1) −1 −1 so Yk ∼ ...β1−1 ). Repeating the above step we obtain that Yk−j ∼ = X(εl−1 ...ε1−1 b−1 aβm = −1 −1 X(βm ...β1 )Lj+1 ). Using the Skowro´ nski - Waschb¨ usch algorithm we conclude there is no

0 / −1 S(1) . In that way we get X(βm ...β1−1 ) ∈ /W arrow Y (1) ∞ −1 −1 1 and so f ∈ rad (X(βm ...β1 ), S(1)). The component C2 is not generalized standard. −1 For i = 2, 3 we may consider modules X(ε2 ...εs a) = X((ε2 ...εs aα1 ...αn βm ...β1−1 )Rn ) −1 and S(1) = X((ε2 ...εs aα1 ...αn βm ...β1−1 )Lm ) in C2i . There is a nonzero homomorphism g : S(1) −→ X(ε2 ...εs a). As in the ﬁrst case, we can prove that g ∈ rad∞ (S(1), X(ε2...εs a)). Analogous arguments show that C2i is not generalized standard for i ∈ {1, 2, 3}. −1 In C31 we have a pair of modules X := X(βm ...β1−1 α1 ...αn δt−1 ...δ1−1 ε1 ...εs ) and Y := X(δ1−1 ε1 ...εs−1 ) with a nonzero morphism f : X −→ Y . They lie on the left and right diagonal {X(ε1 ...εs−1 )Lη }η∈Z and {X(ε1 ...εs−1 )Rμ }μ∈Z respectively. We shall examine a set

f

W = {U ∈ C31 : there exists a ﬁnite path in C31 from U to Y whose composition is nonzero}. The set W assigns a full inﬁnite subquiver in C31 whose vertices are completely determined by WR = {X(δ1−1 ε1 ...εs−1 )R−η }η∈N and WL = {X(δ1−1 ε1 ...εs−1 )L−μ }μ∈N . It is easy to see −1 that neither WR nor WL have modules which lie on walks starting with βm (ending with −1 βm ) or ending with εs (starting with εs ). In this situation X cannot belong to W . Thus f ∈ rad∞ (X, Y ) and C31 is not generalized standard. −1 In C32 , C33 we may take X(α1 ...αn βm ...β1−1 α1 ...αn ε1 ...εs ) = X((α1 ...αn ε1 ...εs−1 )Rm+s−1 ) −1 as X and X(βm−1 ...β1−1 α1 ...αn ε1 ...εs−1 ) = X((α1 ...αn ε1 ...εs−1 )L−m+1 ) as Y . It is obvious that there is a nonzero homomorphism g : X −→ Y . The above consideration can be transferred to show that also C32 , C33 are not generalized standard. For the cases of C41 , C42 we may take nonzero morphisms f : X(δ2 ...δt dc−1 dβ1 ...βm a−1 b) −1 −1 −1 −1 −→ S(3) and g : X(ε2 ...εl ε−1 s ...ε1 d) −→ X(ε1 ...εl εs ...ε1 da ) respectively. Using the given technique one can easily check that both f and g are from rad∞ (mod(T (Ai )) for a relevant i. The next issue we are going to examine is a description of positions and relations between diﬀerent components of pseudotype ZA∞ ∞ in ΓT (Ai ) for i ∈ {1, 2, 3}. In order to do that we look for morphisms between representations from diﬀerent components. Here we give the examples of morphisms from one component to another: • for i = 1:

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from C11 to C11 : X(α2 ...αn δt−1 ...δ1−1 ε1 ...εs εl−1 ...ε2−1 ) −→ X(δp−1 ...δ1−1 ε1 ...εl−1 ) −1 −1 −1 from C11 to C11 : X(α2 ...αq δp−1 ...δ1−1 ε1 ...εl ε−1 s ...ε2 ) −→ X(δt ...δ1 ε1 ...εs−1 ) −1 −1 from C21 to C21 : X(αq−1 ...α1 β1 ...βr b−1 aβm ...β1−1 ) −→ X(α1 ...αq−1 ) −1 −1 1 1 −1 −1 −1 from C2 to C2 : X(αn ...α1 β1 ...βm a bβr ...β1 ) −→ X(α1 ...αn−1 ) −1 ...β1−1 α1 ...αn−1 ) −→ X(ε1 ...εs−1 ) from C21 to C31 : X(ε2 ...εs aβm −1 −1 from C31 to C21 : X(βm ...β2−1 ) −→ X(βm ...β1−1 ) from C31 to C21 : X(β2 ...βr ) −→ X(βr−1 ...β1−1 ) −1 ) −→ X(εl−1 ...ε1−1 ) from C21 to C31 : X(ε2 ...εl bβr−1 ...β1−1 α1 ...αq−1 −1 from C11 to C31 : X(α2 ...αn ) −→ X(βm ...β1−1 α1 ...αn ) −1 −1 −1 from C31 to C11 : X(εl−1 ...ε1 δ1 ...δp αq−1 ...α1−1 β1 ...βr ba−1 βm ...β1−1 d−1 ) −→ S(2) −1 from C11 to C31 : X(α2 ...αq ) −→ X(αq−1 ...α1 β1 ...βr ) −1 from C31 to C11 : X(eβ1 ...βr b−1 aβm ...β1−1 α1 ...αn δt−1 ...δ1−1 ε1 ...εs−1 ) −→ S(2 ) −1 −1 from C11 to C21 : X(δt−1 ...δ1−1 ) −→ X(δt−1 ...δ1−1 βm ...β1−1 ) from C11 to C21 : analogous from C21 to C11 : S(1) −→ X(cd−1) from C21 to C11 : analogous −1 ...β1−1 ) from C11 to C21 : X(αq δp−1 ) −→ X(δp αq−1 ...α1−1 β1 ...βr b−1 aβm −1 from C21 to C11 : X(β2 ...βr δ1 ...δp ez −1 eβ1 ...βr b−1 aβm ...β1−1 α1 ...αn−1 ) −1 −1 −1 −→ X(δp ...δ1 βr−1 ...β2 ) from C11 to C21 : X(δt−1 ...δ1−1 ε1 ...εs εl−1 ...ε1−1 δ1 ...δp αq−1 ...α1−1 ) −→ X(α1 ) from C21 to C11 : X(βr−1 ...β1−1 ) −→ X(α2 ...αn δt−1 ...δ1−1 ε1 ...εs εl−1 ...ε1−1 δ1 ...δp eβ1 ) from C41 to C11 : X(δ2 ...δt d) −→ X(δt−1 ...δ1−1 ) from C41 to C11 : analogous −1 from C11 to C41 : X(δt−1 ...δ1−1 βm ...β2−1 ) −→ X(dβ1 ...βm−1 ) from C11 to C41 : analogous −1 ...β1−1 ) −→ X(dβ1 ...βm−1 ) from C21 to C41 : X(βm from C21 to C41 : analogous −1 −1 from C41 to C21 : X(δ2 ...δt d) −→ X(δt−1 ...δ1−1 βm ...β1−1 ) from C41 to C21 : analogous −1 from C31 to C41 : X(βm ...β2−1 ) −→ X(dβ1 ...βm−1 ) from C41 to C31 : S(3) −→ X(ε1 ...εs εl−1 ...ε1−1 ) • for i = 2: from C12 to C12 : X(α2 ...αn ε1 ...εs εl−1 ...ε2−1 ) −→ X(δp−1 ...δ1−1 ε1 ...εl−1 ) from C12 to C12 : X(αq δp−1 ) −→ X(ε1 ...εs εl−1 ...ε1−1 δ1 ...δp αq−1 ...α1−1 ) ) from C22 to C22 : X(αq−1 ...α1−1 β1 ...βr b−1 a) −→ X(α1 ...αq−1 −1 −1 −1 −1 2 2 −1 −1 from C2 to C2 : X(βm−1 βm αn ...α1 a bβr ...β1 ) −→ X(βm−1 ...β1−1 ) from C22 to C32 : X(ε2 ...εs a) −→ X(α1 ...αn ε1 ε2 ) from C32 to C22 : X(α1 ...αn ) −→ X(aα1 ...αn−1 ) −1 from C32 to C22 : X(εl−1 ...ε1−1 δ1 ...δp αq−1 ...α1−1 ) −→ X(α1 ...αq−1 ) −1 −1 −1 2 2 −1 from C2 to C3 : X(ε2 ...εl bβr ...β1 α1 ...αq−1 ) −→ X(εl−2 ...ε1 ) from C12 to C32 : X(α2 ...αn ε1 ...εs ) −→ X(α1 ...αn ε1 ...εs−1 ) −1 from C32 to C12 : X(εl−1 ...ε1−1 δ1 ...δp αq−1 ...α1−1 β1 ...βr b−1 ad−1 ) −→ S(2)

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from C12 to C32 : X(δp−1 ...δ1−1 ε1 ...εl−1 ) −→ S(4 ) −1 ...β1−1 α1 ...αn ε1 ...εs−1 ) −→ S(2 ) from C32 to C12 : X(eβ1 ...βr b−1 aα1 ...αn βm from C12 to C22 : P (1) −→ S(1) from C12 to C22 : P (1 ) −→ S(1) from C22 to C12 : P (2) −→ S(2) from C22 to C12 : P (2 ) −→ S(2) from C12 to C22 : X(α2 ...αq ) −→ X(αq−1 ...α1−1 β1 ...βr b−1 a) from C22 to C12 : X(δ2 ...δp eβ1 ...βr b−1 aα1 ...αn−1 ) −→ X(δp−1 ...δ1−1 ) from C12 to C22 : X(ε1 ...εs εl−1 ...ε1−1 δ1 ...δp αq−1 ...α1−1 ) −→ X(α1 ...αq−1 ) −1 −1 2 2 −1 −1 −1 from C2 to C1 : X(β2 ...βm da bβr ...β1 α1 ...αq−1 ) −→ X(βm ...β2 ) from C42 to C12 : X(ε−1 s ) −→ X(ε1 ...εs ) −1 2 2 −1 from C1 to C4 : X(βm ...β2−1 ) −→ X(ε2 ...εl ε−1 s ...ε1 dβ1 ...βm−1 ) −1 −1 −1 from C12 to C42 : X(α2 ...αq ) −→ X(α1 ...αq δp−1 ...δ1 ε1 ...εl ε−1 s ...ε1 da ) −1 from C42 to C12 : X(εl ) −→ X(α2 ...αq δp−1 ...δ1 ε1 ...εl ) from C22 to C42 : X(b−1 a) −→ S(3) from C22 to C42 : X(β2 ...βm da−1 bβr−1 ...β1−1 α1 ...αq−1 ) −→ X(β2 ...βm da−1 ) −1 ...β1−1 ) from C42 to C22 : X(β2 ...βm da−1 ) −→ X(βm−1 −1 −1 −1 −1 −1 −1 −1 −1 from C42 to C22 : X(ε−1 s−1 ...ε1 αn ...α1 a bβr ...β1 ) −→ X(εs εs−1 ) from C32 to C42 : P (3) −→ S(3) from C42 to C32 : P (4) −→ S(4 ) • for i = 3: from C13 to C13 : X(ε1 ...εs εl−1 ...ε1−1 αq−1 ...α1−1 ) −→ X(α2 ...αq ) −1 −1 −1 from C13 to C13 : X(ε1 ...εl ε−1 s ...ε1 αn ...α1 ) −→ X(α2 ...αn ) −1 from C23 to C23 : X(αq−1 ...α1 b−1 a) −→ X(bα1 ...αq−1 ) −1 −1 −1 3 3 −1 from C2 to C2 : X(β2 ...βm αn ...α1 a b) −→ X(βm−1 ...β1−1 ) from C23 to C33 : X(ε2 ...εs a) −→ X(α1 ...αn ε1 ε2 ) from C33 to C23 : X(α1 ...αn ) −→ X(aα1 ...αn−1 ) from C33 to C23 : X(αq−1 ...α1−1 ) −→ X(bα1 ...αq−1 ) −1 −1 3 3 −1 from C2 to C3 : X(ε2 ...εl b) −→ X(ε2 ε1 αq ...α1−1 ) from C13 to C33 : X(α2 ...αn ε1 ...εs ) −→ X(α1 ...αn ε1 ...εs−1 ) from C13 to C33 : analogous −1 ...ε1−1 αq−1 ...α1−1 β1 ...βr αq−1 ...α1−1 b−1 ad−1 ε1 ) −→ X(ε1 ) from C33 to C13 : X(εl−1 from C33 to C13 : analogous from C13 to C23 : P (1) −→ S(1) from C13 to C23 : P (1 ) −→ S(1) from C23 to C13 : P (2) −→ S(2) from C23 to C13 : P (2 ) −→ S(2) −1 ...ε1−1 αq−1 ...α1−1 b−1 a) from C13 to C23 : X(ε1 ...εl−1 ) −→ X(εl−1 from C23 to C13 : X(β2 ...βr eb−1 aα1 ...αn−1 ) −→ X(βr−1 ...β2−1 ) −1 −1 −1 −1 from C13 to C23 : X(ε−1 s−1 ...ε1 αn ...α1 a b) −→ X(ε1 ...εs−1 ) −1 from C23 to C13 : X(ε−1 1 da bα1 ...αq−1 ) −→ X(ε1 ) The following proposition summarizes the results of the above analysis.

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Proposition 4.2. Let Cji , Cji be diﬀerent components of pseudotype ZA∞ ∞ in ΓT (Ai ) for i ∈ {1, 2} and j ∈ {1, 1 , 2, 2 , 3, 4 }, respectively i = 3 and j ∈ {1, 1 , 2, 2, 3}. Then HomT (Ai ) (Cji , Cji ) = 0. The graphic representation of our consideration is given by a subgraph of the oriented graph of all components of the Auslander-Reiten quiver. We present only the case of i = 1. The oriented graph of all components of pseudotype ZA∞ ∞ has the following form (omitting loops): 1 C 1 CJ 1P 1 [ qg : F 1P X

'0 1 : CJ 2

z

C21X pg

z

C31 q

' 1

1C

4

References [1] I. Assem, D. Simson and A. Skowro´ nski: Elements of the representation theory of associative algebras, Vol. 1, Techniques of representation theory, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006. [2] M. Auslander and I. Reiten: “Uniserial functors”, Representation theory II, Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., (1979), pp. 1–47, Lecture Notes in Math., 832, Springer, Berlin, 1980. [3] M.C.R. Butler and C.M. Ringel: “Auslander-Reiten sequences with few middle terms and applications to string algebras”, Comm. Algebra, Vol. 15, (1987), no. 1-2, pp. 145–179. [4] P. Gabriel: “Auslander-Reiten sequences and representation-ﬁnite algebras”, Representation theory I, Proc. Workshop, Carleton Univ., Ottawa, Ont., (1979), pp. 1–71, Lecture Notes in Math., 831, Springer, Berlin, 1980. [5] C. Geiss: “On components of type ZA∞ ∞ for string algebras”, Comm. Algebra, Vol. 26, (1998), no. 3, pp. 749–758. [6] A. Jaworska and Z. Pogorzaly: “On trivial extensions of 2-fundamental algebras”, Comm. Algebra, Vol. 34, (2006), no. 11, pp. 3935–3947.

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[7] Z. Pogorzaly and M. Sufranek: “Starting and ending components of the AuslanderReiten quivers of a class of special biserial algebras”, Colloq. Math., Vol. 99, (2004), no. 1, pp. 111–144. [8] A. Skowro´ nski: “Generalized standard Auslander-Reiten components”, J. Math. Soc. Japan, Vol. 46, (1994), no. 3, pp. 517–543. [9] A. Skowro´ nski: Algebras of polynomial growth, Topics in algebra, Part 1 (Warsaw, 1988), pp. 535–568, Banach Center Publ., 26, Part 1, PWN, Warsaw, 1990. [10] A. Skowro´ nski and J. Waschb¨ usch: “Representation-ﬁnite biserial algebras”, J. Reine Angew. Math., Vol. 345, (1983), pp. 172–181.

DOI: 10.2478/s11533-007-0030-4 Research article CEJM 5(4) 2007 686–695

The abelianization of hypercyclic groups B.A.F. Wehrfritz∗ School of Mathematical Sciences Queen Mary, University of London, Mile End Road London E1 4NS England

Received 06 June 2007; accepted 15 August 2007 Abstract: Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G is 2-divisible, then G is 2-divisible. b) If G/G is a 2 -group, then G is a 2 -group. c) If G/G is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O2 (G). c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: hypercyclic group; divisible-by-finite group MSC (2000): 20F19

1

Theorems and examples

The abelianization of a hypercentral group has considerable inﬂuence on the structure of the group itself. For an account of the classical results in this area see Section 9.2 of Robinson’s Ergebnisse [2]. For more recent developments see [1] and [4]. Since hypercentral groups are hypercyclic (a group G is hypercyclic if it has an ascending normal series 1 = G0 ≤ G1 ≤ · · · ≤ Gα ≤ · · · ≤ Gσ = G with cyclic factors) it is natural to ask whether these results extend to hypercyclic groups. The brief answer is that some do precisely, easily and boringly, while some go totally wrong as examples, and often very easy examples, show. This leaves a narrow band in between where things are interestingly and positively diﬀerent. In order to identify where the greatest interest lies we start with the routine positive results, which we group together as a single proposition. ∗

E-mail: [email protected]

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Proposition 1.1. Let G be a hypercyclic group. a) G/G = 1 if and only if G = 1. b) G/G is divisible if and only if G is divisible, and then G is divisible and hypercentral. c) G/G is divisible by finite if and only if G is divisible by finite, and then G is divisiblehypercentral by finite. d) If G/G is finite, then G is periodic and (divisible-abelian by finite). e) If G is periodic the following are equivalent. i) G/G is 2-divisible by finite. ii) G is 2-divisible by finite. iii) G has a normal subgroup of finite index that is a direct product of a divisible abelian 2-group and a 2 -group. Now we turn to some easy examples. Let A be an abelian group and set G = iA, where i2 = 1 and ai = a−1 for all a in A. Then G is hypercyclic and G = [A, i] = A2 . If A is not a 2-group, then G is not hypercentral. If A is inﬁnite cyclic, then G is not divisible by ﬁnite, indeed G is not p-divisible by ﬁnite for any prime p. Thus in the Proposition G/G ﬁnite does not imply that G is divisible by ﬁnite or even that G is divisible by periodic; compare c) and d) of the Proposition and Theorem 1.5 of [4]. Now suppose A is the direct product over all odd primes p of inﬁnite elementary abelian p-groups and again set G = iA as above. Then G is hypercyclic and periodic, G = A and |G/G | = 2. Also G is not p-divisible by ﬁnite for any odd prime p. Compare this with e) of the Proposition. The examples above are all metabelian and parts c) and d) of the Proposition might suggest that metabelian groups are exceptional in this context. This is not the case. Let T = Tr1 (n, Q) be the (lower) unitriangular group of degree n over the rationals Q, so T is divisible, torsion-free and nilpotent of class n − 1. Set D = {diag(±1, ±1, . . . , ±1)} ≤ GL(n, Q), so D is elementary abelian of order 2n . Set G = DT . Then G is hypercyclic (indeed papasoluble of paraheight 1 + n(n − 1)/2, see [3] Chapter 11, especially Pages 156 and 164 for deﬁnitions), G = T and |G/G| = 2n . Again take G = DT , but now with T = Tr1 (n, Z) for Z the integers. Here G = {(tij ) ∈ T : ti,i+1 is even for 1 ≤ i < n}. Thus G is ﬁnitely generated, torsion-free nilpotent of class n − 1 and |G/G | = 22n−1 is again ﬁnite. In the ﬁrst case G is divisible by ﬁnite and in the second case G has no non-trivial divisible subgroups. As a variation let u = diag(−1, 1, −1, 1, . . . , (−1)n ) GL(n, Q) and set G = uT for T = Tr1 (n, R) and R = Q or Z. Then u inverts T /T and so G = T 2 . Thus again G is hypercyclic and G is nilpotent of class n − 1. Set N equal to the direct product ×n≥2 Tr1 (n, R), let v be the automorphism of N that acts on each Tr1 (n, R) as the u above and now put G = vN. Then G is torsion-free, hypercentral with central height ω and G is hypercyclic (of inﬁnite paraheight ω + 1). If R = Q then |G/G| = 2 and G is divisible. If R = Z then G/G is an elementary abelian 2-group and G has no non-trivial divisible subgroups.

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The above examples depend upon the singular nature of the prime 2. If q is any prime the set π of all primes p congruent to 1 modulo q is inﬁnite (Dirichlet’s Theorem). Let A = ×p∈π Ap , where Ap is cyclic of order p, and G = uA, where u is an automorphism of A of order q that acts on each Ap as an automorphism of order q; such u does exist by the choice of π. Then G is hypercyclic, G = A and |G/G| = q. Clearly G contains no non-trivial divisible subgroups. As a modiﬁcation of this let N = ×p∈π Tr1 (p, p) and G = uN, where u acts on each Tr1 (p, p) as diag(αp , αp2 , . . . , αpp ), where αp is an element of order q of the ﬁeld Fp of order p. Again G is hypercyclic, G = N, |G/G | = q and G contains no non-trivial divisible subgroups, but here G although hypercentral is not nilpotent. In spite of these 2 -examples the prime 2 does play a special role here. For any group G and set π of primes, the unique maximal normal π-subgroup of G is denoted by Oπ (G). Theorem 1.2. Let G be a hypercyclic group. Set U = O2 (G) and let V = α γ α G be the intersection of the terms of the lower central series (continued transfinitely) of U. The following are equivalent. a) G/G is divisible by finite-of-odd-order. b) G/V is divisible by finite-of-odd-order. c) G/U is divisible and U/V is divisible by finite. Thus, for example, if in Theorem 1.2 the subgroup O2 (G) is nilpotent, then G is divisible by ﬁnite-of-odd-order if and only if G/G is divisible by ﬁnite-of-odd-order. Suppose G is a hypercyclic group with G/G divisible by ﬁnite-of-odd-order. Then G need not be divisible by periodic and with U and V as in Theorem 1.2, V need not be abelian and G/UD (and hence G/V D) need not be ﬁnite for any divisible subgroup D of G. The following example shows this. Let π be the inﬁnite set of all primes p with p congruent to 1 modulo 3. Set A = p∈π ap and B = ×p∈π ap , where |ap | = p. For each p in π there is an element αp of Fp of order 3. Let S = uT , where T = ×p∈π Tr1 (3, p) and u has order 3 and acts on each Tr1 (3, p) as diag(1, αp , 1). Then u centralizes the centre of each Tr1 (3, p) and hence S contains a central element ep of order p for each p ∈ π. Also S = T = [T, u]. Let G be the central product of A and S amalgamating B and ep : p ∈ π. Speciﬁcally, set G = AS = (A × S)/a−1 p ep : p ∈ π. Then G = S = T and G/G ∼ = (A/B) × u; here A/B is torsion-free, divisible and abelian of inﬁnite rank and u has order 3. In particular G/G is divisible by ﬁnite-ofodd-order. Clearly U = O2 (G) = S and V = α γ α U = T , which is nilpotent of class 2. Each u · Tr1 (3, p) for p ∈ π is supersoluble, so G is hypercyclic. Let D be any divisible subgroup of G. Now 1 is the only divisible subgroup of S/B ∼ = G/A, so D ≤ A. But the divisible radical of the abelian group A is 1 (for if x = (xp ) ∈ A with xq = 1 for some q, then x ∈ / Aq ). Thus D = 1 and G has all the properties claimed for it above.

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In the above group G, set H = AT . Then H is nilpotent of class 2 and H = T = B, so H/H ∼ = A/B × T /B, which is divisible by an abelian π-group. Also O2 (H) = T , which is nilpotent, so the subgroup of H corresponding to the V of G is 1. As with G, the only divisible subgroup of H is 1 and hence H ∼ = H/V is not divisible by periodic, even though H/H is. Since the element u of G plays no role here, in fact there is a group H constructed as the H above for any inﬁnite set π of odd primes. However not all is lost if in Theorem 1.2 we replace ‘ﬁnite-of-odd-order’ by ‘a 2 -group’. Now an abelian group that is divisible by a 2 -group is also a 2 -group by divisible. We use the following in the proof of Theorem 1.2. Theorem 1.3. Let G be a hypercyclic group. a) If G/G is a 2 -group by divisible, then G/O2 (G) is hypercentral and divisible. b) G/G is a 2 -group if and only if G is a 2 -group. If G is the inﬁnite dihedral group, then |G/G | = 4, so G/G is trivially 2-divisible by ﬁnite. Clearly G is not 2-divisible by ﬁnite or even 2-divisible by periodic. Now an abelian group that is divisible by ﬁnite-of-odd-order, while not always divisible (of course), is always 2-divisible. This raises the possibility that a hypercyclic group G is 2-divisible whenever G/G is 2-divisible. This is indeed the case. Since for any set π of primes an abelian group is π-divisible whenever it is π-divisible by a π -group, we can be a little more general. Theorem 1.4. Let G be a hypercyclic group and set π = {2, 3, 5, . . . , pn }, the set of the first n primes. The following are equivalent. a) G/G is π-divisible. b) G is π-divisible. c) G is π-divisible by a π -group. d) G/Oπ (G) is π-divisible. The case n = 1, π = {2} is the case introduced above. If G is dihedral of order 2p for p an odd prime and if π is any set of odd primes containing p, then G/G is π-divisible while G is not π-divisible. Since U = Oπ (G) = 1, nor is G/U. Also 1 is the only π-divisible normal subgroup of G, so G is not even π-divisible by a π -group. Thus Theorem 1.4 looks like the best we can expect in this direction.

2

The lemmas

We begin with a summary of relevant and presumably well-known results. Lemma 2.1. a) Let a ≤ a, b be normal subgroups of the group G with |a| = p and (a, b : a) = q (possibly infinite). i) If p and q are primes with p < q, then a, b = a × b , where b has order q and is normal in G.

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ii) If p is prime and q = ∞ , there exists b characteristic in a, b (and hence normal in G) of infinite order with (a, b : b) dividing p2 (p − 1). If also G centralizes a, b/a, then G centralizes b . iii) Let p = ∞ and let q be prime. Suppose that either [a, b] = 1 or q is odd. Then a, b is abelian and either a, b = a × b , where b is normal in G and has order q, or a, b = a is infinite cyclic. If also G centralizes a, then G centralizes a, b/b in the first case and a in the second. b) Let G be supersoluble. i) G has a normal cyclic series 1 = G0 < G1 < · · · < Gn = G with factors infinite or of prime order such that (Gi : Gi−1 ) ≥ (Gi+1 : Gi ) for 0 < i < n. ii) G has a normal cyclic series 1 = G0 < G1 < · · · < Gn = G with factors infinite or of prime order where (Gi : Gi−1 ) has odd order if and only if i ≤ m and (Gi : Gi−1 ) ≥ (Gi+1 : Gi ) wherever 0 < i < m or m < i < n. In particular Gm = O2 (G) and (Gi : Gi−1 ) = 2 or ∞ for i > m. Proof. All the above are elementary. In a)ii) the subgroup a, bp−1 is abelian and characteristic in a, b, so we can set b = bp(p−1) . In a)iii) if a, b is not torsion-free, clearly b exists. If it is torsion-free, then bq = an for some non-zero n. If q divides n / a, so q does not divide n, 1 = qh + nk for some then a−n/q b has order q as clearly b ∈ integers h and k and we can set a = ah bk . The comment about the centrality of inﬁnite factors in G follows from the fact that the only non-trivial automorphism of an inﬁnite cyclic group acts ﬁxed-point freely. Part b) is a routine consequence of Part a). For any group G let ρ(G) denote the ﬁnite-cyclic residual of G and η(G) the Hirsch– Plotkin radical of G. Also let ζ(G) denote the hypercentre of G and {ζα (G)}0≤α≤σ the upper central series of G. The following is obvious. Lemma 2.2. Let G be a hypercyclic group. Then ρ(G) is a hypercentral group and G ≤ ρ(G). Lemma 2.3. A hypercyclic group G has a unique maximal divisible subgroup δ(G). Also δ(G) = δ(ρ(G)) is hypercentral and δ(G/δ(G)) = 1. Proof. Set R = ρ(G). Then δ(R) exists (e.g. [4] 5.3). If X/δ(R) ≤ G/δ(R) is divisible, then X ≤ R since by deﬁnition G/R is residually ﬁnite. Thus 1 is the only divisible subgroup of G/δ(R). All parts of 2.3 now follow. Lemma 2.4. If G is a hypercyclic, locally nilpotent group, then G is hypercentral. Proof. Let a = 1 be a normal subgroup of G of prime or inﬁnite order and let X be any ﬁnitely generated subgroup of G containing a. Then X is nilpotent, a ∩ζ1(X) = 1 and so a ≤ ζ1 (X), even if |a| is inﬁnite note. Thus a ∈ X ζ1 (X) ≤ ζ1 (G). It follows easily that G is hypercentral.

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Lemma 2.5. Let G be a hypercyclic group. Then ρ(G) ≤ η(G), η(G) is hypercentral and G/η(G) is residually finite-cyclic. Proof. By 2.4 the subgroup N = η(G) has an ascending cyclic series 1 = N0 ≤ N1 ≤ · · · ≤ Nα ≤ · · · ≤ Nv = N of normal subgroups of G with factors central in N. Then CG (Nα+1 /Nα ) N ≤C= α
and G/C is residually ﬁnite-cyclic. By 2.2 we have G ≤ ρ(G) ≤ N. Thus C is hypercentral, so C ≤ N and 2.5 follows easily. Below we make a number of references to [2] 9.23. The main part of this from our point of view is that for G a hypercentral group, G/G is divisible if and only if G is divisible and then the upper central factors of G are all divisible and the elements of G of ﬁnite order are all central. Lemma 2.6. Let N be a finite normal subgroup of the hypercyclic group G with G/N divisible. Then G = N · δ(G). Proof. We may assume that δ(G) = 1. Set C = CG (N). Then G/C is ﬁnite and G = CN. Hence C/(C ∩N) ∼ = G/N is divisible and therefore (e.g. by 2.3) is hypercentral. Set M/(C ∩ N) = ζ1 (C/(C ∩ N)). Then C stabilizes the series 1 ≤ C ∩ N ≤ M, δ(M) ≤ δ(G) = 1 and M/(C ∩ N) is divisible (by [2] 9.23). Thus M has ﬁnite exponent ([4] 5.2), M = C ∩ N and so C = C ∩ N. But then G/N is ﬁnite and divisible. Consequently G = N. Lemma 2.7. Let the group G be hypercyclic and poly (divisible by finite). Then G is divisible by finite. Proof. We may pass to G/δ(G) and assume that δ(G) = 1. Also a very simple induction yields normal subgroups N ≤ H of G with N and G/H ﬁnite and H/N divisible. By 2.6 H = N · δ(H) ≤ N · δ(G) = N. Therefore G is ﬁnite and the lemma is proved.

Lemma 2.8. Let G be a hypercyclic group, π a set of primes and N a normal π -subgroup of G with G/N a divisible π-group. Then N is a direct factor of G and G = N · δ(G). Proof. Note ﬁrst that a divisible hypercyclic group is hypercentral by 2.3, so in our case G/N is divisible abelian (see [2] 9.23). Consequently G ≤ N. Also G/N is a π-group and N is a π -group, so G/G = H/G × N/G , where H/G ∼ = G/N is a divisible abelian π-group.

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Now G is hypercentral (2.2), so G ≤ H ∩η(G) = η(H) and H/η(H) is residually ﬁnite by 2.5. Therefore H = η(H) is locally nilpotent. Hence H = S × G , where S ∼ = G/N is a divisible abelian π-group and G is a π -group. Thus SN = HN = G and S ∩ N = 1. Finally N and S are normal in G, the latter since S is characteristic in H, so G = S × N. Trivially S ≤ δ(G). In the next four lemmas of this section π denotes the set of the ﬁrst n ≥ 1 primes. Our primary interest is in the case n = 1, π = {2}, but it is little extra work to present them in general. Lemma 2.9. Let G be a group with a cyclic normal subgroup A = a of prime order p∈ / π such that G/G is π-divisible. If g ∈ G is a q-th power modulo A for some q in π, then g is a q-th power in G. Proof. By hypothesis there exists h ∈ G and b ∈ A with g = hq b. Now EndA ∼ = Fp ; let h act on A as η ∈ Fp . Since G/G is π-divisible and G acts on A as a ﬁnite abelian group, η is a (non-zero) π -element. Hence η q = 1, so θ = η q−1 + η q−2 + · · · + η + 1 = 0 and therefore b = cθ for some c ∈ A. Consequently (hc)q = hq cθ = hq b = g. Lemma 2.10. Let G be a hypercyclic group with no non-trivial cyclic π-images and set U = Oπ (G). Then G/U is hypercentral. Proof. Now G has an ascending cyclic normal series with factors inﬁnite or of prime order. Necessarily the inﬁnite factors and the π-factors of this series are central in G. If X is any ﬁnitely generated (hence supersoluble, e.g. [3] 11.10) subgroup of G, intersect the terms of this series with X. Thus X has a supersoluble series with its inﬁnite cyclic factors central in X. Denote the unique maximal periodic normal subgroup of any group Y by τ (Y ). Then X/τ (X) is torsion-free nilpotent by 2.1a)iii). Therefore G/τ (G) is torsion-free and locally nilpotent, since here τ (G) = X τ (X) and τ (G) ∩ X = τ (X). Thus G/τ (G) is hypercentral by 2.4. Also τ (G)/U is a π-group by 2.1b). Therefore G/U is hypercentral. Lemma 2.11. Let G be a hypercyclic group and U a normal π -subgroup of G with G/U π-divisible. Then G is π-divisible. Proof. Clearly U and G/G are π-divisible. Let {Uα }0≤α≤σ be an ascending cyclic series of U of subgroups normal in G, such that each factor Uα+1 /Uα has prime order (necessarily in π ). Let g ∈ G and q ∈ π. Pick α minimal with g a q-th power modulo Uα ; α exists since g is a q-th power modulo U by hypothesis. Assume α > 0. Now g = hq u for some h ∈ G and u ∈ Uα . If α is a limit ordinal, then u ∈ Uβ for some β < α and g is a q-th power modulo Uβ . By the choice of α no such β exists and consequently α − 1 exists. But then g is a q-th power modulo Uα−1 by 2.9. This contradiction, again of the minimal choice of α, shows that α = 0 and g = hq . Since this is for all g ∈ G and q ∈ π, so G is π-divisible.

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Although we do not need it here, 2.11 is actually equivalent to the existence of a π-divisible radical δπ (G) in a hypercyclic group G, as the following shows. Lemma 2.12. Let G be a hypercyclic group and let H be its subgroup generated by all the π-divisible subgroups of G. Then H is π-divisible and G/H has no non-trivial π-divisible subgroups. Proof. Certainly H/H is π-divisible. Set U = Oπ (G) = Oπ (H). Then H has no non-trivial cyclic π-images and hence H/U is hypercentral by 2.10. By [2] 9.23 the group H/U is π-divisible. Thus, by 2.11, so is H. Suppose G/H is π-divisible. Then G/G is π-divisible by π-divisible and hence is π-divisible. Consequently G/U is hypercentral (by 2.10) and π-divisible ([2] 9.23 again). Therefore G is π-divisible by 2.11 and so G = H. The lemma follows. Unlike the case of hypercentral groups, there is no analogue of 2.12 for arbitrary sets π of primes. As a trivial example, if G is a dihedral group of order 2p for p any odd prime, then G does not have a unique maximal π-divisible subgroup for any set π of odd primes containing p. Lemma 2.13. Let G be a torsion-free hypercentral group. Then G/δ(G) is torsion-free. Proof. Let δ(G) < X ≤ G, where X/δ(G) is ﬁnite. Then X/X is divisible by ﬁnite. Also X is torsion-free. By Theorem 1.2 of [4] the group X is divisible and hence X = δ(G). The claim follows. Lemma 2.14. Let G be a torsion-free hypercentral group with G/G periodic by divisible. Then G is divisible. Proof. By 2.13 the factor G/δ(G) is torsion-free. Hence we may assume that δ(G) = 1. If G is abelian, then clearly G = 1. If not pick z in ζ2 (G) \ ζ1 (G). Then [z, G] ≤ ζ1 (G), as an image of G/G , is periodic by divisible, as well as non-trivial, torsion-free and reduced. This is impossible and the lemma follows.

3

The main proofs

3.1 The Proof of Proposition 1.1 We omit below reference to the trivial implications in the Proposition. un’s a) If G/G = 1, then G = ρ(G) is hypercentral by 2.2. Therefore G = 1 by Gr¨ Lemma ([2] Vol. 1, p. 48). b) If G/G is divisible, then so is G/ρ(G); hence G = ρ(G) and consequently G is divisible, see 2.2 and [2] 9.23. c) Suppose G/G is divisible by ﬁnite. Then so is G/ρ(G) and hence G/ρ(G) is ﬁnite. Also G ≤ ρ(G) and ρ(G)/ρ(G) is divisible by ﬁnite. Thus ρ(G) is divisible by ﬁnite

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by Theorem 1.2 of [4] and Part c) follows. d) Suppose G/G is ﬁnite. Since G is hypercentral by 2.2, it follows from [4] Theorem 1.5 that G is periodic and divisible-abelian by ﬁnite. Since G/G is ﬁnite, Part d) follows. e) Here G is assumed periodic. Suppose G/G is 2-divisible by ﬁnite. Let U = O2 (G) be the unique maximal 2 -subgroup of G; such exists by 2.1. Then G/U is a hypercyclic 2-group and consequently is hypercentral. Therefore G/U is divisible-abelian by ﬁnite ([4] Theorem 1.1). Set N/U = δ(G/U). Then N has ﬁnite index in G. By 2.8 we have N = S × U for some divisible 2-group S. Necessarily S is abelian.

3.2 The Proof of Theorem 1.3 a) Suppose G/G is a 2 -group by divisible. Then G has no non-trivial image of order 2. Set U = O2 (G). Then G/U is hypercentral by 2.10. By 2.14 we have that G/τ (G) is divisible. Now if A is an abelian group that is a 2 -group by divisible and a 2-group by divisible, then A is 2-divisible and 2 -divisible, so A is divisible. Apply this to G/G U. Hence G/G U is divisible and therefore so is G/U by [2] 9.23. This proves Part a). b) If G/G is a 2 -group, then since G/U is hypercentral by a), so G/U is a 2 -group and consequently G = U is a 2 -group.

3.3 The Proof of Theorem 1.2 Assume G/G is divisible by ﬁnite-of-odd-order. By Theorem 1.3 if U = O2 (G) then G/U is hypercentral and divisible. Suppose U is abelian. If S is a cyclic G-section of U, then U ≤ CG (S) and always G/CG (S) is ﬁnite. Thus G = CG (S); consequently U lies in the hypercentre of G and hence G is hypercentral. In this case G is divisible by ﬁnite by [4] Theorem 1.2 again. Now assume that U is not abelian. Then G/U is divisible by ﬁnite by the previous paragraph. But G/U has no non-trivial elements of odd order and every subgroup of the 2 -group U is necessarily 2-divisible. Therefore U/U is divisible by ﬁnite. Now consider V . Then U/V is hypocentral. Since U is locally ﬁnite, U/V is locally nilpotent. Thus U/V is hypercentral by 2.4. Consequently U/V is divisible by ﬁnite by [4] Theorem 1.2 yet again. We have now proved that a) implies c). Suppose G/U is divisible and U/V is divisible by ﬁnite. To simplify notation assume V = 1. Apply 2.6 to G/δ(U) and its ﬁnite normal subgroup U/δ(U) of odd order. Thus G/δ(U) = U/δ(U) · δ(G/δ(U)). But δ(G/δ(U)) = δ(G)/δ(U), so G = U · δ(G) and G/δ(G) is ﬁnite of odd order. Consequently c) implies b). Since V ≤ G clearly b) implies a). The proof of Theorem 1.2 is complete. Remark 3.1. In Theorem 1.2, assuming a) to c) hold, G/U is divisible, so τ (G)/U is divisible and hence τ (G) = T × U for some divisible abelian 2-group by 2.8. Further G/V is hypercentral; this can be proved from a) by mixing the arguments of the ﬁrst

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and third paragraphs of the above proof. Having done this, one can give an alternative proof of Theorem 1.2. For then G/V is divisible by ﬁnite by [4] Theorem 1.2 and an easy argument shows that G/V is in fact divisible by ﬁnite-of-odd-order. Thus a) and b) are equivalent and imply c). Finally c) implies b) as before.

3.4 The Proof of Theorem 1.4 Suppose G/G is π-divisible and set U = Oπ (G). Then G has no non-trivial cyclic πimages and hence G/U is hypercentral by 2.10. By [2] 9.23 the factor G/U is π-divisible. Also given the latter G is π-divisible by 2.11. Thus so far we have a) ⇒ d) ⇒ b) ⇒ a). If G is π-divisible by a π -group, then so is G/G . Hence G/G is π-divisible and therefore c) ⇒ a) ⇒ b) ⇒ c). The proof is complete.

References [1] L. Heng, Z. Duan and G. Chen: “On hypercentral groups G with |G : Gn | < ∞”, Comm. Algebra, Vol. 34, (2006), no. 5, pp. 1803–1810. [2] D.J.S. Robinson: Finiteness conditions and generalized soluble groups, SpringerVerlag, New York-Berlin, 1972. [3] B.A.F. Wehrfritz: Infinite linear groups, Springer-Verlag, New York-Heidelberg, 1973. [4] B.A.F. Wehrfritz: “On hypercentral groups”, Cent. Eur. J. Math., Vol. 5, (2007), no. 3, pp. 596–606.

DOI: 10.2478/s11533-007-0028-y Research article CEJM 5(4) 2007 696–709

On the Lie algebra of holomorphic infinitesimal isometries of some classical complex symmetric Banach manifolds Jos´e M. Isidro

∗

Facultad de Matem´ aticas, Santiago de Compostela, Spain.

Received 23 June 2007; accepted 04 August 2007 Abstract: The Banach-Lie algebras Hκ of all holomorphic infinitesimal isometries of the classical symmetric complex Banach manifolds of compact type (κ = 1) and non compact type (κ = −1) associated with a complex JB∗ -triple Z are considered and the Lie ideal structure of Hκ is studied. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Classical symmetric complex Banach manifolds, JB∗ -triples, Cartan factors, Banach-Lie algebras, Complete holomorphic vector fields, MSC (2000): Primary 32M15, 22E65, Secondary 17C65, 17B65.

1

Introduction and preliminaries

In this paper we are concerned with classical symmetric complex Banach manifolds M κ of compact type (κ = +1) and non compact type (κ = −1), and with the Lie algebra Hκ of all holomorphic infinitesimal isometries of M κ . We refer to [20] sections 17,18 and 23 for the basic ideas and background on this topic, where one can ﬁnd the notions and relations that are used but not explained here. Let M κ be a manifold of the mentioned class. Take any point o ∈ M κ . According to above references, there is a local canonical chart of M κ at o in which the tangent space Z := To M κ is a complex Banach space which is naturally endowed with a uniquely determined ternary algebraic-metric structure known as JB∗ triple. Moreover, with respect to this local canonical chart, every inﬁnitesimal isometry ∂ on M κ can be represented in a unique way in the form X = p(z) ∂z where p : Z → Z ∂ is a continuous polynomial of degree ≤ 2. To be precise, we have X = δ(z) + pκc (z) ∂z ∗

E-mail: [email protected]

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where δ ∈ Der(Z) is a derivation of the JB∗ -triple Z and pκc is explicitly given in terms of the triple product by pκc (z) = c + κ{zcz} for z ∈ Z and ﬁxed c ∈ Z. In this paper we use this simple expression of the inﬁnitesimal isometries of M κ to analyze the Lie ideal structure of Hκ . Recall that any inﬁnitesimal isometry of M κ is a holomorphic vector ﬁeld X that is complete on M κ , and that for κ = −1 the converse is also true: every complete holomorphic vector ﬁeld on M −1 is an inﬁnitesimal isometry. We establish a bijection between the set of all closed triple ideals P in Z and the set of all closed Lie ideals in the algebra Hκ . This correspondence is given by ∂ : δ ∈ D, c ∈ P }. P ←→ J = {X = δ(z) + pκc (z) ∂z where D := L(P Z) is the closed Lie ideal generated in Der(Z) by the set of inner derivations P Z := {c z : c ∈ P, z ∈ Z} and c z := cz − zc is the map w → {czw} − {zcw}, w ∈ Z. Let us ﬁrst remark that the JB∗ -triple Z and its dual (denoted indistinctly by Z κ ) have the same closed triple ideals and the same derivations. Remark also that the Banach-Lie algebra associated to Z κ by the Kantor-K¨ocher-Tits construction (the KKT construction, see [13],[14] and [16],[17]) is Hκ . Given a closed triple ideal P in Z κ , we can apply the KKT construction to P viewed as a JB∗ -triple itself; this gives rise to another Lie algebra Jκ which turns out to be a closed Lie ideal of Hκ . Moreover, every closed Lie ideal Jκ in Hκ can be obtained in this way, and we have Jκ = JD ⊕ JκP where D, which does not depend on κ, is the closed Lie ideal generated in Der(Z) by the set of inner triple derivations P Z, ∂ and JD consists of the vector ﬁelds Xδ = δ(z) ∂z for δ ∈ D. The quadratic summand JκP , ∂ for c ∈ P . Since which depends on κ, consists of the vector ﬁelds Yc = (c + κ{z, c, z}) ∂z D is uniquely determined by P , we get the mentioned bijection which we use to study the Lie ideal structure of the algebras Hκ in some classical cases. Namely, we assume that Z is a special Cartan factor. Under this assumption a good knowledge of Der(Z) is available with which an operator-theoretic characterization of the Lie ideal L(P Z) is given and a full description of the family of closed Lie ideals in Hκ is presented. In particular, the case of Cartan factors of type IV requires a study of the closed Lie ideals in the von Neumann algebra L(H), a problem in which there has been a long standing interest since the early results of Topping [19] for a separable Hilbert space H.

2

Ideals in the Banach-Lie algebras Hκ associated with a JB∗ triple.

Let Z be a complex JB∗ -triple, and denote by M −1 and M +1 the simply connected complex symmetric Banach manifolds (with base point) associated with the hermitian Jordan triple system Z and with its opposite, respectively. Then M −1 and M +1 are complex symmetric Banach manifolds of non-compact and compact type, respectively. We refer to M κ for κ ∈ {−1, +1} as the classical complex symmetric Banach manifolds associated to Z. It is known that Aut(M κ ), the group of all biholomorphic isometries of M κ , is a real

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Banach-Lie group when endowed with the topology of local uniform convergence over M κ . Its Lie algebra Hκ consists of all holomorphic infinitesimal isometries of M κ , which endowed with the topology of local uniform convergence becomes a Banach-Lie algebra. In particular, H−1 consists of all complete holomorphic vector ﬁelds on M −1 . Our goal is to ﬁnd the closed Lie ideals in these two real Banach-Lie algebras. Recall that Der(Z) is the Lie algebra of all derivations of the JB∗ -triple Z, and that ∂ Hκ consists of the polynomial vector ﬁelds X = δ(z) + pκc (z)) ∂z where δ ∈ Der(Z) κ and, for each c ∈ Z, the polynomial pc is given in terms of the triple product in Z by pκc (z) = c + κ{zcz}, z ∈ Z. It will be convenient to distinguish between the linear map ∂ δ ∈ Der(Z) and the vector ﬁeld Xδ := δ(z) ∂z . Similarly, we distinguish between the ∂ κ κ . Moreover, we have the direct polynomial pc : Z → Z and the vector ﬁeld Yc := pκc (z) ∂z Banach space sum decomposition Hκ = D ⊕ Pκ

(1)

where D := {Xδ : δ ∈ Der(Z)} and Pκ := {Ycκ : c ∈ Z}, and the following rules hold [D, D] ⊂ D,

[D, Pκ ] ⊂ Pκ ,

[Pκ , Pκ ] ⊂ D.

Lemma 2.1. Let J be any closed Lie ideal in Hκ . Then i) D := {δ ∈ Der(Z) : Xδ ∈ J} is a closed Lie ideal in Der(Z) and P := {c ∈ Z : Ycκ ∈ J} is a closed ideal in the JB∗ -triple Z. ii) If X = Xδ + Ycκ ∈ J then Xδ ∈ J and Ycκ ∈ J. Proof. Clearly D is a closed real vector subspace of Der(Z). Let δ ∈ D and Δ ∈ Der(Z) be arbitrarily given. Then Xδ ∈ J and XΔ ∈ Hκ , therefore [Xδ , XΔ ] = X[δ,Δ] ∈ J, which shows that [δ, Δ] ∈ D. Thus D is a closed Lie ideal in Der(Z). ∂ Clearly P is a closed real vector subspace of Z. The circular vector ﬁeld I := iz ∂z ∂ belongs to Hκ . Let c ∈ P ; then Ycκ ∈ J and as J is an ideal, [I, Ycκ ] = pκic (z) ∂z = Yicκ ∈ J, whence i P ⊂ P which shows that P is a complex subspace of Z. By ([5] proposition 1.4,[11] proposition 1.8) in order to prove that it is an ideal in Z, it suﬃces to see that P is invariant under each derivation of Z. Let c ∈ P and δ ∈ Der(Z) be arbitrarily given. κ Then Ycκ ∈ J and therefore [Xδ , Ycκ ] = Yδ(c) ∈ J which shows δ(c) ∈ P as required. κ κ Let X = Xδ + Yc ∈ J. Then [Xδ + Yc , I] = Yicκ ∈ J, hence ic ∈ P and by the previous step c ∈ P , that is Ycκ ∈ J. Now Xδ = X − Ycκ ∈ J. κ Thus, the ideal J is invariant under the canonical projections πD : H → D and πP : Hκ → P and hence we have the direct Banach space sum decomposition J = JD ⊕ JP

(2)

where JD := J ∩ D and JP := J ∩ P . To shorten the notation we introduce the following symbols c c := c c − c c (3)

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where cc is the operator z → {cc z} for z ∈ Z. Proposition 2.2. Let Hκ = D ⊕ P be the real Banach–Lie algebra associated with a JB∗ -triple Z. Let JD ⊂ D be a closed real vector subspace and let JP ⊂ P be a closed complex vector subspace. Then J := JD ⊕ JP is a closed Lie ideal in Hκ if and only if the following two conditions hold: i) JD is a closed Lie ideal in D and P Z := {Xcc : c ∈ P, c ∈ Z} ⊂ JD . ii) P := {c ∈ Z : Ycκ ∈ JP } is a closed triple ideal in Z and δ(Z) ⊂ P if Xδ ∈ JD . Proof. X = Xδ + Ycκ ∈ J and W = Xδ + Ycκ ∈ Hκ be arbitrarily given.Then [X, W ] = [Xδ , Xδ ] + [Xδ , Ycκ ] + [Ycκ , Xδ ] + [Ycκ , Ycκ ] κ = X[δ,δ ] + 2Xcc + Yδ(c )−δ (c) .

(4)

Assume J is a closed Lie ideal in Hκ . Then [X, W ] ∈ J and by (2.1) we have [Xδ , Xδ ] ∈ J, hence equation (4) gives κ 2Xcc + Yδ(c )−δ (c) ∈ J which is equivalent to the set of conditions Xcc ∈ JD ,

κ Yδ(c )−δ (c) ∈ JP .

The ﬁrst of these relations gives part of i). The second one is equivalent to δ(c )−δ (c) ∈ P . By lemma (2.1) we know that P is a closed triple ideal in Z, therefore by ([1], remark 4.4) it is invariant under each derivation in Der(Z). Thus δ (c) ∈ P , and therefore δ(c ) ∈ P whenever δ ∈ D and c ∈ Z as stated. Conversely, assume that JD and JP satisfy conditions i) and ii), and set J := JD ⊕ JP . Let J be the closure of J in Hκ , and take any X ∈ J and any sequence (Xn )n∈N in J such that limn→∞ Xn = X. Let Xn = Xδn + Ycκn and X = Xδ + Ycκ be the decompositions of Xn and X given by (1), respectively. Since the projectors in (1) are continuous, we have lim Xδn = Xδ

n→∞

and

lim Ycκn = Yc .

n→∞

Since JD and JP are closed, we have Xδ ∈ JD and Ycκ ∈ JP and X = Xδ + Ycκ ∈ J, which is therefore closed. By ([1], remark 4.4) every closed triple ideal in Z is invariant under each δ ∈ Der(Z), thus δ(P ) ⊂ P . But then i) and ii) simply say that the bracket [X, W ] ∈ J whenever X ∈ J and W ∈ Hκ , hence J is a Lie ideal in Hκ . κ For a given closed Lie ideal J in H , we set D := {δ ∈ Der(Z) : Xδ ∈ J} and P := {c ∈ Z : Ycκ ∈ J}. It is now clear that each closed Lie ideal J in Hκ determines (and is determined by) a unique pair (D, P ), where D is a closed Lie ideal in Der(Z), P is a closed triple ideal in Z and the components of (D, P ) are linked by the conditions i) δ(Z) ⊂ P

whenever

δ ∈ D,

i i) c c ∈ D

whenever

c ∈ P and c ∈ Z.

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The ideal J associated with (D, P ) is ∂ : δ ∈ D, c ∈ P }. J = {X = δ(z) + pκc (z) ∂z

(5)

It is easy to see that the mapping (D, P ) ←→ J given by (5) is a bijection. Proposition 2.3. Let Z be a complex JB∗ -triple and P a closed triple ideal in Z. Then there is a unique closed Lie ideal D in Der(Z) such that the pair (D, P ) satisfies conditions i) and ii) in (2.2). In fact D = L(P Z). Proof. Assume that P is a closed triple ideal in Z and let us seek for a closed Lie ideal D in Der(Z) such that the pair (D, P ) satisﬁes conditions i) and ii) in (2.2). By ii), D contains the set of inner derivations P Z and, as D is a closed Lie ideal, it must contain the closed Lie real ideal generated by P Z, which is L(P Z). Thus we must have D ⊃ L(P Z). Let us take the smallest possible choice for D and check that it satisﬁes condition ii), for which we recall that L(P Z) = spanR [P Z, Der(Z)]. Each element in P Z is of the form δ = cz − zc for some c ∈ P and some z ∈ Z, therefore, since P is a triple ideal, we have δ(w) = {czw} − {zcw} ∈ P for all w ∈ Z, that is δ(Z) ⊂ P . Thus each generator of D satisﬁes condition i) in (2.2). For δ ∈ P Z and Δ ∈ Der(Z) we have [δ, Δ](Z) ⊂ δ Δ(Z) − Δ δ(Z) ⊂ δ(Z) − Δ(P ) ⊂ P − P ⊂ P since δ(Z) ⊂ P and closed triple ideals are invariant under all derivations. Then each f ∈ spanR [P Z, Der(Z)] satisﬁes f (Z) ⊂ P and, as P is closed, also the elements g ∈ spanR [P Z, Der(Z)] satisfy g(Z) ⊂ P and conditions i) and ii) hold. Hence the pair (D, P ) is associated with a closed Lie ideal in the algebra Hκ . If P and P are closed triple ideals in Z with P = P then the corresponding ideals in Hκ are diﬀerent since they have diﬀerent projections onto the quadratic component P. Thus ∂ (6) P ←→ J := {X = (δ(z) + pκc (z)) : δ ∈ L(P Z), c ∈ P } ∂z establishes a bijection between the set of closed triple ideals P in Z and the set of closed Lie ideals in the algebra Hκ . As an immediate consequence we get Lemma 2.4. Let Z be a JB∗ -triple and let J = JD ⊕ JP be a closed Lie ideal in Hκ . Then JD = {0} ⇐⇒ JP = {0} ⇐⇒ J = {0}. Besides, if all derivations of Z are inner then J = Hκ ⇐⇒ P = Z.

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Proof. Assume P = {0}. Then D = L(P Z) = {0}. Now assume D = {0}. By condition ii) we have P Z ⊂ D = {0}, and so c c = 0 for all c ∈ P and c ∈ Z. Thus c c = cc − c c = 0. Replacing c with ic we get i cc + c c) = 0, and therefore cc = 0. Since this holds for all c ∈ Z, we get c = 0 and P = {0}. Assume P = Z therefore by condition ii) we have Z Z ⊂ D. Recall that the real Lie algebra generated by Z Z is Inder(Z), the algebra of all inner derivations of Z. Since we have assumed that all derivations of Z are inner, we get Der(Z) ⊂ D and so D = Der(Z). But then (D, P ) = (Der(Z), Z) and J = Hκ . The converse is clear. We have seen that each closed triple ideal P in Z gives rise to (or comes from) a closed Lie ideal J in the algebra Hκ . However, the analogous statement for closed Lie ideals D in Der(Z) is not true. Indeed, those D appearing in the process satisfy D ⊂ Inder(Z) ⊂ Der(Z), and there are JB∗ -triples with non-inner derivations [10]. Thus the following question arises naturally: does every closed Lie ideal D ⊂ Inder(Z) come from a closed Lie ideal J ⊂ Hκ ? Lemma 2.5. Let Z be a complex JB∗ -triple and let D be a closed Lie ideal in Der(Z). Then D is associated by (2.3) with a closed triple ideal P in Z if and only if Pm := P D(Z) , the closed triple ideal generated by D(Z), satisfies Pm Z ⊂ D. Proof. Recall that there is at most one triple ideal P such that the pair (D, P ) satisﬁes conditions i) and ii) in (2.2). If such a P exists, by i) we must have δ(Z) ⊂ P whenever δ ∈ D, therefore P ⊃ δ∈D δ(Z) = P D(Z) , and as P is a closed triple ideal we have P ⊃ Pm . It is immediate to check that if Pm does not satisfy condition ii) then no other ideal P does. Hence D is associated with D if and only if Pm Z ⊂ D. κ We look for conditions on (D, P ) for J to be topologically complemented in H . Lemma 2.6. Let (D, P ) be the pair associated by (2.2) with a closed Lie ideal J in Hκ . Then the following conditions are equivalent: i) D and P are topologically complemented in Der(Z) and in Z, respectively. ii) There is a continuous projection f from Hκ onto J = JD ⊕ JP which commutes with ∂ the adjoint of the circular field I = iz ∂z . Proof. i) ⇒ ii). Let π : Der(Z) → D and π : P → P be continuous projections onto D and P , respectively, and deﬁne f : Hκ → J by f := ππD + π πP, that is f (Xδ + Ycκ ) := Xπ(δ) + Yπκ (c) for all X = Xδ + Ycκ ∈ Hκ . It is immediate to check that f is a continuous idempotent that maps Hκ onto J and satisﬁes f ([I, X]) = [I, f (X)] for X ∈ Hκ . ii) ⇒ i). If f : Hκ → J is a projection that commutes with adI, then f (D) ⊂ D and f (P) ⊂ P. Indeed, let Xδ ∈ D; then [I, Xδ ] = 0, hence 0 = f ([I, Xδ ]) = [I, f (Xδ )] which shows that f (Xδ ) ∈ D, therefore we have f (Xδ ) = Xδ for some uniquely determined δ ∈ Der(Z). The map δ → δ is a continuous projection from Der(Z) onto D. In the

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same way, using Ycκ ∈ P ⇐⇒ [I, Ycκ ] = Ycκ we prove that f (Ycκ ) = Yc for a uniquely determined c ∈ Z and the map c → c is a continuos projection from Z onto P . Lemma 2.7. Let Z be an irreducible JB∗ -triple with the inner derivation property, and let J be a closed Lie ideal in Hκ . If (D, P ) is the pair associated to J by (2.2) and both D and P are split subspaces in Der(Z) and Z, then either J = {0} or J = Hκ . Proof. By assumption P is a complemented JB∗ -triple ideal in Z which is irreducible, hence we have either P = {0} or P = Z. If P = {0} then D = {0} and so J = {0}. Now consider the case P = Z. By condition ii) in (2.2) we have Z Z ⊂ D. Since Z Z consists of inner derivations and D is a closed Lie ideal in Der(Z), we must have Inder(Z) ⊂ D ⊂ Der(Z), where Inder(Z) denotes the closed real Lie ideal generated by Z Z. But Z has the inner derivation property, hence Der(Z) = Inder(Z) and so D = Der(Z). Thus (D, P ) = (Der(Z), Z) and J = Hκ . As an immediate consequence we get the following classical result: Corollary 2.8. Let Z be an irreducible JB∗ -triple with dim(Z) < ∞. Then the Lie algebras Hκ of all holomorphic infinitesimal isometries of the classical complex symmetric manifolds M κ are algebraically simple, i.e., they have no proper Lie ideals. Proof. Since dim(Z) < ∞, we have dim(Hκ ) < ∞ and every Lie ideal J in Hκ is automatically closed and complemented. Moreover Z has ﬁnite rank. Let (D, P ) be the pair associated with J by (2.2). It is known ([15], chapter 8) that all irreducible JB∗ triples of ﬁnite rank are algebraically simple (i.e, they have no proper triple ideals) and that they have the inner derivation property. Hence {0} and Z are the only possible choices for P , but then we have either J = {0} or J = Hκ due to (2.4).

3

Lie ideals in the algebras Hκ associated with Cartan factors

We are now in a position to tackle the problem of characterizing the closed Lie ideals in the Banach-Lie algebras Hκ of inﬁnitesimal isometries on the classical symmetric complex Banach manifolds Mκ associated to the especial Cartan factors. We make a type by type discussion. Suppose Z is a type I Cartan factor. Thus Z is the space L(E, F ) of bounded linear operators z : E → F , where E and F are complex Hilbert spaces with dim E ≤ dim F , endowed with the operator norm and the canonical triple product 2{x, y, z} := xy ∗ z+zy ∗ x for x, y, z ∈ Z. For X = E, F , let Her(X) denote the set of hermitian elements in L(X). The algebra of derivations Der(Z) consists of the linear maps δa,b (z) = i(za + bz),

z∈Z

where a = a∗ ∈ Her(E) and b = b∗ ∈ Her(F ) and the bracket is given by [δa1 ,b1 , δa2 ,b2 ] = δ−i[a1 ,a2 ], i[b1 ,b2 ] .

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This shows that the map (ia, ib) → δa,b from iHer(E) × iHer(F ) onto Der(Z) is a Lie homomorphism whose kernel is Z = {(ia, ib) ∈ iHer(E) × iHer(F ) : bz = za ∀z ∈ Z}. By ([12] lemma 3.1) we have Z = {iρ(IE , IF ) : ρ ∈ R} and so Der(Z) ≈ i Her(E) × Her(F ) /Z

(7)

as Lie algebras. Remark that, for a derivation δ = δa,b + Z ∈ Der(Z) and an operator = δa,b (z) which does not depend on the representative we take in the z ∈ Z, we have δ(z) class to compute it. Since Der(Z) is endowed with the operator norm we have ˆ = sup i(za + bz) ≤ a + b, δ z≤1

which shows that (ia, ib) → δa,b + Z is continuous. Since it is surjective and linear, is also an open map. Lemma 3.1. Each closed triple ideal P in a Cartan factor of type I is a left L(E)-module and a right L(F )- module. Proof. Let P be a closed triple ideal in Z. For a ∈ Her(E), the map δa,0 (z) := iza, z ∈ Z, is a derivation of Z. Since P is invariant under all derivations we have iza ∈ P and, as P is a complex space, za ∈ P . But then zw ∈ P for all z ∈ P and all w ∈ L(E). In a similar manner, by considering δ0,b ∈ Der(Z) we get L(F )P ⊂ P . 1 1 ∗ ∗ As usual, Re(w) := 2 (w + w ) and Im(w) := 2i (w − w ) stand for the real and the imaginary part of the operator w ∈ L(H), where H is a complex Hilbert space, and for a non void subset S ⊂ L(H), the closed bilateral associative ideal generated by S in L(H) is denoted byJ(S). Proposition 3.2. If Z is a type I Cartan factor and P is a closed triple ideal in Z then the Lie ideal D associated with P by (2.3) is the closure (in the operator norm in Der(Z)) of the set (8) Δ := {δa,b : a ∈ Im J(Z ∗ P ), b ∈ Im J(P Z ∗)}. Proof. In our case J(Z ∗ P ) ⊂ L(E) and J(P Z ∗ ) ⊂ L(F ) are closed bilateral associative ideals, in particular they are selfadjoint spaces, and therefore the sets iIm J(Z ∗ P ) and iImJ(P Z ∗ ) are closed Lie ideals in iHer(E) and iHer(F ), respectively. Hence Δ as given by (8) is a closed real Lie ideal in iHer(E) × iHer(F ), and so D := πΔ, the closure of its image under the canonical quotient map in (7), is a closed Lie ideal in Der(Z). Moreover, for c ∈ P and z ∈ Z we have (c z)w = {czw} − {zcw} = 2iIm (cz ∗ ) w + w 2iIm (z ∗ c)

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that is, c z = δa,b where a := Im (z ∗ c) ∈ Im J(Z ∗ P ) ⊂ Her(E) b := Im (cz ∗ ) ∈ Im J(P Z ∗ ) ⊂ Her(F ). Thus P Z ⊂ D which shows that the pair (P, D) satisﬁes condition ii) in (2.2). On the other hand, let a ∈ Im J(Z ∗ P ) ⊂ L(E) and b ∈ Im (P Z ∗ ) ⊂ L(F ) be arbitrarily given. Since P is a right L(E)-module and a left L(F )-module, we have δa,b (w) = i(wa + bw) ∈ P

∀w ∈ Z,

and therefore the pair (P, D) satisﬁes condition i) in (2.2). Hence D is the ideal associated to P . Since we have already established that D is the closed Lie ideal generated by P Z, indirectly we have established L(P Z) = D. Summing up we get Theorem 3.3. Let Z = L(E, F ) a Cartan factor of type I. Then the closed Lie ideals of the algebras Hκ of holomorphic infinitesimal isometries of the symmetric Banach manifolds associated with Z are L := {X = (δ(z) + pκc (z))

∂ : δ ∈ D, c ∈ P } ∂z

where P is a closed triple ideal in Z and D is the closure (in the operator norm) of the set Δ := {δa,b : a ∈ Im J(Z ∗ P ), b ∈ Im J(P Z ∗)}. Our next step will be to study the Lie algebras associated with a Cartan factor of type II or III. Let H be a complex Hilbert space with a conjugation ξ → ξ, and let z denote the transpose of z ∈ L(H), given by z (ξ) := z ∗ (ξ), ξ ∈ H. For ε ∈ {−1, +1}, let us deﬁne Z ε := {z ∈ L(H) : z = εz}. The spaces Z ε endowed with the operator norm and the usual triple product, are JB∗ triples known as Cartan factors of types II and III respectively. In these two cases, the algebra of derivations Der(Z ε ) consists of the linear maps δa,a (z) = i(az + za ),

z ∈ Z ε.

(9)

where a ∈ Her(H). To shorten the notation we write δa := δa,a . The bracket is given by [δa δb ] = δi[a,b] , thus the map ia → δa from iHer(H) onto Der(Z ε ) is a Lie homomorphism whose kernel is Z = {ia ∈ iHer(H) : i(az + za ) = 0 ∀z ∈ Z ε }. By ([12] lemma 3.1) we have Z = {iρIH : ρ ∈ R} and so Der(Z ε ) ≈ iHer(H)/Z

(10)

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as Lie algebras. Remark that, for a derivation δˆ = δa + Z ∈ Der(Z ε ) and an operator ˆ = δa (z) which does not depend on the representative we take in the z ∈ Z ε , we have δ(z) class to compute it. Since Der(Z ε ) is endowed with the operator norm we have ˆ = sup i(az + za ) ≤ 2a, δ z≤1

which shows that ia → δa + Z is continuous. Since it is surjective and linear, is also an open map. Let Z ε ⊂ L(H) be a Cartan factor of type II or III, and let P be a closed triple ideal in Z. Then P is a closed selfadjoint and selftransposed subspace of Z. Indeed by ([5] proposition 1.4, [11] propositions 1.8 and 1.7), P is an M-ideal in Z ε hence it is invariant under all surjective complex linear isometries of Z ε , and in particular under the map z → z , z ∈ L(H). For c ∈ P and z, w ∈ Z ε we have (c z)w = 2iIm (cz ∗ ) w + w 2iIm (z ∗ c) that is, c z = δa where a := Im (cz ∗ ) ∈ Im J(S), where J(S) stands for the closed complex bilateral associative ideal generated by the set S := P Z ε∗ in the C∗ -algebra L(H). We claim that each element x ∈ J(S) satisﬁes δx (Z ε ) ⊂ P. Indeed, for a generator of J(S), say s = cz ∗ where c ∈ P and z ∈ Z ε∗ , we have δs (Z ε ) = (c z)Z ε = {czZ ε } − {zcZ ε } ⊂ P since P is a triple ideal in Z ε . Thus p(δs ) Z ε ⊂ P for any polynomial p ∈ C[X] with p(0) = 0. The set {p(δs ) : s ∈ S, p ∈ C[X], p(0) = 0} is norm dense in J(S) and P is closed, hence our claim is proved. As a consequence, the set Δ := {δa : a ∈ Im J(P Z ∗ )} is a closed Lie ideal in iHer(H) that satisﬁes Δ(Z ε ) ⊂ P. Therefore D := πΔ, the closure (in the operator norm on Der(Z)) of the image of Δ under the canonical quotient map in (10) is a closed Lie ideal in Der(Z) which satisﬁes conditions i) and ii) in (2.2) and hence D is the Lie ideal associated to P by (2.3). Summing up we get Theorem 3.4. Let Z ε ⊂ L(H) a Cartan factor of type II or III. Then the closed Lie ideals of the algebras Hκ of holomorphic infinitesimal isometries of the symmetric Banach manifolds associated with Z ε are L := {X = (δ(z) + pκc (z))

∂ : δ ∈ D, c ∈ P } ∂z

where P is a closed triple ideal in Z ε and D is the closure (in the operator norm) of the set Δ := {δa : a ∈ Im J(P Z ε∗) }.

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Remark 3.5. 1. As a byproduct, the above argument shows that the closed Lie ideal L(P Z ε∗ ) coincides with D = πΔ. 2. Symmetric Cartan factors (i.e ε = +1) are unital JB∗ -algebras, and by ([11]lemma 1.6) each closed triple ideal P is a closed bilateral associative ideal in L(H). Therefore in this case we have S := P Z ∗ = P and so J(S) = P . In order to make a similar study for Cartan factors of type IV, we need to establish some results concerning the Lie structure of the von Neumann algebra Z := L(H), where H is a (non necessarily separable) inﬁnite-dimensional complex Hilbert space. We shall consider Z from several points of view: as a C∗ -algebra, as a JB∗ -triple with the canonical triple product {x, y, z} := 12 (xy ∗ z + zy ∗ x), and as a Banach-Lie algebra in the commutator product [x, y] := xy − yx; in all cases Z is endowed with the operator norm. These mathematical structures give rise to diﬀerent notions of ideal, and our goal now is to study the relations between them. Since these three structures are closely related to one another, and the various types of binary and ternary products come all from the original product in L(H), one can expect that the corresponding ideals should also be closely related, as indeed occurs. It is known ([11] lemma 1.6) that the closed two-sided associative ideals in the C∗ algebra Z are exactly the same as the closed triple deals in the JB∗ -triple Z. We turn to the closed Lie ideals in the Banach-Lie algebra L(H). We shall use the fact that whenever dim(H) = ∞, the von Neumann algebra L(H) is properly infinite. The results we need on properly inﬁnite von Neumann algebras can be found in [4] example 48.12, deﬁnition 49.5 and characterization in page 302. Proposition 3.6. Let H be a (not necessarily separable) infinite-dimensional Hilbert space. For every closed Lie ideal L in L(H) there exists a uniquely determined closed two-sided ideal J in L(H) such that J ⊂ L ⊂ J + 1C. Proof. Due to the assumption dim(H) = ∞, L(H) is a properly inﬁnite von Neumann algebra. By ([18], theorem 1), there is a closed two-sided ideal J in L(H) such that J ⊂ L ⊂ J + 1C.

(11)

By ([7], theorem 1), there exists a (possibly non closed) uniquely determined two-sided ideal J in L(H) such that [L(H), J] ⊂ L ⊂ J + 1C. (12) By ([7], proposition 3), J is the largest of the two-sided ideals I in L(H) satisfying [L(H), I] ⊂ L, hence we must have J ⊂ J which, combined with (11) gives J ⊂ L ⊂ J + 1C ⊂ J + 1C. On the other hand, again by ([7] proposition 3), J is the smallest of the bilateral ideals I in L(H) satisfying L ⊂ I + 1C, hence we must have J ⊂ J, and so J = J which,

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again by ([7], proposition 3), is the two-sided associative ideal generated by [L(H), L] and therefore is uniquely determined by L. The following is an extension to the non separable case of a classical result of Topping [19] Corollary 3.7. Let H be a (not necessarily separable) infinite-dimensional Hilbert space. Then there are exactly two families of closed Lie ideals L in L(H): those of the form L = J and those of the form L = J + 1C, where J ranges over the closed two-sided ideals in L(H). Proof. For a given closed Lie ideal L in L(H) we let J be the only closed bilateral ideal of L(H) satisfying (12). Then it is clear that L coincides either with J + 1C or with J depending on whether or not 1 ∈ L. Remark 3.8. >From the Lie point of view there is a strong diﬀerence between the ﬁnite and the inﬁnite-dimensional situations when dealing with Z = L(H). Indeed, L(H) consists of (continuous) linear transformations δ : H → H hence, for dim(H) = n < ∞, we are dealing with a Lie algebra of square n × n matrices which has no non-trivial associative ideals, but always has the non-trivial Lie ideal sl(n, C) of the matrices with zero trace. When dim(H) = ∞ the situation is just the opposite: there is no trace ideal, but L(H) always contains non trivial associative ideals J (for instance, the ideal of compact operators). Each of these associative ideals J gives rise to two Lie ideals which are: J itself and CI + J. Next we consider Cartan factors of type IV. Let X be a complex Hilbert space with a conjugation x → x¯ and inner product (· | ·). Deﬁne a triple product by {xyz} := (x | y)z − (z | x¯)¯ y + (z | y)x,

x, y, z ∈ X.

(13)

Then X with the conjugation ¯·, the triple product {· · ·} and the norm · ∞ given by x2∞ := x2 + [ x4 − |(x | x¯)|2 ]1/2 ,

x∈X

is a Cartan factor of type IV that we denote by Z. The algebra Der(Z) of all triple derivations of Z can easily be described in terms of the space X [9]: it consists of the linear maps δρ,a (x) := iρx + ia(x) x ∈ X, where ρ ∈ R and a ∈ Her(X) is a hermitian element in L(X). Thus Der(Z) = iRIX ⊕ iHer(X) as a direct vector-space sum. Assume that dim(X) = ∞. Let L be a closed Lie ideal in Hκ and let (D, P ) be the pair associated with it by (2.2). By ([15] chapter 8), ﬁnite rank Cartan factors have no non-trivial closed triple ideals, hence we have either P = {0} (and then L = {0}) or P = Z. In the latter case we can not apply (2.4) to conclude that L = Hκ since

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dim(X) = ∞ and hence Z does not have the inner derivation property by ([10] theorem 3). By (13), each inner derivation of Z is a continuous ﬁnite rank operator on X, hence the real norm-closed Lie ideal D = L(Z Z) is contained in K(X) ∩ iHer(X), where K(X) denotes the ideal of compact operators on X. The complexiﬁcation D ⊗R C of D is a complex norm closed Lie ideal in L(X). On the other hand, by (3.7) the only closed Lie ideals in L(X) are {0},

C · IX ,

J,

C · IX ⊕ J

where J ranges over the family of closed two-sided associative ideals in L(X). Thus in our case D = K(X) ∩ iHer(X). Summing up we get the following result Theorem 3.9. Let Z be the Cartan factor of type IV associated with a complex separable infinite-dimensional Hilbert space X. Then the only norm-closed proper Lie ideal in the algebra Hκ of vector fields is ∂ : a ∈ K(X) ∩ Her(X), v ∈ Z}. L = { ia(z) + pκv (z) ∂z

Acknowledgements Supported by Ministerio de Educaci´on y Cultura of Spain, Research Project MTM 200502541.

References [1] T.J. Barton and Y. Friedman: “Bounded derivations of JB∗ -triples”, Quart. J. Math. Oxford Ser. 2, Vol. 41, (1990), pp. 255–268. [2] D. Beltit˘a and M. Sabac: Lie algebras of bounded operators, Operator Theory: Advances and Applications, Vol. 120, Birkh¨auser Verlag, Basel, 2001. [3] P. Civin and B. Yood: “Lie and Jordan structures in Banach algebras”, Pacific J. Math., Vol. 15, (1965), pp. 775–797. [4] J.B. Conway: A course in operator theory, Graduate Studies in Mathematics, Vol. 21, American Mathematical Society, Providence RI, 2000. [5] S. Dineen and R.M. Timoney: “The centroid of a JB∗ -triple system”, Math. Scand., Vol. 62, (1988), pp. 327–342. [6] C.K. Fong, C.R. Miers and A.R. Sourour: “Lie and Jordan ideals of operators on Hilbert space”, Proc. Amer. Math. Soc., Vol. 84, (1982), pp. 516–520. [7] C.K. Fong and G.J. Murphy: “Ideals and Lie ideals of operators”, Acta Sci. Math., Vol. 51, (1987), pp. 441–456. [8] L.A. Harris: “Bounded symmetric homogeneous domains in inﬁnite dimensional spaces”, Proceedings on Infinite Dimensional Holomorphy, Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973, pp. 13–40, Lecture Notes in Math., Vol. 364, Springer, Berlin, 1974.

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[9] F.J. Herv´es and J.M. Isidro: “Isometries and automorphisms of the spaces of spinors”, Rev. Mat. Univ. Complut. Madrid, Vol. 5, (1992), pp. 193–200. [10] T. Ho, J. Martinez-Moreno, A.M. Peralta and B. Russo: “Derivations on real and complex JB∗ -triples”, J. London. Math. Soc.(2), Vol. 65, (2002), pp. 85–102. [11] J.M. Isidro and W. Kaup: “Weak continuity of holomorphic automorphisms in JB∗ triples”, Math. Z., Vol. 210, (1992), pp. 277–288. [12] W. Kaup: “On real Cartan factors”, Manuscripta Math., Vol. 92, (1997), pp. 191–222. [13] M. Koecher: “Imbedding of Jordan algebras into Lie algebras I”, Amer. J. Math., Vol. 89, (1967), pp. 787–816. [14] M. Koecher: “Imbedding of Jordan algebras into Lie algebras II”, Amer. J. Math., Vol. 90, (1968), pp. 476–510. [15] O. Loos: Bounded symmetric domains and Jordan pairs, University of California at Irvine, Lecture Notes, 1997. [16] K. Meyberg: “Jordan–Triplesysteme und die Koecher-Konstruktion von LieAlgebren”, Math. Z., Vol. 115, (1970), pp. 58–78. [17] K. Meyberg: “Zur Konstruktion von Lie-Algebren aus Jordan-Triplesystemen”, Manuscripta Math., Vol. 3, (1970), pp. 115–132. [18] C.R. Miers: “Closed Lie ideals in operator algebras”, Canad. J. Math., Vol. 33, (1981), pp. 1271–1278. [19] D.M. Topping: “On linear combinations of special operators”, J. Algebra, Vol. 10, (1968), pp. 516–521. [20] H. Upmeier: Symmetric Banach manifolds and Jordan C∗ -algebras, North Holland Mathematics Studies, Vol. 104, North-Holland Publishing Co., Amsterdam, 1985.

DOI: 10.2478/s11533-007-0026-0 Research article CEJM 5(4) 2007 710–719

On canonical screen for lightlike submanifolds of codimension two K.L. Duggal∗ Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario N9B3P4, Canada

Received 03 May 2007; accepted 30 July 2007 Abstract: In this paper we study two classes of lightlike submanifolds of codimension two of semiRiemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions subject to some reasonable geometric conditions and support the results through examples. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Half lightlike submanifold, coisotropic submanifold, canonical screen distribution, screen conformal fundamental forms MSC (2000): 53B25, 53C50, 53B50.

1

Introduction

The general theory of lightlike submanifolds (see for example [7]) uses a non-degenerate screen distribution which (due to the degenerate induced metric) is not unique. Therefore, the induced objects of the submanifold depend upon the choice of a screen. Thus, it is reasonable to look for a canonical screen in lightlike geometry. Following [7] considerable work has been pursued to deal with the interdependence of induced objects and now there are large classes of lightlike hypersurfaces of semi-Riemannian manifolds with the choice of a canonical screen distribution (see [1, 2, 5, 10]), in some cases subject to a reasonable geometric condition. Recently, the present author has introduced the concept of induced scalar curvature for a class of lightlike hypersurfaces of Lorentzian manifolds [4] using the choice of a canonical screen distribution. Continuing our study in this direction, a next step is to ﬁnd canonical screens for ∗

E-mail: [email protected]

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lightlike submanifolds of codimension two [6]. There are two such classes of submanifolds explained as follows. Let (M, g) be a codimension two lightlike submanifold of an (m+2) ¯ g¯ ) of constant index q ≥ 1, where g is the dimensional semi-Riemannian manifold ( M, induced degenerate tensor ﬁeld of g¯ on M [6] and m > 1. All manifolds are paracompact and smooth. Denote by F (M) the algebra of smooth functions on M and by Γ(E) the F (M) module of smooth sections of a vector bundle E over M and the same notation for any other vector bundle. There exists a vector ﬁeld ξ ∈ Γ (T M), ξ = 0, such that g ( ξ , X ) = 0, for any X ∈ Γ (T M). For each tangent space Tx M we consider ¯ : g¯ ( X , U ) = 0, ∀ U ∈ Tx M } Tx M ⊥ = {X ∈ Tx M ¯ . The a degenerate 2 - dimensional orthogonal (but not complementary) subspace of Tx M radical subspace Rad Tx M ⊆ Tx M ⊥ is either a 1-dimensional or 2-dimensional subspace of Tx M. There exists a complementary non-degenerate distribution S(T M) to Rad T M in T M, called a screen distribution of M, with the following orthogonal distribution T M = Rad T M ⊕orth S(T M). The submanifold (M, g, S (T M)) is called a half lightlike submanifold [6, 8] if dim(Rad T M) = 1. We use the term half lightlike since for this class T M ⊥ is half lightlike. On the other hand, if dim(Rad T M) = 2, then, Rad T M = T M ⊥ and (M, g, S(T M)) is called a coisotropic submanifold [11]. The objective of this paper is to show that there exist canonical screen distributions for a large variety of both the above stated classes. We deal with these two classes separately in sections 2 and 3 respectively, prove one main theorem for each class and support the results through examples.

2

Half lightlike submanifolds

Let (M, g, S(T M)) be a half lightlike submanifold of a semi-Riemannian manifold ¯ (M , g¯). Then, there exist vector ﬁelds ξ , u ∈ Tx M ⊥ such that g¯ ( ξ, v ) = 0,

g¯ ( u , u ) = 0,

∀ v ∈ Tx M ⊥ .

The above relations imply that ξ ∈ Rad Tx M. Consider the orthogonal complementary ¯ Certainly ξ and u belong to Γ (S(T M ⊥ ). Choose distribution S(T M ⊥ ) to S(T M) in T M. u as a unit vector ﬁeld, with g¯ ( u, u ) = = ± 1. We brieﬂy summarize the following results (for details see [8]). Let D = span{u} be a supplementary distribution to Rad T M in S(T M ⊥ ). Hence we have the following orthogonal decomposition S(T M ⊥ ) = D ⊥ D⊥ , where D ⊥ is the orthogonal complementary distribution to D in S(T M ⊥ ). Let F be a 1dimensional non-null subbundle of D ⊥ . Then, for any local null section ξ of Rad(T M) on a

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K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

coordinate neighborhood U ⊂ M, there exists a uniquely deﬁned vector ﬁeld N ∈ Γ ( D⊥ ) satisfying g¯(N, ξ) = 1,

∀ W ∈ Γ(S(T M)|U )

g¯(N, N) = g¯(N, W ) = 0,

(2.1)

if and only if N is given by g¯(V, V ) 1 {V − ξ }, V ∈ Γ(F|U ) (2.2) g¯(ξ, V ) 2 g¯(ξ, V ) ¯ and ∇ be the Levi-Civita connection on M ¯ and a linear such that g¯(ξ, V ) = 0. Let ∇ connection on M respectively and P the projection of T M on S(T M). The local Gauss and Weingarten formulas are: N=

¯ X Y = ∇X Y + D1 (X , Y ) N + D2 (X , Y )u , ∇ ¯ X N = − AN X + τ (X) N + ρ(X) u , ∇ ¯ X u = − Au X + ψ(X) N, ∇ ∇X P Y = ∇X ξ =

∇∗X P Y − A∗ξ X

(2.3) (2.4) (2.5)

+ E(X , P Y ) ξ,

(2.6)

− τ (X) ξ,

(2.7)

∀X , Y ∈ Γ (T M)

where D1 and D2 are the lightlike and the screen second fundamental forms of M respectively, τ, ρ and ψ are 1-forms on M. Both AN and Au are the shape operators of M. Also E is the local second fundamental form of S(T M) with respect to Rad T M, A∗ξ is the shape operator of the screen distribution and ∇∗ is the metric connection on S(T M) but, in general, ∇ is not a metric connection on M. Indeed, ∀ X, Y, Z ∈ Γ(T M), we have (∇X g)(Y, Z) = D1 (X, Y )η(Z) + D1 (X, Z)η(Y ),

η(X) = g(X, N).

(2.8)

Using (2.1) and (2.3) - (2.7) we obtain D1 (X, ξ) = 0,

D1 (X, P Y ) = g(A∗ξ X, P Y ),

τ (X) = g¯(∇X N, ξ),

(2.9)

ρ(X) = ¯ g (∇X N, u),

ψ(X) = −D2 (X, ξ),

(2.10)

E(X, P Y ) = g¯(∇X P Y, N).

(2.11)

Suppose a screen S(T M) changes to another screen S(T M) , where {ξ, N, Wa , u } and {ξ, N , Wa , u } respectively are two quasi-orthonormal frame ﬁelds for the same null section ξ. The following are the transformation equations due to this change (for details see [7, pages 164-165]). Wa

=

m−1

Aba (Wb − b f b ξ) ;

b =1

1 N =N− 2

m−1

u = u − f ξ,

a fa2

+ f

2

a =1

1 ∇X P Y = ∇X P Y + D1 (X, P Y ) 2

ξ+ m−1

m+1

fa Wa + fu,

a =1

+D2 (X, P Y )fξ − D1 (X, P Y )

m−1 a=1

(2.13)

a fa2 + f 2

a=1

(2.12)

ξ

fa Wa

(2.14)

K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

D1 (X, Y ) = D1 (X, Y ), D2 (X, Y ) = D2 (X, Y ) − D1 (X, Y )f, 1 ||W ||2 − f 2 D1 (X, P Y ) E (X, P Y ) = E(X, P Y ) − 2 + g(∇X P Y, W ) + D2 (X, P Y )f, where W =

m a=1

713

(2.15)

(2.16)

fa Wa . Let ω be the dual 1-form of W given by ω(X) = g(X, W ),

∀ X ∈ Γ(T M).

(2.17)

Denote by S the ﬁrst derivative of a screen distribution S(T M) given by S(x) = span{[X, Y ]x ,

Xx , Yx ∈ S(T M),

x ∈ M},

(2.18)

where [, ] denotes the Lie-bracket. If S(T M) is integrable, then, S is a sub bundle of S(T M). We state and prove the following theorem:

Theorem 2.1. Let (M, g, S(T M)) be a half lightlike submanifold of a semi-Riemannian ¯ m+2 , g¯) with m > 1. Suppose the subbundle F of D⊥ admits a covariant manifold (M constant non-null vector ﬁeld. Then, with respect to a section ξ of Rad T M, M can admit an integrable screen S(T M). Moreover, if the ﬁrst derivative S deﬁned by (2.18) coincides with S(T M), then, S(T M) is a canonical screen of M, up to an orthogonal transformation with a canonical lightlike transversal vector bundle and the screen second fundamental form E is independent of a screen distribution. Proof. By hypothesis, consider (without any loss of generality), along M, a unit covariant constant vector ﬁeld V ∈ Γ(F|U ), that is, g¯(V, V ) = e = ±1. To satisfy the condition given in (2.2), we choose a section ξ of Rad T M such that g¯(V, ξ) = 0. For convenience in calculations, we set g¯(V, ξ) = θ−1 . Using this and (2.2), the null transversal vector bundle of M takes the form N = θ (V −

eθ ξ ). 2

(2.19)

Then using (2.19) in (2.4) and (2.7) we get ¯ X N, ξ) = X(θ) g¯(V, ξ) − τ (X) = g¯(∇

e ¯ X ξ, ξ) (θ)2 g¯(∇ 2

= X(θ) (θ)−1 = X(log θ).

e¯ 2 ¯ ¯ ρ(X) = g¯(∇X N, u) = g¯ ∇X (θV ) − ∇X (θ ξ), u) 2 e θ2 ψ(X). = 2

(2.20)

(2.21)

Using above value of τ , (2.19) and (2.7) we obtain 2 ¯ X N = X(θ)V − e θX(θ)ξ + e θ2 A∗ X + eθ ψ(X)u. ∇ ξ 2 2 2

(2.22)

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K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

On the other hand, substituting the value of τ and ρ = 0 in (2.4), we get 2 ¯ X N = −AN X + X(θ)V − e θ X(θ) ξ + eθ ψ(X)u. ∇ 2 2

(2.23)

Equating (2.21) and (2.22) we obtain AN X = −

e θ2 ∗ A X, 2 ξ

∀X ∈ Γ(T M|U ).

(2.24)

Since A∗ξ is symmetric with respect to g, the Eq. (2.24) implies that AN is also symmetric with respect to g, which further follows from [8, page 128] that the screen distribution S(T M) is integrable. This means that S is a subbundle of S(T M). Using (2.24) in the second Eq. of (2.11) we get E(X, P Y ) = −

e θ2 D1 (X, Y ), 2

∀X, Y ∈ Γ(T M|U ).

(2.25)

Using (2.24), (2.16) and D1 = D1 we obtain g(∇X P Y, W ) =

1 ||W ||2 − f 2 D1 (X, Y ) − D2 (X, Y )f 2

(2.26)

∀ X, Y ∈ Γ(T M|U ). Since the right hand side of (3.17) is symmetric in X and Y , we have g([X, Y ], W ) = ω([X, Y ]) = 0, ∀X, Y ∈ Γ(S(T M)|U ), that is, ω vanishes on S. By hypothesis, if we take S = S(T M), then, ω vanishes on this choice of S(T M) which implies from (2.17) that W = 0. Therefore, the functions fa vanish. Finally, substituting this data in (2.16) it is easy to see that the function f also vanishes. Thus, the transformation b Eqs. (2.12), (2.13) and (2.14) become Wa = bm−1 =1 Aa Wb (1 ≤ a ≤ m − 1), N = N and E = E where (Aba ) is an orthogonal matrix of S(T M) at any point x ∈ M. Therefore, S(T M) is a canonical screen up to an orthogonal transformation with a canonical transversal vector ﬁeld N and the screen fundamental form E is independent of a screen distribution. This completes the proof. To understand some examples of half lightlike submanifolds, satisfying theorem 2.1, we ﬁrst quote the following result. Proposition 2.2 [6]. Let (M, g, S(T M)) be a half-lightlike submanifold of a semi– ¯ , with g¯ of index q ∈ {1, . . . , m+1}. Then we have the following: Riemannian manifold M (i) If u is spacelike then S(T M) is of index q − 1. In particular S(T M) is Riemannian for q = 1 and Lorentzian for q = 2. (ii) If u is timelike then S(T M) is of index q − 2. In particular S(T M) is Riemannian for q = 2 and Lorentzian for q = 3. It follows from proposition 2.2 (i) that M can be a half lightlike submanifold of a Lorentzian manifold for which g¯(u, u) = = 1. Thus, it is obvious from the structure equations that we choose g¯(V, V ) = e = −1, a covariant constant timelike unit vector ﬁeld. There are many examples of n-dimensional product Lorentzian spaces (such as warped product globally hyperbolic spacetime [3]) which posses at least one timelike covariant

K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

715

constant vector ﬁeld and, therefore, can satisfy the hypothesis of the theorem 2.1. In particular, M provides a physical model of null 2-surfaces in a 4-dimensional space time of general relativity. For the remaining cases when q = 2, we refer to two examples given in [8, 9], both of which can satisfy the theorem 2.1.

3

Coisotropic submanifolds

For this case, dim(Rad T M) = 2 and Rad T M = T M ⊥ which implies that T M ⊥ is totally lightlike. There exist ﬁelds of frames {ξ1 , ξ2 , W1 , . . . , Wm−2 } and {ξ1 , ξ2 , W1 , . . . , Wm−2 , N1 , N2 } ¯ respectively such that Rad T M = span{ξ1 , ξ2} and the canonical normal on M and M null bundle NM = span{N1 , N2 } satisfying g¯(ξi , ξj ) = g¯(Ni , Nj ) = 0,

∀i, j = 1, 2.

g¯(Ni , ξj ) = δij ,

Following are the Gauss and Weingarten equations [11]: ¯ X Y = ∇X Y + ∇ Di (X , Y ) Ni ,

(3.1)

i

¯ X Ni = − AN X + τij (X) Nj , ∇ i ∗ Ei (X , P Y ) ξi, ∇X P Y = ∇X P Y + i

∇X ξi = − A∗ξi X − τij (X) ξj ,

(3.2) (3.3)

∀X , Y ∈ Γ (T M),

(3.4)

where i, j = 1, 2, Di are the local second fundamental forms of M with respect to the normals Ni , ANi are the respective shape operators of M and τij are 1-forms on M. Also Ei are the local second fundamental forms of S(T M) with respect to Rad T M, A∗ξi are the respective shape operators of the screen distribution and ∇∗ is the metric connection on S(T M). D1 (X, ξ1 ) = D2 (X, ξ2 ) = 0,

Di (X, P Y ) = g(A∗ξi X, P Y ),

Ei (X, P Y ) = g¯(∇X P Y, Ni ) = g(ANi X, P Y ).

(3.5) (3.6)

Suppose a screen S(T M) changes to another screen S(T M) , where , N1 , N2 } is another quasi-orthonormal frame ﬁelds for the same pair {ξ1 , ξ2 , W1 , . . . , Wm−2 of null sections {ξ1 , ξ2}. The following are the transformation equations due to this change.

Wa =

m−2

Aba Wb − b

b =1

Ni

= Ni +

2 j=1

Nij ξj +

2

fib ξi

(3.7)

fia Wa ,

(3.8)

i=1 m−2 a =1

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K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

with the conditions 2Nii = −

m−2

a (fia )2 ,

N12 + N21 +

a=1

∇X P Y = ∇X P Y − −

a=1

a f1a f2a = 0.

(3.9)

a=1

Di (X, Y ) = Di (X, Y ),

m−2

m−2

∀X , Y ∈ Γ (T M), 2 Di (X, P Y )Nij ξj

(3.10)

2 j=1

2

i=1

Di (X, P Y )fia Wa ,

(3.11)

i=1

1 E1 (X, P Y ) = E1 (X, P Y ) − ||Z1 ||2 D1 (X, P Y ) + N21 D2 (X, P Y ) 2 + g(∇X P Y, Z1 ) − g(Z2 , Z2 ))D2 (X, P Y ) (3.12) 1 E2 (X, P Y ) = E2 (X, P Y ) − ||Z2 ||2 D2 (X, P Y ) + N12 D1 (X, P Y ) 2 + g(∇X P Y, Z2 ) − g(Z1 , Z1 )D1 (X, P Y ) (3.13) where Zi = m a=1 fia Wa are two characteristic vector ﬁelds of the screen change. Let ωi be the respective dual 1-forms of Zi given by ωi (X) = g(X, Zi ),

∀ X ∈ Γ(T M).

(3.14)

It is known that the second fundamental forms and their respective shape operators of a non-degenerate submanifold are related by means of the metric tensor. Contrary to this we see from Eqs. (3.5) and (3.6) that there are interrelations between the lightlike and the screen second fundamental forms and their respective shape operators. This interrelation indicates that the lightlike geometry depends on a choice of screen distribution. While we know from Eq. (3.10) that the lightlike second fundamental forms are independent of a screen, the same is not true for the screen fundamental forms (see Eqs. (3.12) and (3.13)), which is the root of non-uniqueness anomaly in the lightlike geometry. Since, in general, it is impossible to remove this anomaly, we consider a class of coisotropic submanifolds M whose lightlike and screen fundamental forms are related by two conformal non-vanishing smooth functions in F (M). The motivation for this geometric restriction comes from the classical geometry of non-degenerate submanifolds for which there are only one type of fundamental forms with their one type of respective shape operators. Thus, we make the following deﬁnition. Definition 3.1. A coisotropic submanifold (M, g, S(T M)) of a semi-Riemannian man¯ g¯) is called a screen locally conformal submanifold if its screen fundamental ifold (M, forms Ei are conformally related to the corresponding lightlike fundamental forms Di by Ei (X, P Y ) = ϕi Di (X, Y ),

∀X, Y, Γ(T M|U ),

i ∈ {1, 2},

where ϕi s are non-vanishing smooth functions on a neighborhood U in M.

(3.15)

K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

717

In order to avoid trivial ambiguities, we will consider U to be connected and maximal in the sense that there is no larger domain U ⊃ U on which Eq. (3.15) holds. In case U = M the screen conformality is said to be global. Moreover, the above deﬁnition will also hold for a coisotropic submanifold of codimension higher than two.

Theorem 3.2. Let (M, g, S(T M)) be a codimension two coisotropic screen conformal ¯ m+2 , g¯) with m > 1. Then, submanifold of a semi-Riemannian manifold (M (a) any choice of a screen distribution is integrable and (b) the two 1-forms ωi in (3.14) vanish identically on the ﬁrst derivative S given by (2.18). (c) If S coincides with S(T M), then, there exists a pair of null sections {ξ1 , ξ2 } of Γ(Rad T M) with respect to which S(T M) is a canonical screen distribution of M, up to an orthogonal transformation with a canonical pair {N1 , N2 } of lightlike transversal vector bundle and the screen fundamental forms Ei are independent of a screen distribution. Proof. Using (3.15) in (3.6) and the second equation of (3.5) we get ANi X = ϕi A∗ξi X,

∀X ∈ Γ(T M|U ).

(3.16)

Since each A∗ξi is symmetric with respect to g, Eq. (3.16) implies that each ANi is also symmetric with respect to g, which further follows from [11, page 38] that any choice of a screen distribution is integrable proving (a). Choose an integrable screen S(T M). This means that S is a subbundle of S(T M). Using (3.15) in (3.12) and Di = Di we obtain 1 g(∇X P Y, Z1 ) = ||Z1 ||2 D1 (X, Y ) + (g(Z2 , Z2 ) − N21 ) D2 (X, Y ) 2

(3.17)

∀ X, Y ∈ Γ(T M|U ). Since the right hand side of (3.17) is symmetric in X and Y , we have g([X, Y ], Z1 ) = ω1 ([X, Y ]) = 0, ∀X, Y ∈ Γ(S(T M)|U ), that is, ω1 vanishes on S. Similarly, repeating above steps for Eq. (3.13) we claim that ω2 vanishes on S. If we take S = S(T M), then, both ωi vanish on this choice of S(T M) which implies from (3.14) that both the characteristic vector ﬁelds Zi vanish. Therefore, all the functions fia vanish. Finally, substituting this data in (3.9), (3.12) and (3.13) it is easy to see that all the functions Nij also vanish. Thus, the transformation Eqs. (3.7), (3.8), (3.12) (3.13) b b become Wa = bm−1 =1 Aa Wb (1 ≤ a ≤ m − 1), Ni = Ni and Ei = Ei where (Aa ) is an orthogonal matrix of S(T M) at any point x ∈ M. Therefore, S(T M) is a canonical screen up to an orthogonal transformation with canonical transversal vector ﬁelds Ni and the screen fundamental forms Ei are independent of a screen distribution. This completes the proof. To present some examples of coisotropic submanifolds, satisfying theorem 3.2, we ﬁrst quote the following result:

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K.L. Duggal / Central European Journal of Mathematics 5(4) 2007 710–719

Theorem 3.3. [11, page 43]. Let (M, g, S(T M)) be a proper totally umbilical coisotropic ¯ (c), g¯). Then the submanifold of a semi-Riemannian manifold of constant curvature (M screen distribution S(T M) is integrable, if and only if, each 1-form τij induced by S(T M) satisﬁes d(Tr(τij )) = 0, where Tr(τij ) is the trace of the matrix (τij ). Since the primary result of theorem 3.2 is the existence of an integrable screen, it follows from the above theorem that a large class of totally umbilical coisotropic lightlike ¯ submanifolds of (M(c), g¯) are candidates for the existence of a canonical screen distribution.

Acknowledgement This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

References [1] M.A. Akivis and V.V. Goldberg: “On some methods of construction of invariant normalizations of lightlike hypersurfaces”, Diﬀerential Geom. Appl., Vol. 12, (2000), pp. 121–143. [2] C. Atindogbe and K.L. Duggal: “Conformal screen on lightlike hypersurfaces”, Int. J. Pure Appl. Math., Vol. 11, (2004), pp. 421–442. [3] J.K. Beem and P.E. Ehrlich: Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Math., Vol. 67, Marcel Dekker, New York, 1981. [4] K.L. Duggal: “On scalar curvature in lightlike geometry”, J. Geom. Phys., Vol. 57, (2007), pp. 473–481. [5] K.L. Duggal: “A report on canonical null curves and screen distributions for lightlike geometry”, Acta Appl. Math., Vol. 95, (2007), pp. 135–149. [6] K.L. Duggal and A. Bejancu: “Lightlike submanifolds of codimension two”, Math. J. Toyama Univ., Vol. 15, (1992), pp. 59–82. [7] K.L. Duggal and A. Bejancu: Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and its Applications, Vol. 364, Kluwer Academic Publishers Group, Dordrecht, 1996. [8] K.L. Duggal and D.H. Jin: “Half lightlike submanifolds of codimension 2”, Math. J. Toyama Univ., Vol. 22, (1999), pp. 121–161. [9] K.L. Duggal and B. Sahin: “Screen conformal half-lightlike submanifolds”, Int. J. Math. Math. Sci., Vol. 68, (2004), pp. 3737–3753. [10] K.L. Duggal and A. Gim´enez: “Lightlike hypersurfaces of Lorentzian manifolds with distinguished screen”, J. Geom. Phys., Vol. 55, (2005), pp. 107–122. [11] D.H. Jin: “Geometry of coisotropic submanifolds”, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., Vol. 8, no. 1, (2001), pp. 33–46.

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[12] B. O’Neill: Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, New York, 1983.

DOI: 10.2478/s11533-007-0024-2 Research article CEJM 5(4) 2007 720–732

Decay rates of Volterra equations on RN Monica Conti1∗, Stefania Gatti2† and Vittorino Pata1‡ 1 2

Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano Via Bonardi 9, 20133 Milano, Italy

Dipartimento di Matematica, Universit` a di Modena e Reggio Emilia via Campi 213/B, 41100 Modena, Italy

Received 26 March 2007; accepted 06 July 2007 Abstract: This note is concerned with the linear Volterra equation of hyperbolic type t μ(s)Δu(t − s)ds = 0 ∂tt u(t) − αΔu(t) + 0

on the whole space RN . New results concerning the decay of the associated energy as time goes to inﬁnity were established. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Integro-diﬀerential equations, memory kernel, polynomial decay MSC (2000): 35B40, 45K05, 45M05

1

Introduction

For α > 0, we consider the linear homogeneous Volterra integro-diﬀerential equation arising in linear viscoelasticity (see [2, 3, 5, 7, 15]) for the unknown variable u = u(x, t) :

∗ † ‡

[email protected] [email protected] [email protected]

M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

RN × [0, ∞) → R ⎧ t ⎪ ⎪ ⎪ ∂tt u(t) − αΔu(t) + μ(s)Δu(t − s)ds = 0, ⎪ ⎪ ⎪ 0 ⎪ ⎨ u(x, 0) = u0 (x), ⎪ ⎪ ⎪ ∂t u(x, 0) = u1 (x), ⎪ ⎪ ⎪ ⎪ ⎩lim |x|→∞ u(x, t) = 0.

721

t > 0, (1.1)

The memory kernel μ is assumed to be a (nonnegative) piecewise smooth decreasing summable function on R+ = (0, ∞), possibly unbounded in a neighborhood of zero, of total mass ∞ κ= μ(s)ds < α. 0 2

Setting H = L (R ), endowed with the norm · and the inner product ·, ·, and V = H 1 (RN ), problem (1.1), with initial data N

(u0 , u1 ) ∈ V × H, admits a unique generalized solution (see the subsequent Sec.3) u(t) ∈ C([0, ∞), V )

with

∂t u(t) ∈ C([0, ∞), H).

This paper focused on the analysis of the decay of the energy associated with (1.1) t t 2 2 E(t) = α − μ(s)ds ∇u(t) + ∂t u(t) + μ(s)∇u(t) − ∇u(t − s)2 ds, 0

0

depending on the decay properties of the kernel μ. The asymptotic behavior of E(t) has been previously investigated in [2, 5, 10, 11], where the authors obtained polynomial decay rates for the energy depending on the space dimension N, under the requirement that μ must be exponentially stable (cf. [10]) or polynomially stable (cf. [11]), for more regular initial data (u0 , u1 ) which also belong to L1 (RN ) × L1 (RN ). A common assumption of these works is that μ satisﬁes some diﬀerential inequalities, such as μ (s) + δ[μ(s)](p−1)/p ≤ 0, for some δ > 0 and p ∈ (2, ∞] (see also [3, 4, 6, 9]). This is clearly a rather demanding limitation on the choice of the possible kernels μ. For instance, μ must be strictly decreasing and cannot possess horizontal inﬂection points. Here, following some methods developed in the recent papers [1, 12], and using the Fourier transform techniques successfully employed in [10, 11], we are able to provide uniform decay results under quite general assumptions on μ, without appealing to unsatisfactory diﬀerential inequalities. The following two technical lemmas shall be needed in establishing the main result in the succeeding section. For the reader’s convenience, simple proofs will be presented.

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M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

Lemma 1.1. Let k ≥ 1, t > 0. Then 1 2 I(t) = r k−1 e−r t dr ∼ c(k)t−k/2 , 0

(t → ∞).

Proof. With the change of variable r 2 t = τ , the integral turns into t 1 I(t) = k/2 τ (k−2)/2 e−τ dτ. 2t 0 ∞ Choose then c(k) = (1/2) 0 τ (k−2)/2 e−τ dτ < ∞. Lemma 1.2. Let k ≥ 1, m > 0, t > 0. If m = k, 1 r k−1 dr ∼ c(m, k)t− min{m,k} , J(t) = m (1 + rt) 0

(t → ∞).

If m = k, J(t) ∼ t−k log t,

(t → ∞).

Proof. Assume ﬁrst m < k. Then, it is convenient to write 1 1 r k−1 dr. J(t) = m t 0 (t−1 + r)m Note that r k−1(t−1 + r)−m ≤ r k−1−m ∈ L1 (0, 1), by the Lebesgue dominated convergence theorem, we have 1 1 r k−1 . dr = lim t→∞ 0 (t−1 + r)m k−m Let then m ≥ k. Performing the change of variable rt = τ , we obtain 1 t τ k−1 J(t) = k dτ. t 0 (1 + τ )m ∞ If m > k, the conclusion follows by taking c(m, k) = 0 τ k−1 (1 + τ )−m dτ < ∞. If m = k, an integration by parts gives t t k−1 t k τ k−1 τ log τ dτ = log t − k dτ, k k+1 1+t 0 (1 + τ ) 0 (1 + τ ) and the integral in the right-hand side remains ﬁnite as t → ∞.

2

The Main Result

Assume that there exists a strictly increasing sequence {sn }, with s0 = 0, either ﬁnite (possibly reduced to s0 only) or converging to s∞ ∈ (0, ∞] such that, for all n > 0, μ has jumps at s = sn , and it is absolutely continuous on each interval In = (sn−1 , sn ) and on the interval I∞ = (s∞ , ∞), unless I∞ is not deﬁned. If s∞ < ∞, then μ may or may

M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

723

not have a jump at s = s∞ . Thus, μ can be singular at s = 0, and μ exists almost everywhere. The ﬂatness rate of μ is deﬁned as (see [1, 12]) Rμ = mμ ({s ∈ R+ : μ(s) > 0 and μ (s) = 0}), where

1 mμ (P) = κ

P ⊂ R+ .

μ(s)ds, P

As in [10, 11], we obtain polynomial decay results for the energy assuming that the initial data are summable and the memory kernel has an exponential or a polynomial decay at inﬁnity, provided that the ﬂatness rate of the kernel is not too large. In the following statements, Q stands for a generic increasing positive function. Theorem 2.1. Assume that Rμ < 1/2 and μ(s) ≤ Ce−δs ,

for all s ≥ 1,

for some C ≥ 0 and δ > 0. Then, E(t) ≤ Q(R)(1 + t)−N/2 , whenever E(0) + u0 + u1 L1≤ R

if N ≤ 2,

E(0) + u0 L1 + u1 L1≤ R

if N > 2.

Remark 2.2. As it will be clear from the proof of Theorem 2.1, if N > 2 and we do not require the boundedness of u0 L1 (replaced by the boundedness of u0), we still have the weaker decay estimate E(t) ≤ Q(R)(1 + t)−1 . Theorem 2.3. Assume that Rμ < 1/2 and μ(s) ≤ C(1 + s)−1−p ,

for all s ≥ 1,

for some C ≥ 0 and p > 0. Then, ⎧ ⎨(1 + t)− min{p,N p/(2p+2)} E(t) ≤ Q(R) · ⎩(1 + t)−p log(2 + t)

if 2p = N − 2, if 2p = N − 2,

whenever E(0) + u0 L1 + u1 L1 ≤ R. As for the case of bounded domains, investigated in [1], the interesting question of what happens when Rμ ≥ 1/2 remains open.

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The History Space Setting

Following the approach of [1], set the problem in the so-called history space setting. Namely, we view (1.1) as an ordinary diﬀerential equation in a proper Hilbert space accounting for the past history of the variable u (cf. [3, 4]). Extending the solution to (1.1) for all times, by setting u(t) = 0 when t < 0, and considering for t ≥ 0 the auxiliary variable η t (x, s) = u(x, t) − u(x, t − s), s ∈ R+ , the integro-diﬀerential equation of problem (1.1) reads ∞ μ(s)Δη t (s)ds = 0, ∂tt u(t) − κωΔu(t) −

t > 0,

(3.1)

0

having ω = (α − κ)/κ > 0. Note that η 0 (s) = u0 . Introducing the μ-weighted L2 -space M = L2μ (R+ ; V ), the variable η is the (unique) mild solution in the sense of [14, §4] of the diﬀerential equation in M ∂t η t = T η t + ∂t u(t),

t > 0,

(3.2)

with initial data η 0 (s) = u0 ,

(3.3)

where T is the inﬁnitesimal generator of the right-translation semigroup on M, that is, the linear operator T η = −η with domain D(T ) = {η ∈ M : η ∈ M, η(0) = 0}, the prime being the distributional derivative. Arguing, for example, as in [8], exploiting the Lumer-Phillips theorem [14], one can show that (3.1)-(3.2) generate a strongly continuous semigroup of linear contractions on V × H × M. In particular, choosing the initial datum for η as in (3.3), we recover the well-posedness result for the original problem (1.1) stated in the Introduction. Accordingly, the energy takes the simpler form E(t) = κω∇u(t)2 + ∂t u(t)2 + η t 2M .

4

The Transformed Equation

For v ∈ H, let

v denote the usual Fourier transform of v 1 e−ixξ v(x)dx. v (ξ) = H- lim

M →∞ (2π)N/2 |x|≤M Applying the Fourier transform to (3.1) and (3.2), we obtain, for every ξ ∈ RN , the system ∞ ⎧ 2 2 ⎪ ⎪ ∂tt u

(t) + κω|ξ| u

(t) + |ξ| μ(s)

η t (s)ds = 0, t > 0, ⎪ ⎪ ⎨ 0 (4.1)

(t), t > 0, ∂t η t = T η t + ∂t u ⎪ ⎪ ⎪ ⎪ ⎩

(0) = u

1 , η 0 (s) = u

0 , u

(0) = u

0, ∂t u

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725

in the transformed variables u

(ξ, t) and η t (ξ, s), where now T is the inﬁnitesimal generator of the right-translation semigroup on L2μ (R+ ; RN ), and | · | stands for the standard euclidean norm in RN . The energy density function is given by ∞ 2 2 2 2 E(ξ, t) = κω|ξ| |

u(ξ, t)| + |∂t u

(ξ, t)| + |ξ| μ(s)|

η t(ξ, s)|2ds. 0

In particular, E(ξ, 0) = α|ξ|2|

u0 (ξ)|2 + |

u1(ξ)|2 . Moreover, by the Plancherel theorem,

E(t) = RN

E(ξ, t)dξ.

Performing standard multiplications in (4.1), the energy density is seen to satisfy the diﬀerential equality (cf. [1]) d E + |ξ|2 Θ = 0, (4.2) dt for every ﬁxed ξ ∈ RN , where ∞ + Θ(ξ, t) = − μ (s)|

η t (ξ, s)|2ds + [μ(s− η t (ξ, sn )|2 ≥ 0. n ) − μ(sn )]|

0

n

The above sum, accounting for the jumps of μ, includes the value n = ∞ if s∞ < ∞. Remark 4.1. Here and in the sequel, the calculations hold for regular initial data, and, in particular, for η 0 (ξ, s) ∈ D(T ). Unfortunately, η 0 (ξ, 0) = 0 unless u

0 (ξ) = 0. However, we can perform formal estimates, which can be rigorously justiﬁed in a suitable approximation scheme (see [1]).

5

Energy Functionals

Along the lines of [12], given ν ∈ (0, 1) and a measurable set P ⊂ R+ , we consider the functionals 1 ∞ Φ1 (ξ, t) = − ϕν (s)∂t u

(ξ, t)

η t(ξ, s)ds, κ 0

(ξ, t)

u(ξ, t), Φ2 (ξ, t) = ∂t u ∞ ∞ μ(σ)χP (σ)dσ |

η t (ξ, s) − u

(ξ, t)|2ds, Φ3 (ξ, t) = 0

s

+

where the function ϕν : R → [0, ∞) is deﬁned as ϕν (s) = μ(sν )χ(0,sν ] (s) + μ(s)χ(sν ,∞] (s), s for some ﬁxed sν > 0 such that 0 ν μ(s)ds ≤ ν/2. Denoting + t 2 − μ(s)|

η (ξ, s)| ds, and ΓP (ξ, t) = ΓP (ξ, t) = P

R+ \P

μ(s)|

η t(ξ, s)|2 ds,

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we have Lemma 5.1. For any ν ∈ (0, 1), there exist constants εν > 0 and cν > 0, depending only on ν, with εν → 0 as ν → 0, such that, for every measurable set P ⊂ R+ , the inequalities d 2 − Φ1 ≤ εν |ξ|2|

u|2 − (1 − ν)|∂t u

|2 + (εν + mμ (P))|ξ|2 Γ+ P + cν |ξ| ΓP dt + ω|ξ|2

P

μ(s)

uη (s)ds + cν Θ,

|ξ|2 − d Φ2 ≤ −κ(ω − ν)|ξ|2|

Γ − |ξ|2 u|2 + |∂t u

|2 + dt 4ν P d + Φ3 = −ΓP + 2 μ(s)

uη (s)ds dt P

(5.1)

P

μ(s)

uη (s)ds,

(5.2) (5.3)

hold for every ﬁxed ξ ∈ RN . Inequalities (5.1)-(5.2) are minor modiﬁcations of similar ones proved in [12], to which we address the reader, whereas (5.3) is a straightforward calculation. Lemma 5.2. There exist a measurable set P ⊂ R+ , constants a, ν ∈ (0, 1) and M > 0 such that the functional L(ξ, t) =

ω+a 2 M(1 + |ξ|2 ) |ξ| Φ3 (ξ, t) E(ξ, t) + Φ1 (ξ, t) + aΦ2 (ξ, t) + 2 |ξ| 2

satisﬁes the diﬀerential inequality d L + 2ε0E ≤ 0, dt

(5.4)

for some ε0 > 0 and every ﬁxed ξ ∈ RN . Proof. Again, the proof follows the lines of [1]. Thus, we limit ourselves to provide the essential details. Since Rμ < 1/2, there exists n > 0 large enough such that, setting P = {s ∈ R+ : nμ (s) + μ(s) > 0}, the inequality mμ (P) < 1/2 holds. Choosing a=

1 + mμ (P) < 1, 2

collecting estimates (5.1)-(5.3), and ﬁxing ν small enough, the functional Φ(ξ, t) = Φ1 (ξ, t) + aΦ2 (ξ, t) +

ω+a 2 |ξ| Φ3 (ξ, t) 2

satisﬁes the diﬀerential inequality d Φ + 2ε0 E ≤ c|ξ|2Γ− P + cΘ, dt

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for some ε0 > 0 and some c > 0. Since nμ (s) + μ(s) ≤ 0, for almost every s ∈ R+ \ P, we have that ∞ − ΓP ≤ −n μ (s)|

η(s)|2 ds ≤ nΘ. 0

Therefore, d Φ + 2ε0 E ≤ cn|ξ|2Θ + cΘ. dt Adding M(1 + |ξ|2)|ξ|−2 dtd E to both sides, in light of (4.2), we ﬁnally obtain d L + 2ε0 E ≤ cn|ξ|2Θ + cΘ − M(1 + |ξ|2)Θ ≤ 0, dt for M > 0 large enough. Now, denote

⎧ ⎨e−δt Υp (t) = 1 ⎩ (1 + t)p

if p = ∞, if p < ∞,

and we introduce the functional Ψp (ξ, t) =

t 0

Υp (t − s)|

u(ξ, s)|2ds.

Lemma 5.3. For every t ≥ 0 and every ξ ∈ RN , k(1 + |ξ|2) 1 + |ξ|2 E(ξ, t) ≤ L(ξ, t) ≤ E(ξ, t) + k|ξ|2Ψp (ξ, t), 2 2 k|ξ| |ξ|

(5.5)

for some k ≥ 1. In particular, for t = 0, L(ξ, 0) ≤

k(1 + |ξ|2) E(ξ, 0). |ξ|2

(5.6)

The proof can be easily obtained by means of straightforward calculations, making use of the representation formula for η t devised in [13]. Notation. Till the end of the paper, Q ≥ 1 will stand for a generic constant, which depends (increasingly) only on R.

6

Proof of Theorem 2.1

Due to (5.4), for every ε ∈ (0, ε0 ], we have d L + εE ≤ −ε0 κω|ξ|2|

u|2 . dt Assume ﬁrst that |ξ| ≥ 1. From (5.5) and (6.1), the inequality d L + εL ≤ −ε0 κω|ξ|2|

u|2 + εk|ξ|2Ψ∞ dt

(6.1)

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M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

holds for every ε > 0 small enough. Choosing ε < δ, the Gronwall lemma leads to t t

−εt −εt 2 ετ 2 L(ξ, t) ≤ L(ξ, 0)e − e |ξ| ε0 κω e |

u(ξ, τ )| dτ − εk eετ Ψ∞ (ξ, τ )dτ . 0

0

Exchanging the order of integration, t t t ετ δτ 2 e Ψ∞ (ξ, τ )dτ = e |

u(ξ, τ )| e−(δ−ε)s dsdτ ≤ 0

0

τ

1 δ−ε

0

t

eετ |

u(ξ, τ )|2dτ.

Therefore, for ε small, we recover L(ξ, t) ≤ L(ξ, 0)e−εt, which, in light of (5.5)-(5.6), gives E(ξ, t) ≤ k 2 E(ξ, 0)e−εt. Thus, an integration on |ξ| ≥ 1 yields 2 −εt E(ξ, t)dξ ≤ k e |ξ|≥1

|ξ|≥1

E(ξ, 0)dξ ≤ k 2 E(0)e−εt ≤ Qe−εt .

(6.2)

Consider then |ξ| < 1. In that case, (5.5) and (6.1) entail d L + ε|ξ|2L ≤ −ε0 κω|ξ|2|

u|2 + εk|ξ|2Ψ∞ , dt for all ε > 0 small enough. Reasoning as before, we obtain 2

E(ξ, t) ≤ k 2 E(ξ, 0)e−ε|ξ| t , and an integration on |ξ| < 1 provides 2 E(ξ, t)dξ ≤ k |ξ|<1

|ξ|<1

2

E(ξ, 0)e−ε|ξ| t dξ.

The assumption u0L1 + u1 L1 ≤ R guarantees that sup E(ξ, 0) ≤ Q.

|ξ|<1

Hence, passing to polar coordinates, −ε|ξ|2t E(ξ, 0)e dξ ≤ Q

1

0

|ξ|<1

and applying Lemma 1.1, we conclude that 1 2 E(ξ, t)dξ ≤ Q r N −1 e−εr t dr ≤ |ξ|<1

2

r N −1 e−εr t dr,

0

Q . (1 + t)N/2

(6.3)

M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

729

If we do not have information on the boundedness of u0 L1 , we need to control a further integral; namely (passing to polar coordinates),

2 −ε|ξ|2 t

2

|ξ|<1

where f ≥ 0 satisﬁes

|ξ| |

u0(ξ)| e

1

f (r)r

N −1

0

Setting F (r) = −

1 r

dξ =

1

2

f (r)r N +1 e−εr t dr,

0

dr ≤

∞

0

f (r)r N −1dr = u0 2 ≤ Q.

f (s)sN −1 ds, and integrating by parts, we get

1

f (r)r

N +1 −εr 2 t

e

0

1 2 2 dr = −2 F (r) re−εr t − εr 3 te−εr t dr 0 1 1 3 −εr 2 t −εr 2 t ≤Q t r e dr + re dr . 0

0

By applying Lemma 1.1 to each term, we conclude that Q 2 . |ξ|2|

u0(ξ)|2 e−ε|ξ| t dξ ≤ 1+t |ξ|<1

(6.4)

Collecting (6.2) and (6.3), along with (6.4) when u0 L1 is not bounded, we ﬁnd the required decay estimates for E(t).

7

Proof of Theorem 2.3

For easier computation, we set q = (p + 1)/p. Lemma 7.1. For every ﬁxed ξ ∈ RN , we have 1 g d [|ξ|2 gΨp ] + [|ξ|2gΨp ]q ≤ E, dt Q κω having set g = g(ξ) =

|ξ|2 . (1 + |ξ|2)2

Proof. By a direct computation, t t 1 d 2 2 Ψp = Υp (t − s)|

u(s)| ds + |

u| ≤ −p [Υp (t − s)]q |

u(s)|2 ds + E. dt κω|ξ|2 0 0 The H¨older inequality entails the control t 1/q t (q−1)/q q 2 [Υ(t − s)] |

u(s)| ds |

u(s)|2 ds . Ψp ≤ 0

0

(7.1)

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M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

Integrating (5.4) over [0, ∞), and using (5.6) and the bound u0 L1 +u1L1 ≤ R, provides the integral estimate ∞ Q |

u(s)|2 ds ≤ 2 . |ξ| g 0 Therefore, we obtain

t

0

[Υp (t − s)]q |

u(s)|2 ds ≥

1 [|ξ|2g]q−1 [Ψp ]q , Q

which, substituted into the diﬀerential inequality, yields (7.1). For ε > 0, we introduce the further functional

0

L0 (ξ, t) = g(ξ)L(ξ, t) + ε|ξ|2g(ξ)Ψp(ξ, t). Collecting (5.4) and (7.1), for a ﬁxed ε small enough, we have d 1 L0 + gE + [|ξ|2 gΨp ]q ≤ 0. dt Q

(7.2)

Besides, it is clear from (5.5) that 2k(1 + |ξ|2) 1 E(ξ, t) ≤ L g(ξ)E(ξ, t) + |ξ|2g(ξ)Ψp(ξ, t) , (ξ, t) ≤ 0 2 2 k(1 + |ξ| ) |ξ|

(7.3)

and, due to (5.6), L0 (ξ, 0) = g(ξ)L(ξ, 0) ≤

k E(ξ, 0) ≤ Q. 1 + |ξ|2

(7.4)

Since (as E is decreasing in time) g(ξ)E(ξ, t) ≤ g(ξ)E(ξ, 0) ≤ Q, it follows that [L0 (ξ, t)]q ≤

Q(1 + |ξ|2q ) 2 q g(ξ)Ψ (ξ, t)] g(ξ)E(ξ, t) + [|ξ| , p |ξ|2q

which, replaced in (7.2), leads to |ξ|2q d L0 (ξ, t) + [L0 (ξ, t)]q ≤ 0. dt Q(1 + |ξ|2q ) If |ξ| ≥ 1, on account of (7.4), the generalized Gronwall lemma yields

−p t Q + [L0 (ξ, 0)]−q+1 E(ξ, 0). ≤ L0 (ξ, t) ≤ p Q (1 + t) (1 + |ξ|2) Hence, from (7.3)-(7.4),

Q E(ξ, t)dξ ≤ (1 + t)p |ξ|≥1

|ξ|≥1

E(ξ, 0)dξ ≤

Q . (1 + t)p

(7.5)

M. Conti et al. / Central European Journal of Mathematics 5(4) 2007 720–732

731

Conversely, if |ξ| < 1, using again (7.4), we ﬁnd the estimate L0 (ξ, t) ≤

−p Q Q |ξ|2q t −q+1 + [L (ξ, 0)] ≤ ≤ . 0 Q(1 + |ξ|2q ) (1 + |ξ|2q t)p (1 + |ξ|t1/(2q) )2qp

Thus, exploiting (7.3) and using polar coordinates,

1 E(ξ, t)dξ ≤ Q dξ ≤ Q 1/(2q) )2qp |ξ|<1 |ξ|<1 (1 + |ξ|t

and by applying Lemma 1.2 we conclude that ⎧ ⎨(1 + t)− min{p,N p/(2p+2)} E(ξ, t)dξ ≤ Q · ⎩(1 + t)−p log(2 + t) |ξ|<1

0

1

r N −1 dr, (1 + rt1/(2q) )2qp

if 2p = N − 2, if 2p = N − 2.

(7.6)

Collecting (7.5) and (7.6), we establish the required estimate for E(t).

References [1] M. Conti, S. Gatti, V. Pata: “Uniform decay properties of linear Volterra integrodiﬀerential equations”, Math. Models Methods Appl. Sci. (to appear). [2] C.M. Dafermos: “An abstract Volterra equation with applications to linear viscoelasticity”, J. Diﬀerential Equations, Vol. 7, (1970), pp. 554–569. [3] C.M. Dafermos: “Asymptotic stability in viscoelasticity”, Arch. Rational Mech. Anal., Vol. 37, (1970), pp. 297–308. [4] C.M. Dafermos: “Contraction semigroups and trend to equilibrium in continuum mechanics”, in Applications of Methods of Functional Analysis to Problems in Mechanics, P. Germain and B. Nayroles Eds.), Lecture Notes in Mathematics no.503, Springer-Verlag, Berlin-New York, 1976, pp.295–306 [5] G. Dassios, F. Zaﬁropoulos: “Equipartition of energy in linearized 3-D viscoelasticity”, Quart. Appl. Math., Vol. 48, (1990), pp. 715–730. [6] M. Fabrizio, B. Lazzari: “On the existence and asymptotic stability of solutions for linear viscoelastic solids”, Arch. Rational Mech. Anal., Vol. 116, (1991), pp. 139–152. [7] M. Fabrizio, A. Morro, “Mathematical problems in linear viscoelasticity”, SIAM Studies in Applied Mathematics no.12, SIAM Philadelphia, 1992. [8] M. Grasselli, V. Pata: “Uniform attractors of nonautonomous systems with memory”, in Evolution Equations, Semigroups and Functional Analysis, A. Lorenzi and B. Ruf (Eds.), Progr. Nonlinear Diﬀerential Equations Appl. no.50, Birkh¨auser Boston, 2002, pp.155–178. [9] Z. Liu, S. Zheng: “On the exponential stability of linear viscoelasticity and thermoviscoelasticity”, Quart. Appl. Math., Vol. 54, (1996), pp. 21–31. [10] J.E. Mu˜ noz Rivera: “Asymptotic behaviour in linear viscoelasticity”, Quart. Appl. Math., Vol. 52, (1994), pp. 629–648.

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[11] J.E. Mu˜ noz Rivera, E. Cabanillas Lapa: “Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels”, Comm. Math. Phys., Vol. 177, (1996), pp. 583–602. [12] V. Pata: “Exponential stability in linear viscoelasticity”, Quart. Appl. Math., Vol. 64, (2006), pp. 499–513. [13] V. Pata, A. Zucchi: “Attractors for a damped hyperbolic equation with linear memory”, Adv. Math. Sci. Appl., Vol. 11, (2001), pp. 505–529. [14] A. Pazy, Semigroups of linear operators and applications to partial diﬀerential equations, Springer-Verlag, New York, 1983. [15] M. Renardy, W.J. Hrusa, J.A. Nohel, Mathematical problems in viscoelasticity, John Wiley & Sons, Inc., New York, 1987.

DOI: 10.2478/s11533-007-0027-z Research article CEJM 5(4) 2007 733–740

On splitting up singularities of fundamental solutions to elliptic equations in C2 T.V. Savina∗ Department of Mathematics 321 Morton Hall, Ohio University Athens, OH 45701, U.S.A.

Received 20 July 2007; accepted 02 August 2007 Abstract: It is known that the fundamental solution to an elliptic diﬀerential equation with analytic coeﬃcients exists, is determined up to the kernel of the diﬀerential operator, and has singularities on characteristics of the equation in C2 . In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Complex Variables, Elliptic Equations, Fundamental Solution MSC (2000): 32S70, 35J15, 32W50

1

Introduction and preliminaries

The aim of this paper is to construct a simple representation for fundamental solutions to linear elliptic diﬀerential equations that is convenient for applications, including construction of reﬂection formulas. Consider a homogeneous linear elliptic diﬀerential equation, written in its canonical form [11] (with the Laplacian, Δx,y , in the principal part), in a domain D ⊂ R2 ˆ ≡ Δx,y u + a ∂u + b ∂u + cu = 0, Lu ∂x ∂y 2

2

(1.1)

∂ ∂ where Δx,y = ∂x 2 + ∂y 2 , and the coeﬃcients a, b, c are real-analytic functions of (x, y) in D ⊂ R2 that could be continued to V e(D) = D × D in C2 as holomorphic functions. ∗

E-mail: [email protected]

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T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

The adjoint equation has the form ˆ ∗ v ≡ Δx,y v − ∂(av) − ∂(bv) + cv = 0. L ∂x ∂y

(1.2)

Since characteristics of elliptic equations are complex, consideration of (1.1)–(1.2) in C2 is natural [1], [4], [6]–[11] and helps to understand even applied problems initially formulated in R2 (radio-physics, optics etc). Equations (1.1)–(1.2) could be rewritten in characteristic variables z = x + iy, ζ = x − iy: 2 ˆ C U ≡ ∂ U + A ∂U + B ∂U + CU = 0, L ∂z∂ζ ∂z ∂ζ

(1.3)

2 ˆ ∗C V ≡ ∂ V − ∂(AV ) − ∂(BV ) + CV = 0, L ∂z∂ζ ∂z ∂ζ

(1.4)

where z+ζ z−ζ 1 z + ζ z − ζ , ) + ib( , ) , A(z, ζ) = a( 4 2 2i 2 2i z+ζ z−ζ z+ζ z−ζ 1 , ) − ib( , ) , B(z, ζ) = a( 4 2 2i 2 2i

1 z+ζ z−ζ C(z, ζ) = c( , ) 4 2 2i

(1.5)

are holomorphic functions of (z, ζ), and ζ = z¯ describes the real space R2 . ˆ ∗ G(x0 , y0, x, y) = δ(x0 , y0), where δ(x0 , y0) is the Dirac funcA solution of equation L ˆ Fundamental solution for L ˆ exists and tion, is called a fundamental solution of operator L. has been constructed by many authors [1]–[3], [6]–[10] speciﬁcally the following Vekua’s theorem [10] holds: Theorem 1.1. There exists an entire function g0 (z, ζ, z0 , ζ0 ) such that G(z, ζ, z0 , ζ0) = −

1 R(z0 , ζ0 , z, ζ) ln[(z − z0 )(ζ − ζ0 )] + g0 (z, ζ, z0 , ζ0 ) 4π

(1.6)

is a solution of equation (1.3) with respect to variables (z, ζ) and a solution of (1.4) with ˆ that is an respect to (z0 , ζ0). Here R(z0 , ζ0 , z, ζ) is the Riemann function of operator L, entire function deﬁned as the solution of the Goursat problem: ⎧ ˆ ∗ R = 0, ⎪ L ⎪ C ⎪ ⎪ ζ

⎪ ⎨ A(z0 , τ )dτ , R|z=z0 = exp (1.7) ζ0 ⎪ ⎪

⎪ z ⎪ ⎪ B(t, ζ0 )dt . ⎩ R|ζ=ζ0 = exp z0

A representation for the Riemann function as a series is constructed below (in the simplest case of the Laplace equation, A = B = C = 0, the Riemann function is equal to unity). Note that G(z, ζ, z0 , ζ0) is a multiple-valued function in C2 , whose restriction to R2 , ˆ however, is a single-valued fundamental solution of L.

T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

2

735

Main result

ˆ can be represented in the form Lemma 2.1. The fundamental solution of L G=− Gj =

∞

1 (G1 + G2 ), 4π

(2.1)

αkj (x0 , y0; x, y)fk (ψj ),

j = 1, 2 ,

(−1)−k−1 (−k − 1)!ξ k ,

k ≤ −1,

ξk (ln ξ k!

k = 0, 1, ... ,

(2.2)

k=0

fk (ξ) =

C0 = 0,

− Ck ),

Ck =

k 1 l=1

ψ1 = (x − x0 ) + i(y − y0 ),

l

,

(2.3)

k = 1, 2, ... ,

ψ2 = (x − x0 ) − i(y − y0 ).

Proof. Functions Gj written in characteristic variables (z, ζ) are solutions of the equation 2 ˆ ∗C Gj = ∂ Gj − ∂AGj − ∂BGj + CGj = δ(z0 , ζ0 ). (2.4) L ∂z∂ζ ∂z ∂ζ Substituting (2.2) into (2.4) and using the following property [5] of functions (2.3) d fk (ξ) = fk−1 (ξ) dξ

(2.5)

result in the Hamilton-Jacobi equations for the functions ψj ∂ψj ∂ψj · =0 ∂z ∂ζ

(2.6)

and transport equations for functions αkj j ˆ ∗C αj , Lα0j = 0, Lαk+1 = −L k ∂ψ ∂ ∂ψ ∂ · −A + · −B . L= ∂z ∂ζ ∂ζ ∂z

(2.7)

To ensure logarithmic singularity at the point (z0 , ζ0 ) we chose ψ1 = z − z0 ,

ψ2 = ζ − ζ0 .

(2.8)

ˆ intersecting R2 at point (x0 , y0 ) are given by equations Therefore, the characteristics of L ψ1 = 0 and ψ2 = 0. To ﬁnd αkj we set the initial conditions for system (2.7): ⎧ z

⎪ 1 1 ⎪ α = exp B(t, ζ )dt , α0| = 0, k = 1, 2, . . . , ⎪ 0 0|ζ=ζ0 ⎨ ζ=ζ0 z0 (2.9) ζ

⎪ 2 2 ⎪ α = exp A(z , τ )dτ , α = 0, k = 1, 2, . . . . ⎪ 0 0|z=z k|z=z ⎩ 0

ζ0

0

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T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

Functions αkj are uniquely determined by recursion (2.7) and (2.9), speciﬁcally, α01

ζ = exp

z A(z, τ )dτ +

B(t, ζ0 )dt ,

α02

ζ = exp

z0

ζ0

z A(z0 , τ )dτ +

ζ0

B(t, ζ)dt . (2.10)

z0

Thus, formal series (2.2) are constructed. To complete the proof we need to show convergence of series Gj and the absence of branching point of their sum, G, in D ⊂ R2 . The later is ensured if logarithms in both expressions (2.2) are multiplied by the same function, R(z0 , ζ0, z, ζ). Indeed, it is easy to check that both series ∞

(z αk1 (z0 , ζ0 , z, ζ)

k=0

− z0 )k k!

∞

and

αk2 (z0 , ζ0 , z, ζ)

k=0

(ζ − ζ0 )k k!

(2.11)

are formal solutions of (1.7). Unfortunately, uniqueness of the solution to problem (1.7) does not ensure convergence of series (2.11), because they can not be interpreted as the Taylor series with respect to one of the variables since the coeﬃcients αkj depend on all of the variables z0 , ζ0 , z, ζ. To prove convergence of the ﬁrst series (2.11) let us consider an auxiliary family of problems depending on parameter ξ: ⎧ ˆ ∗ Vξ (z0 , ζ0, z, ζ, ξ) = 0, ⎪ L ⎪ C ⎪ ⎪ ζ

z0 +ξ ⎪ ⎨ A(z0 + ξ, τ )dτ + B(t, ζ0 )dt , Vξ |z=z +ξ = exp (2.12) 0 z0 ζ0 ⎪ ⎪

⎪ z ⎪ ⎪ B(t, ζ0 )dt . ⎩ Vξ |ζ=ζ = exp 0

z0

This problem coincide with (1.7) when ξ = 0, and its formal solution, ∞

αk1 (z0 , ζ0, z, ζ)

k=0

(z − z0 − ξ)k , k!

(2.13)

has the same coeﬃcients αk1 as the ﬁrst series (2.11). Convergence of the later, therefore, followed from convergence (2.13). To show existence and uniqueness of solutions to (2.12) in the class of analytic functions we use the method similar to [10]. The ﬁrst equation (2.12) is equivalent to the following integral equation: z Vξ (z0 , ζ0 , z, ζ, ξ) −

B(t, ζ)Vξ (z0 , ζ0, t, ζ, ξ)dt

z0 +ξ

ζ −

z A(z, τ )Vξ (z0 , ζ0 , z, τ, ξ)dτ +

dt z0 +ξ

ζ0

ζ

(2.14) C(t, τ )Vξ (z0 , ζ0 , t, τ, ξ)dτ

ζ0

= Φ(z0 , ζ0 , z, ξ) + Ψ(z0 , ζ0 , ζ, ξ), where Φ and Ψ are entire functions in C4 . We use the rest two conditions of (2.12) to ﬁnd the right hand side of (2.14). Substituting z = z0 + ξ into equation (2.14) yields Φ(z0 , ζ0, z0 + ξ, ξ) + Ψ(z0 , ζ0 , ζ, ξ) = F1 (z0 , ζ0 , ζ, ξ),

(2.15)

T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

where

ζ F1 (z0 , ζ0 , ζ, ξ) = exp

z0 +ξ

−

B(t, ζ0 )dt

A(z0 + ξ, τ )dτ + z0

ζ0

ζ

A(z0 + ξ, τ ) exp

ζ0

737

τ

A(z0 + ξ, τ )dτ +

z0 +ξ

B(t, ζ0 )dt dτ.

(2.16)

z0

ζ0

Substituting ζ = ζ0 into (2.14) gives Φ(z0 , ζ0 , z, ξ) + Ψ(z0 , ζ0 , ζ0 , ξ) = F2 (z0 , ζ0 , z, ξ),

(2.17)

where z F2 (z0 , ζ0, z, ξ) = exp

z

t B(t, ζ0 )dt − B(t, ζ0 ) exp B(t , ζ0 )dt dt. z0 +ξ

z0

(2.18)

z0

From (2.17)-(2.18) Φ(z0 , ζ0 , z0 + ξ, ξ) = −Ψ(z0 , ζ0 , ζ0 , ξ) − exp

z

B(t, ζ0 )dt .

(2.19)

z0

Substitution of (2.19) into (2.15) results in z0 +ξ

B(t, ζ0 )dt . Ψ(z0 , ζ0 , ζ, ξ) = F1 (z0 , ζ0 , ζ, ξ) + Ψ(z0 , ζ0, ζ0 , ξ) − exp

(2.20)

z0

From (2.17) and (2.20) ﬁnally we have Φ(z0 , ζ0 , z, ξ) + Ψ(z0 , ζ0 , ζ, ξ) = F (z0 , ζ0 , z, ζ, ξ),

(2.21)

where z0 +ξ

B(t, ζ0 )dt . F (z0 , ζ0, z, ζ, ξ) = F1 (z0 , ζ0 , ζ, ξ) + F2 (z0 , ζ0 , z, ξ) − exp

(2.22)

z0

Thus, the auxiliary problem (2.12) is reduced to the Volterra type integral equation z Vξ (z0 , ζ0 , z, ζ, ξ) −

B(t, ζ)Vξ (z0 , ζ0, t, ζ, ξ)dt

z0 +ξ

ζ −

z A(z, τ )Vξ (z0 , ζ0 , z, τ, ξ)dτ +

ζ0

= F (z0 , ζ0 , z, ζ, ξ),

ζ dt

z0 +ξ

(2.23) C(t, τ )Vξ (z0 , ζ0 , t, τ, ξ)dτ

ζ0

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T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

studied by Vekua [10], who proved the uniqueness and existence theorem in the class of analytic functions. Thus, there exists unique solution of (2.12), which has unique Taylor expansion (2.13) with respect to variable ξ at point ξ = z − z0 . Therefore, series (2.13) converges, and, consequently, so does the ﬁrst series in (2.11); the later is a representation of the Riemann function. Analogous consideration of the following problem depending on parameter η: ⎧ ˆ ∗ Vη (z0 , ζ0 , z, ζ, η) = 0, ⎪ L ⎪ C ⎪ ⎪ ⎪ ζ

⎪ ⎨ A(z0 , τ )dτ , Vη |z=z = exp 0 (2.24) ζ0 ⎪ ⎪

ζ0 +η z ⎪ ⎪ ⎪ ⎪ A(z0 , τ )dτ + B(t, ζ0 + η)dt , ⎩ Vη |ζ=ζ0 +η = exp z0

ζ0

k 2 (ζ−ζ0 −η) with solution Vη = ∞ , proves convergence of the second series (2.11) to k=0 αk k! ˆ the Riemann function of operator L. Thus, the multiplier of the logarithms in the ﬁrst terms in the both expansions G1 =

∞

(z αk1 (z0 , ζ0, z, ζ)

k=0

G2 =

∞

(ζ αk2 (z0 , ζ0, z, ζ)

k=0

− z0 )k (z − z0 )k ln(z − z0 ) − Ck , αk1 (z0 , ζ0 , z, ζ) k! k! k=0 ∞

∞ − ζ 0 )k (ζ − ζ0 )k ln(ζ − ζ0 ) − Ck αk2 (z0 , ζ0 , z, ζ) k! k! k=0

(2.25)

converges to the entire function. This fact, in turn, ensures convergence of the second terms as well. That ﬁnishes the proof. 2

∂ Corollary 2.2. In the case of the Helmholtz equation ∂z∂ζ uH + real number, functions (2.25) reduce to the form used in [8], [7]

λ2 H u 4

= 0, where λ is a

∞ 1 [−λ2 (z − z0 )(ζ − ζ0 )]k ln(z − z ) − C 0 k , 4π k=0 4k (k!)2 ∞ 1 [−λ2 (z − z0 )(ζ − ζ0 )]k H ln(ζ − ζ ) − C G2 = − 0 k 4π k=0 4k (k!)2

GH 1 = −

(2.26)

H Summing up GH 1 and G2 one obtains known fundamental solution of the Helmholtz equation: H GH 1 + G2 =

1 c + ln λ/2 J0 λ (z − z0 )(ζ − ζ0 ) − N0 λ (z − z0 )(ζ − ζ0 ) , 2π 4

(2.27)

where c is the Euler constant, and J0 and N0 are the Bessel and the Neumann functions of zero order respectively. Note, that for the Laplace’s equation, λ = 0, functions (2.26) become GL1 = −

1 1 ln(z − z0 ) and GL2 = − ln(ζ − ζ0 ) 4π 4π

(2.28)

T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

739

respectively. Remark 2.3. For the iterated Laplacian, Δp , where p is a positive integer, function G has the form G(z, ζ, z0 , ζ0 ) = − G1 =

1 4p π

(G1 (z, ζ, z0 , ζ0 ) + G2 (z, ζ, z0 , ζ0 )),

where,

(z − z0 )p−1 (ζ − ζ0 )p−1 (z − z0 )p−1 (ζ − ζ0 )p−1 ln(z − z ), G = ln(ζ − ζ0 ). 0 2 (p − 1)!2 (p − 1)!2

Remark 2.4. Consider linear elliptic equation of order 2m with constant coeﬃcient in the principal part [6] ∼

Lu(x, y) ≡ L2m ( where

2m−1 ∂ ∂ ∂ ∂ , )u(x, y) + Ln (x, y, , )u(x, y) = 0, ∂x ∂y ∂x ∂y n=0

∂ ∂ ∂ ∂ L2m ( , ) = aα ( )α ( )2m−α , ∂x ∂y ∂x ∂y α=0 n ∂ ∂ ∂ ∂ anα (x, y)( )α ( )n−α . Ln (x, y, , ) = ∂x ∂y ∂x ∂y α=0

(2.29)

2m

(2.30)

Assume also that functions anα (x, y) are real-analytic, and equation (2.29) has exactly 2m distinct characteristics (all characteristics are simple). Then, in a neighborhood of ∼

point (x0 , y0 ) fundamental solution of operator L has the following representation G=

m

Gj (x, y, x0 , y0 ) + G¯j (x, y, x0 , y0 ) ,

(2.31)

j=1

where Gj = G¯j =

∞ k=2m−2 ∞

bjk (x, y, x0 , y0 )fk x − x0 + λj (y − y0 ) ,

¯ j (y − y0 ) . ¯bj (x, y, x0 , y0 )fk x − x0 + λ k

(2.32)

k=2m−2

¯ j are the complex-conjugate roots Here functions fk are determined by (2.3) and λj and λ of the characteristic equation L2m (p, q) = 0.

References [1] D. Colton and R.P. Gilbert: “Singularities of solutions to elliptic partial diﬀerential equations with analytic coeﬃcients”, Quart. J. Math. Oxford Ser. 2, Vol. 19, (1968), pp. 391–396. [2] F. John: “The fundamental solution of linear elliptic diﬀerential equations with analytic coeﬃcients”, Comm. Pure Appl. Math., Vol. 3, (1950), pp. 273–304.

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T.V. Savina / Central European Journal of Mathematics 5(4) 2007 733–740

[3] F. John: Plane waves and spherical means applied to partial diﬀerential equations, Springer-Verlag, New York-Berlin, 1981. [4] D. Khavinson: Holomorphic partial diﬀerential equations and classical potential theory, Universidad de La Laguna, 1996. [5] D. Ludwig: “Exact and Asymptotic solutions of the Cauchy problem”, Comm. Pure Appl. Math., Vol. 13, (1960), pp. 473–508. [6] T.V. Savina: “On a reﬂection formula for higher-order elliptic equations”, Math. Notes, Vol. 57, no. 5–6, (1995), pp. 511–521. [7] T.V. Savina: “A reﬂection formula for the Helmholtz equation with the Neumann condition”, Comput. Math. Math. Phys., Vol. 39, no. 4, (1999), pp. 652–660. [8] T.V. Savina, B.Yu. Sternin and V.E. Shatalov: “On a reﬂection formula for the Helmholtz equation”, Radiotechnika i Electronica, (1993), pp. 229–240. [9] B.Yu. Sternin and V.E. Shatalov: Diﬀerential equations on complex manifolds, Mathematics and its Applications, Vol. 276, Kluwer Academic Publishers Group, Dordrecht, 1994. [10] I.N. Vekua: New methods for solving elliptic equations, North Holland, 1967. [11] I.N. Vekua: Generalized analytic functions, Second edition, Nauka, Moscow, 1988.

DOI: 10.2478/s11533-007-0029-x Research article CEJM 5(4) 2007 741–750

Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations B.E. Eshmatov1∗ and E.T. Karimov2† 1 2

Karshi State University, 700430 Karshi, Uzbekistan

Institute of Mathematics, Uzbek Academy of Sciences, 700125 Tashkent, Uzbekistan

Received 11 December 2006; accepted 14 August 2007 Abstract: In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Parabolic-hyperbolic type equations, gluing conditions, boundary value problems, integral equation, ”abc” method MSC (2000): 35M10

1

Introduction

Boundary value problems (BVP) for mixed parabolic-hyperbolic type equations have numerous applications in physics, biology and in other material sciences. For example, the movement of gas in a channel surrounded by a porous environment can be described by parabolic-hyperbolic equation [4]. It is known that some multi dimensional analogues of the main BVP can be reduced by Fourier transformation to the BVP for equations with parameters. This is why interest to study these types of equations grew. In this direction note the works [10], [11] and the ∗ †

E-mail: [email protected] E-mail: [email protected]

742 B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750

references therein. Many BVP for mixed type equations were considered in mixed domains. Note that with “mixed domain” we mean a domain in which the equation will change its type. To determine the solution of the considered BVP in the whole domain, the desired function and its derivatives should be glued continuously or by a special gluing condition on the lines, which separates various parts of the mixed domain. Special gluing conditions sometimes generalize continuous gluing conditions; sometimes they are used because of their physical meaning. The special gluing condition used in the present paper generalizes the following gluing condition x

eat uy (t, −0)dt, 0 < x < 1,

ax

e uy (x, +0) = b 0

for the equation 0=

⎧ ⎪ ⎪ ⎨ ux

− uyy , y > 0,

⎪ ⎪ ⎩ uxx

− uyy , y < 0.

This is used in the work of N.Yu. Kapustin and E.I. Moiseev [5]. This special gluing condition, according to the theory of parabolic-hyperbolic type heat equations [12], represents the equality of temperatures and streams on the boundary of oscillation bodies with diﬀerent tenses. In section 2 we prove the unique solvability of a non-local BVP for parabolic-hyperbolic equations with non-characteristic line of changing type with spectral parameter. We note the works [2], [6] and [13], which have results similar to ours. The result, which is obtained in the second section, shows that when we give data on left characteristics in any form (local and non-local, inside the integral or integrodiﬀerential operator, see [2], [7]) of the hyperbolic part of the equation, we can get the unique solution at any values of the parameter λ. This means that the problem does not have eigenvalues. In section 3 we consider another non-local BVP with special gluing condition. This gluing condition on one hand generalizes the known gluing condition [5], and on the other hand allows us to prove the unique solvability of the considered problem with less conditions on the given functions and parameters, which is more suitable in an application.

2

A non-local problem with continuous gluing condition

We consider the equation Lu = λu, in a domain Ω, in which λ = λ1 + iλ2 (∀λ1 , λ2 ∈ R) and Lu =

⎧ ⎪ ⎪ ⎨u

x

− uyy , y > 0,

⎪ ⎪ ⎩ uxx

− uyy , y < 0.

(1)

B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750 743

Here Ω is a simply-connected domain, bounded at y > 0 by AB = {(x, y) : x = 0, 0 < y < q}, BC = {(x, y) : y = q, 0 < x < p}, CD = {(x, y) : x = p, 0 < y < q}, and at y < 0 by characteristics AE : x + y = 0, DE : x − y = p of the equation (1), where p, q > 0. We introduce the following notations: Ω1 = Ω ∩ (y > 0), Ω2 = Ω ∩ (y < 0), AD = {(x, y) : y = 0, 0 < x < p}. θ is a set of points of intersections of characteristics, outgoing from the point (x, 0) ∈ AD with characteristic AE, i.e., θ(x/2, −x/2). The Problem NP1 . Find a solution of the equation (1) from the set of functions 2,1 W1 ≡ C(Ω) ∩ C 1 (Ω) ∩ Cx,y (Ω1 ) ∩ C 2 (Ω2 ), satisfying boundary conditions u(x, y)|AB = ϕ(y), 0 ≤ y ≤ q,

(2)

u(x, y)|BC = ψ(x), 0 ≤ x ≤ p,

(3)

u(θ) + a(x)u(x, 0) = b(x), 0 ≤ x ≤ p.

(4)

and the non-local condition

Here a(x), b(x), ϕ(y), and ψ(x) are given, generally speaking complex-valued functions, and moreover ϕ(q) = ψ(0), ϕ(0)(1 + a(0)) = b(0), a(x) = − 12 , and a(0) = −1. Theorem 2.1. If ϕ(y) ∈ C 1 [0, q], ψ(x) ∈ C 1 [0, p], and a(x), b(x) ∈ C 1 [0, p] ∩ C 2 (0, p), then there exists a unique solution to the Problem NP1 . Proof. The solution of the second BVP for the equation (1) at y > 0 can be written as [3]: q

x

ϕ(y1)G(x, y, 0, y1)dy1 +

u(x, y) = 0

x

ψ(x1 )Gy1 (x, y, x1 , 1)dx1 + 0

ν(x1 )G(x, y, x1 , 0)dx1 , 0

(5)

in which eλ(x−x1 )

∞

−

e G(x, y, x1, y1 ) = 2 π(x − x1 ) n=−∞

(2n+y−y1 )2 4(x−x1 )

−

+e

(2n+y+y1 )2 4(x−x1 )

is a Green function of the second BVP for the equation at y > 0. From (5) we obtain q

u(x, 0) =

x

ϕ(y1 )G(x, 0, 0, y1)dy1 + 0

x

ψ(x1 )Gy1 (x, 0, x1 , 1)dx1 + 0

ν(x1 )G(x, 0, x1 , 0)dx1 . 0

(6) and it is a functional relation between the functions u(x, 0) = τ (x) and uy (x, 0) = ν(x) on AD.

744 B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750

Now we ﬁnd another functional relation between these functions on AD. For this we use a solution of the Cauchy problem for the equation (1) at y < 0, which has the form [9]: ⎧ ⎪ ⎨

1 u(x, y) = τ (x + y) + τ (x − y) + 2⎪ ⎩ x+y

I1

x+y

λ ((x − t)2 − y 2 ) dt

ν(t)I0 x−y

λ ((x − t)2 − y 2)

⎫ ⎪ ⎬

τ (t) dt , ⎪ ⎭ λ ((x − t)2 − y 2 )

+λy x−y

(7)

s

where Is (x) = (−i) Js (ix) is the Bessel function of the ﬁrst kind with imaginary argument [1]. Using condition (4) we have x

τ (0) + [1 + 2a(x)]τ (x) +

τ (t) 0

−

x

ν(t)I0

x ∂ I0 λt(t − x) dt t ∂x

λt(t − x) dt = 2b(x).

(8)

0

Considering u(x, y) ∈ C(Ω) in (8) instead of τ (x), we substitute in its representation (6): ⎧ q ⎨

ϕ(0) + [1 + 2a(x)] x

+ 0

⎛ q ⎝

⎩

⎫ ⎬

x

ϕ(y1)G(x, 0, 0, y1)dy1 + 0

ψ(x1 )Gy1 (x, 0, x1 , 1)dx1 0

t

ϕ(y1 )G(t, 0, 0, y1)dy1 +

0

⎞

ψ(x1 )Gy1 (t, 0, x1 , 1)dx1 ⎠ ·

0

−

x

0

0

x ∂ · I0 λt(t − x) dt t ∂x

λz(z − x) − (1 + 2a(x))G(x, 0, z, 0) dz

ν(z) I0

⎛ x t + ⎝

⎭

⎞

ν(x1 )G(t, 0, x1 , 0)dx1 ⎠ ·

0

x ∂ · I0 λt(t − x) dt = 2b(x). t ∂x

Granting the following equality x

⎛ t ⎝

0

ν(x1 )G(t, 0, x1 , 0)dx1 ⎠ ·

0

x

= 0

we obtain that

⎞

⎛ x ⎝ ν(z) z

x ∂ · I0 λt(t − x) dt t ∂x ⎞

x ∂ · I0 λt(t − x) G(t, 0, z, 0)dt⎠ dz, t ∂x x

ν(z)K1 (x, z)dz = Φ1 (x). 0

(9)

B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750 745

Diﬀerentiating (9) once with respect to x and taking G(x, 0, x, 0) = 0 and I0 (0) = 1 into account we get x

ν(z)K2 (x, z)dz = Φ2 (x),

ν(x) +

(10)

0

in which ∂ λz K1 (x, z) = − K2 (x, z) = I0 λz(z − x) + I2 λz(z − x) ∂x 2 x

∂ −(1+2a(x)) G(x, 0, z, 0)−λxa (x) ∂x z ⎧ ⎪ ⎨ λx

λ G(x, 0, z, 0) + −[1 + 2a(x)] ⎪ 2 ⎩ 4

λz(z − x) + I2

I0

⎡

x

⎢ I0

λz(z − x)

G(t, 0, z, 0) ⎣

λz(z − x) + I2

G(t, 0, z, 0)dt

λz(z − x)

2

z

λtx − I0 λt(t − x) + 2I2 λt(t − x) + I4 λt(t − x) dt , 4

Φ2 (x) = Φ1 (x) =

λx + 2a (x) γ(x) + [1 + 2a(x)] ⎩ 4 x

+ψ(x)Gy1 (x, 0, x1 , 1) +

+2I2

⎧ q ⎨

0

λt(t − x) + I4

⎫ ⎬

x λ ψ(x1 )Gy1 x (x, 0, x1 , 1)dx1 + γ(t) t I0 λt(t − x) ⎭ 4

λt(t − x)

0

λt(t − x) + I2

+ I0

q

λt(t − x)

dt,

x

ϕ(y1 )G(x, 0, 0, y1)dy1 +

γ(x) =

ϕ(y1 )Gx (x, 0, 0, y1)dy1 0

0

ψ(x1 )Gy1 (x, 0, x1 , 1)dx1 . 0

(10) is the second kind Volterra type integral equation. From the representations of the functions K2 (x, z) and Φ2 (x), applying known properties of G(x, y, x1 , y1 ) [3], and using some properties of the Bessel function [1] based on the general theory of integral equations [8], one can easily be sure that (10) has a unique solution, which is represented as x

ν(x) = Φ2 (x) +

Φ2 (t)R(x, z)dz,

(11)

0

where R(x, z) is the resolvent kernel of K2 (x, z). Since the Problem NP1 reduced to the equivalent integral equation (10), from the unique solvability of the equation (10) we can conclude that Problem NP1 has a unique solution. This solution can be represented by the formulas (5), (7) in Ω1 , Ω2 respectively. The theorem is proved. Corollary 2.2. The Problem NP1 does not have eigenvalues.

746 B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750

3

A non-local problem with special gluing condition

Consider the equation 0=

⎧ ⎪ ⎪ ⎨ uxx

− uy − μ1 u,

⎪ ⎪ ⎩ uxx

− uyy + μ2 u, y < 0,

y > 0,

(12)

in the domain Ω. The Problem NP2 . Find a solution of (12) from the set of functions

2,1 (Ω1 ∪ AD) ∩ C 2 (Ω2 ) , W ≡ C Ω ∩ C 1 (Ω2 ∪ AD) ∩ Cx,y

satisfying the boundary condition ux (0, y) = f (y), 0 < y < q,

(13)

u(0, y) = u(p, y), 0 ≤ y ≤ q,

(14)

the non-local conditions √ 0, μ A0x 2

[u(θ)] + c(x)u(x, −0) = d(x), 0 ≤ x ≤ p,

(15)

and the gluing condition α uy (x, +0) = 1 + 2c(x)

x

√ uy (t, −0)J0 [ μ2 (x − t)] dt, 0 < x < p.

(16)

0

Here α ∈ R\{0}, c(x), d(x), and f (y) are given real-valued functions, c(x) = − 12 , and √ 0, μ A0x 2

[g(x)] = g(x) −

x

g(t) 0

∂ J0 μ2 x(x − t) dt. ∂x

We introduce the following notations: u(x, +0) = τ+ (x), u(x, −0) = τ− (x), uy (x, +0) = ν+ (x), uy (x, −0) = ν− (x). Since u(x, y) ∈ C(Ω) one can easily see that τ+ (x) = τ− (x). Theorem 3.1. If p

μ2 > 0, α + μ1 > 0, 0

√ √ f (0)sh α + μ1 d(t) sh α + μ1 (t − p)dt = − , 1 + 2c(t) α

d(0) = 0, c(0) = −1, c(x) ∈ C[0, p] ∪ C 1 (0, p), and f (y) ∈ C 1 [0, q], then there exists a unique solution to the Problem NP2 . Proof. Consider equation (12) in the domain Ω2 . The solution of the Cauchy problem for (12) at y < 0 can be written as ⎧ ⎪

1⎨ u(x, y) = ⎪τ− (x + y) + τ− (x − y) + 2⎩

x+y

ν− (t)J0 x−y

μ2 ((x −

t)2

−

y 2)

dt

B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750 747 x+y

+μ2 y

τ− (t)

J1

μ2 ((x − t)2 − y 2)

dt .

(17)

⎪ ⎭

μ2 ((x − t)2 − y 2)

x−y

⎫ ⎪ ⎬

Using condition (15), we obtain after some calculations ν− (x) = τ∗ (x)+μ2

x

τ∗ (t) 0

J1

√

√

μ2 (x − t)

μ2 (x − t)

dt−2d (x)−2μ2

x

d(t)

J1

0

√

√

μ2 (x − t)

μ2 (x − t)

dt, (18)

where τ∗ (x) = (1 + 2c(x))τ− (x). Note that from (15), setting d(0) = 0, and c(x) = − 12 , we get that τ− (0) = 0. Substituting (18) into (16) gives ⎧ x ⎨

α √ [τ∗ (t) − 2d (t)]J0 [ μ2 (x − t)]dt ⎩ 1 + 2c(x)

uy (x, +0) = ν+ (x) =

0

x

+μ2 0

⎫ ⎛ t ⎞ √ ⎬ J [ μ (t − z)] √ ⎝ [τ∗ (z) − 2d(z)] 1√ 2 dz ⎠ J0 [ μ2 (x − t)]dt ⎭ μ2 (t − z)

=

0

α (I1 + μ2 I2 ). 1 + 2c(x)

We consider each piece separately. For I1 , using integration by parts and taking d(0) = 0 and τ− (0) = 0 into account, we have I1 = τ∗ (x) − 2d(x) −

√

x

μ2

√ [τ∗ (t) − 2d(t)]J1 [ μ2 (x − t)]dt.

0

For I2 , we change the order of integration, and then with the change of variable ξ = t − z in the inner integral we get 1 I2 = √ μ2

x

[τ∗ (z) − 2d(z)]dz

0

x−z 0

dξ √ √ √ J1 [ μ2 ξ]J0 [ μ2 (x − z) − μ2 ξ] . ξ

Considering the following formula [1] η

Jp (cξ)Jq (cη − cξ)

0

we obtain 1 I2 = √ μ2 Hence we have

1 dξ = Jp+q (cη), (η, (p) > 0, (q) > −1), ξ p

x

√ [τ∗ (z) − 2d(z)]J1 [ μ2 (x − z)]dz.

0

!

2d(x) ν+ (x) = α τ+ (x) − . 1 + 2c(x)

(19)

Now we prove the uniqueness of the solution. For this we suppose that the Problem NP2 has two solutions (u1 (x, y), u2 (x, y)). Setting U = u1 − u2, we obtain the homogeneous problem in which d(x) = 0 and f (y) = 0.

748 B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750

Passing to the limit in the equation Uxx − Uy − μ1 U = 0 as y → +0, we have

τ+ (x) − ν+ (x) − μ1 τ+ (x) = 0.

(20)

Multiplying (20) by the function τ+ (x), integrating along Aε Bε = {(x, y) : y = 0, ε < x < p − ε}, and then passing to the limit as ε → 0 we get x

τ+ (x)τ+ (x)dx

−

0

p

τ+ (x)ν+ (x)dx − μ1

0

p

τ+2 (x)dx = 0.

(21)

0

Substituting (19) at d(x) = 0 into (21) and considering τ+ (0) = 0 we obtain p

τ+2 (x)dx

p

+ (α + μ1 )

0

τ+2 (x)dx = 0.

0

From this and setting α + μ1 > 0, we get that τ+ (x) = 0. Since U(x, y) is the solution to the equation Uxx − Uy − μ1 U = 0 satisfying conditions Ux (0, y) = 0, U(0, y) = U(p, y), and U(x, +0) = τ+ (x) = 0, we have U(x, y) = 0 in Ω1 . Taking U(x, y) ∈ C(Ω) into account we get U(x, y) = 0 in Ω. This means that the Problem NP2 has a unique solution. Now we prove existence of the solution. Substitute (19) into (20) to get τ+ (x) − (α + μ1 )τ+ (x) = d1 (x),

(22)

2αd(x) . The solution of (22) at α + μ1 > 0, satisfying condition 1 + 2c(x) τ+ (0) = τ+ (p) = 0, has the form

in which d1 (x) = −

p

τ+ (x) =

G∗ (x, t)d1 (t)dt,

0

where ⎧ √ ⎪ ⎪ ⎨ sh α + μ

√ (t − p)sh α + μ1 x, 0 ≤ x ≤ t, √ 1 1 G∗ (x, t) = √ sh α + μ1 √ √ ⎪ α + μ1 ⎪ ⎩ sh α + μ1 tsh α + μ1 (x − p), t ≤ x ≤ p, is the Green-function of the problem ⎧ ⎪ ⎪ ⎨ τ (x) − (α + μ +

⎪ ⎪ ⎩ τ+ (0)

1 )τ+ (x)

= τ+ (p) = 0.

= 0,

(23)

B.E. Eshmatov and E.T. Karimov / Central European Journal of Mathematics 5(4) 2007 741–750 749

We satisfy (23) with condition ux (0, y) = τ+ (0) = f (0). For this it is enough to set p 0

√ √ f (0)sh α + μ1 d(t) sh α + μ1 (t − p)dt = − . 1 + 2c(t) α

We can rewrite (23) as ⎧ ⎨

√ √ 1 √ sh α + μ1 (x − p) d1 (t)sh α + μ1 tdt τ+ (x) = τ− (x) = √ α + μ1 sh α + μ1 ⎩ x

0

⎫

⎬ √ √ +sh α + μ1 x d1 (t)sh α + μ1 (t − p)dt⎭ . p

(24)

x

The solution of the Problem NP2 in Ω1 has the form p

u(x, y) =

y

τ+ (x1 )G∗ (x, y, x1 , 0)dx1 +

f (y1)G∗ (x, y, 0, y1)dy1

0

0

−

y

u(p, y1)G∗x1 (x, y, p, y1)dy1 , 0

where

eμ1 (y−y1 )

∞

−

G∗ (x, y, x1 , y1 ) = e 2 π(y − y1 ) n=−∞ −

−e

(x−x1 −2p+2n)2 4(y−y)

−

−e

(x−x1 +2n)2 4(y−y)

(x+x1 −2p+2n)2 4(y−y)

−

+e

(x+x1 +2n)2 4(y−y)

is the Green-function of the mixed BVP ux (0, y) = f (y), u(p, y) = g(y), 0 ≤ y ≤ q for (12) at y > 0. The solution of the Problem NP2 in Ω2 is determined by the formula (17), where the function τ+ (x) can be deﬁned by (24) and the function ν− (x) by the formula (19).

References [1] G. Bateman and A. Erdelyi: Higher transcendental functions. Bessel functions, functions of parabolic cylinder, orthogonal polynomials (in Russian), Russian translation of extracts from Volume II of the original English edition (McGraw Hill, New York, 1953), Nauka, Moscow, 1966. [2] A.S. Berdyshev and E.T. Karimov: “Some non-local problems for the parabolichyperbolic type equation with non-characteristic line of changing type”, Cent. Eur. J. Math., Vol. 4, (2006), no. 2, pp. 183–193. [3] A. Friedman: Partial diﬀerential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliﬀs, N.J., 1964.

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[4] I.M. Gel’fand: “Some questions of analysis and diﬀerential equations” (in Russian), Uspehi Mat. Nauk, Vol. 14, (1959), no. 3, pp. 3–19. [5] N.Yu. Kapustin and E.I. Moiseev: “On spectral problems with a spectral parameter in the boundary condition” (in Russian), Diﬀer. Uravn., Vol. 33, (1997), no. 1, pp. 115–119, 143. [6] E.T. Karimov: “About the Tricomi problem for the mixed parabolic-hyperbolic type equation with complex spectral parameter”, Complex Var. Theory Appl., Vol. 50, (2005), no. 6, pp. 433–440. [7] E.T. Karimov: “Non-local problems with special gluing condition for the parabolichyperbolic type equation with complex spectral parameter”, Panamer. Math. J., Vol. 17, (2007), no. 2, pp. 11–20. [8] M.L. Krasnov: Integral equations. Introduction to the theory (in Russian), Nauka, Moscow, 1975. [9] J.M. Rassias: Lecture notes on mixed type partial diﬀerential equations, World Scientiﬁc Publishing Co., Inc., Teaneck, NJ, 1990. [10] K.B. Sabytov: “On the theory of equations of mixed parabolic-hyperbolic type with a spectral parameter”, Diﬀerentsial’nye Uravneniya, Vol. 25, (1989), no. 1, pp. 117– 126, 181–182. [11] M.S. Salakhitdinov and A.K. Urinov: Boundary value problems for equations of mixed type with a spectral parameter (in Russian), Fan, Tashkent, 1997. [12] A.G. Shashkov: System-structural analysis of the heat exchange proceses and its application, Moscow, 1983. [13] G.D. Tojzhanova and M.A. Sadybekov: “Spectral properties of an analogue of the Tricomi problem for an equation of mixed parabolic-hyperbolic type” (in Russian), Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat., (1990), no. 5, pp. 48–52.

DOI: 10.2478/s11533-007-0031-3 Research article CEJM 5(4) 2007 751–763

A Numerical Approach of the sentinel method for distributed parameter systems Aboubakari Traore1∗ , Benjamin Mampassi2† , Bisso Saley3‡ 1

Dept. of Mathematics and Computer Science, Cheikh Anta Diop University, Dakar, Senegal 2

Dept. of Mathematics Computer Science, Cheikh Anta Diop University, Dakar, Senegal 3

Dept. of Mathematics, Niamey University, Niamey, Niger.

Received 03 May 2007; accepted 28 August 2007 Abstract: In this paper we consider the problem of detecting pollution in some non linear parabolic systems using the sentinel method. For this purpose we develop and analyze a new approach to the discretization which pays careful attention to the stability of the solution. To illustrate convergence properties we give some numerical results that present good properties and show new ways for building discrete sentinels. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Sentinel method, collocation method, Chebyshev diﬀerentiation matrix, Gauss–Legendre points and weight matrix MSC (2000): 65M70; 65N22

1

Introduction

Let us consider the following boundary value problem: ⎧ ∂2y ∂y ⎪ x) in Q ⎪ (t, x) − (t, x) + F (y(t, x)) = ξ(t, x) + λξ(t, ⎪ ⎨ ∂t ∂x2 y(t, x) = 0 on ⎪ ⎪ ⎪ ⎩ y(0, x) = y 0 (x) + τ y0 (x) on Ω

(1)

where • Q =]0, T [×Ω, with Ω =] − 1, 1[ , T > 0, and = (0, T ) × ∂Ω, • F : R −→ R is a locally lipschitzian function satisfying the following: there exists ∗ † ‡

E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

752

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

positive constants c1 , c2 , α such that |F (ζ)| ≤ c1 |ζ|α+1 + c2

∀ζ∈R

• ξ and y 0 are given functions respectively in L2 (Q) and L2 (Ω), • λξ is the ”pollution” term such that |λ| << 1 and ξ being given in L2 (Q) satisfying <1 (2) ξ L2 (Q)

• τ y0 is the ”missing” term such that |τ | << 1 and y0 being in L2 (Ω). With above assumptions, it is well known that there exists a unique solution y(t, x; λ, τ ) in L2 (0, T, H01 (Ω)) of the system (1). For more details on problems of type (1) we refer the reader to [6] and [15]. In this paper we are concerned with the following type of inverse problem. Having some knowledge about the solution y(t, x; λ, τ ) over a subdo x). Such a problem can be main ϑ ⊂ Ω, we want to determine the pollution term λξ(t, solved by the sentinel method which was developed by J.L.Lions (1992). This consists in building a so-called sentinel function by setting T S(λ, τ ) = (h0 + w) × y(t, x; λ, τ )dxdt (3) ϑ

0

for a given h0 in L2 (]0, T [×ϑ) and, where w is a solution of the following constrained optimization problem

∂S(λ, τ )

=0 (4) ∂τ λ=τ =0

∂S(λ, τ )

=c (5) ∂λ λ=τ =0 w L2 (Qϑ ) −→ min

(6)

where Qϑ = (0, T ) ×ϑ, c being an arbitrary constant. The existence and the uniqueness of the function w was proved in [12, 13]. Notably, it has been established that the problem (4 - 6) is equivalent to an exact null controllability problem from which the Hilbert Uniqueness Method (HUM) [12] was used to establish the existence and the uniqueness of the control function w. Therefore the sentinel function is determined in the unique way. If the sentinel function is determined then the pollution estimated parameter is given by 1 λ (Sobs − S(0, 0)) (7) c where T

Sobs = ϑ

0

(h0 + w) × yobs (t, x)dtdx

(8)

yobs (t, x) being the observed solution over the domain ϑ. Since the introduction of the sentinel method many authors have developed several applications to this method, such as, in environment, in epidemiology, in ecology and so on (see for instance [8], [9], [16] and [17]). Unfortunately, due to the success of the least square approximation for solving inverses problems, numerical aspects of sentinel methods have not yet been suﬃciently investigated. However, we can refer to the works

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

753

of J.P. Kern´evez et. al [4], [3] for some numerical approximations of the sentinel method where they have used ﬁnite diﬀerences method. We should also mention that in [4], the authors have concentrated their eﬀort in solving an equivalent problem that is an exact controllability problem. Contrary to [4], [3] we develop a new idea to approximate the problem (4-6) using pseudo spectral approximation. Then we construct a sequence of sentinel functions associated to an approximate discrete system of the problem (1) and we prove its convergence to the continuous problem ones. This paper is organized as follows. In section 2, a semi discretization of the state system is performed, leading to a sequence of approximated sentinels problems. The convergence of the numerical scheme is presented in section 3. Section 4 is devoted to numerical experiments.

2

Formulation of the approximated problem

In this section we develop a numerical method to solve the following parameter identiﬁcation problem: Given observations yobs ti , x ti and position x j at time j ∈ ϑ, ﬁnd a which is computed ﬁrst by approximating the state system and, secondly by parameter λ solving a constrained optimization problem.

2.1 Chebyshev pseudo spectral semi-discretization of the problem (1) Let consider the collocation Chebyshev points xi = cos (iπ/N), i = 0, ..., N and the space X N = L2 (0, T, PN (Ω)) where PN (Ω) denotes the polynomial space of degree at most N over the interval Ω =] − 1, 1[. We are interested in ﬁnding the approximated solution of the system (1) by deﬁning the function y N (t, x) ∈ X N that satisﬁes the following collocation discretized systems ⎧ N ∂y (t, xi ) ∂ 2 y N (t, xi ) ⎪ xi ) ⎪ − + (y N (t, xi ))2 = ξ(t, xi ) + λξ(t, ⎪ ⎨ ∂t ∂x2 (9) y N (t, −1) = y N (t, 1) = 0 ⎪ ⎪ ⎪ ⎩ y N (0, xi ) = y 0 (xi ) + τ y0 (xi ) for i = 0, ..., N. Let y N (t) = (y(t, x1 ), y(t, x2 ), · · ·, y(t, xN −1))t We now introduce the derivative vector of y at Chebyshev collocation points

t ∂y ∂y ∂y N (t, x1 ), (t, x2 ), · · ·, (t, xN −1 ) y x (t) = ∂x ∂x ∂x

(10)

(11)

There exists ( see [19]) a (N + 1) by ( N + 1) matrix DN such that yN (t) ≈ DN × y N (t) x then, the system 9 can be written in the following compact form ⎧ N ⎪ ⎨ d y (t) − D 2 × y N (t) + L(y N (t)) = f N (t, λ) N dt ⎪ ⎩ y N (0) = g N (β)

(12)

(13)

754

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

where L(y N (t)) is a vector whose components are y (t, xi )2 and where f N (t, λ) and g N (β) xi ) respectively y 0 (xi ) + τ y0 (xi ) and, are the vector-values of components ξ(t, xi ) + λξ(t, 2 denotes the (N − 1) by (N − 1) matrix obtaining by stripping (DN )2 of its where D N ﬁrst and last rows and columns.

2.2 Description of the approximated sentinel function The second level of the numerical scheme involves approximating the control space by a M sequence Xϑ M >0 of ﬁnite dimensional space. In the framework of this paper we have chosen XϑM = L2 (0, T, PN (ϑ)) where PN (ϑ) designs the space of polynomials of degree at most M over ϑ. For the sake of simplicity we shall let M = N for the remainder of this paper. We are led to construct a family of sentinels deﬁned by T N S (λ, τ ) = (h0 (t, x) + w N (t, x)) × y N (t, x; λ, τ )dxdt (14) 0

ϑ

where • y N (t, x; λ, τ ) is the solution of the approximated state system (9) • w N is the solution of the minimization problem (4 - 6) over the control space XϑM and where the sentinel function is replaced by its approximation (14). Thus, for the sake of computation, we consider the Gauss - Legendre quadrature to approximate the expression (14). This can be done by ﬁxing a constant q. If we denote by ti and x j the Gauss - Legendre quadrature points respectively associated with the interval (0, T ) and the domain ϑ, and by hi the Gauss Legendre weights, then we can approximate (14) by the following Gauss-Legendre quadrature formula |ϑ| × T × S (λ, τ ) ≈ (h0 ( tk , x j ) + w N ( tk , x j )) × hk × hj × y N ( tk , x j ; λ, τ ) (15) 2 j=0 k=0 q

q

N

a (q + 1)2 - square diagonal matrix with entries hk × hj : We now consider H 2 2 H = diag h0 , h1 h0 , · · ·, hq h0 , h0 h1 , · · ·, hq h1 , · · ·, hq and the vectors

and

(16)

t H 0 = h0 ( t0 , x 0 ), · · ·, h0 ( tq , x 0 ), · · ·, h0 ( tq , x q )

(17)

t t0 , x 0 ), · · ·, w N ( tq , x 0 ), · · ·, w N ( tq , x q ) W N = w N (

(18)

t Y N (λ, τ ) = y N ( t0 , x 0 ; λ, τ ), · · ·, y N ( tq , x 0 ; λ, τ ), · · ·, y N ( tq , x q ; λ, τ )

(19)

With these notations we can rewrite the expression (15) as S N (λ, τ ) ≈

|ϑ| × T × Y N (λ, τ ) × (H 0 + W N )t × H 2

Replacing S by S N in (4-5) and using ﬁnite diﬀerences approximations yields S N (0, Δτ ) = S N (0, −Δτ )

(20)

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

and

755

S N (Δλ, 0) − S N (−Δλ, 0) = 2cΔλ

with given values of Δτ and Δλ that are assumed to be small enough. Due to equation (16), the constraints to the approximated problem became

t × W N = − Y N (0, Δτ ) t × H ×H Y N (0, Δτ ) × H 0

(21)

N t × W N = 2 × Δλ × c − Y N (Δλ, 0) t × H ×H Y (Δλ, 0) × H 0 |ϑ| × T

(22)

On the other hand, the objective function (6) can be approximated by N 2 |ϑ| × T × WN w ≈ (W N )t × H 4

(23)

Finally, we consider the following approximation to the inﬁnite dimensional constrained minimization problem (4-6) W

N

min

2 ∈R(q+1)

× WN (W N )t × H

(24)

subject to (21) and (22). The convergence of this approximation is analyzed in the next section.

3

The convergence study

In this section we ﬁrst analyze the convergence of the scheme (9) for the solution of the state system (1) and then we will prove the convergence of the discrete sentinel function S N (λ, τ ) to the continuous ones S(λ, τ ) as the mesh-sizes tends to zero. Let consider πN the Chebyshev interpolation operator which is deﬁned from L2 (0, T, H01 (Ω)) to X0N such that πN φ(xj ) = φ(xj ) for all Chebyshev collocation points xj . Here X0N denotes the space of functions φ ∈ X0 that satisﬁes the boundary condition φ = 0 on Σ. It was previously established (see [1]) that • π N is a bounded linear operator satisfying πN = 1

(25)

• For any function φ ∈ L2 (0, T, H01 (Ω)), πN (φ) → φ in L2 (0, T, H01 (Ω)) as N → ∞. In this setting, the approximate system (9) takes the form ⎧ N ⎨ ∂y (t, x) = F (y N )(t, x) + π (h )(t, x) N λ ∂t ⎩ N y (0, x) = πN (gτ )(x)

(26)

(27)

∂ 2 y(t, x) x) and + (y(t, x))2, hλ (t, x) = ξ(t, x) + λξ(t, where we have set F (y)(t, x) = ∂x2 gτ (x) = y 0 (x) + τ y0 (x). The following holds

756

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

Theorem 3.1. Suppose that gτ ∈ L2 (Ω) and F (.) is uniformly Lipschitzian in L2 ((0, T )× Ω). Then, the semi-discrete solution of the system (27) converges in . L2 ((0,T )×Ω) to the unique solution of (1), uniformly on the interval (0, T ). The proof of this theorem is similar to that of the theorem 2.1 in [1]. This is based on reformulating (1) as well as (27) by t y(t, x) = gτ (x) + (F (y(t, x) + hλ (t, x))dt (28) 0

and N

t

N

y (t, x) = π (gτ )(x) +

0

F (y N )(t, x) + π N hλ (t, x) dt

(29)

Clearly, combining (28) and (29) we obtain N y − y 2 ≤ π N (gτ ) − gτ L2 ((0,T )×Ω) + c1 F (y N ) − F (y) L ((0,T )×Ω) + c2 π N hλ − hλ with c1 and c2 positive constants. We then prove the theorem using properties of the operator πN . We now analyze the convergence of the sentinel approximation (15). First, we have to prove the following result. Theorem 3.2. Under assumptions of the theorem 1, the approximated sentinel function S N (λ, τ ) associated to the system (9) converges to the sentinel function S(λ, τ ) associated to the continuous ones (1), i.e.

N

S (λ, τ ) − S(λ, τ ) → 0 as N → ∞, ∀ (λ, τ ) ∈ R2 (30) Proof. We will prove this theorem in 2 stages. Stage 1: For the sake of simplicity, we let y(t, x) = y, Q = (0, T ) × ϑ and . 2 the usual norm in L2 ((0, T ) × ϑ). From deﬁnitions (3) and (15) we deduce

N

N N

S (λ, τ ) − S(λ, τ ) =

(h0 + w ) × y (λ, τ )dxdt − (h0 + w) × y(λ, τ ) dQ

Q

≤ Q h0 (y N (λ, τ ) − y(λ, τ )) dQ + Q w N (y N (λ, τ ) − y(λ, τ )) dQ

+ Q (w N − w)y(λ, τ ) dQ ≤ h0 2 . y N (λ, τ ) − y(λ, τ )2 + w N 2 . y N (λ, τ ) − y(λ, τ )2 + y(λ, τ ) 2 . w N − w 2 ≤ h0 2 + w N 2 . y N (λ, τ ) − y(λ, τ )2 + y(λ, τ ) 2 . w N − w 2 Since y N (λ, τ ) − y(λ, τ )2 → 0, there exists a positive constant K such that

N

S (λ, τ ) − S(λ, τ ) ≤ K. w N − w ∀ λ, τ ∈ R 2

This last inequality shows that the convergence of S N depends on that of w N . Stage 2: In this stage we have to prove that w N → w as N → ∞.

(31)

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757

It is well known that the determination of the control function w satisfying requirements (3-6) is equivalent to a system of an exactly null controllability problem given by the following system −

∂z(t, x) ∂ 2 z(t, x) + + 2y0 × z(t, x) = wχϑ in (0, T ) × Ω ∂t ∂x2 z(t, −1) = z(t, 1) = 0 on (0, T ) z(T, x)

= 0

(32)

in Ω

subject to z(0, x) = −q0 (0, x)

(33)

where q0 (t, x) is the solution of (32) by replacing the control w(t, x) by a given function h0 (t, x) and we design by y0 (t, x) the solution of system (1) taking λ = τ = 0. It should be noted that the system (32) is obtained by applying the adjoint method to the system (4)-(6). It is known that (32) is a well-posed problem and, through the HUM (Hilbert Uniqueness Method) method developed in ([13], [12]) it is proved that a such w exist. Note that the issue of existence and uniqueness of the control was investigated by J.J.Lions and others authors. One can see ([7], [4], [13], [12], [18]) for more details on the sentinel theory. Thus we deﬁne over the approximated space XN the approximation of (32) by ⎧ N ⎨ ∂z (t, x) = π N F (z(t, x)) + w N (t, x) ∂t ⎩ N z (0, x) = π N (−q0 (x))

(34)

∂ 2 z(t, x) − 2y0.z(t, x) F (z(t, x)) = − ∂2x

(35)

where

and where π N the interpolation operator in the Chebyshev basis. Clearly for a known w, the system (32) is well posed and z N (t, x) → z(t, x) as N tends to ∞. So,combining (32) and (34) yields N N ∂z(t, x) ∂z N (t, x) w − w 2 − F (z(t, x)) + = π F (z(t, x)) − 2 L (0,T,θ) ∂t ∂t L (0,T,θ) ∂ zN − z N π F (z) − F (z) 2 ≤ + L (0,T,θ) 2 ∂t L (0,T,θ)

≤

0(z) as N → 0

The convergence of w N is then proved and we can deduce through (31) the convergence of discrete sentinels functions.

4

Numerical experiments

For our numerical experiments we have chosen Ω =] − 1, 1[ and ϑ =] − 1/2, 1/2[. To numerically test the sentinel method, computational data yobs (ti , xj ) are generated. The

758

A. Traore et al. / Central European Journal of Mathematics 5(4) 2007 751–763

given functions in the system (1) are chosen as follows: ⎧ ⎨ ξ(t, x) + λξ(t, x) = −10−4 (x2 + 1) exp(−t) + λ(x2 − 1)2 exp(−2t) ⎩ y 0 (x) + τ y0 (x) = τ (x2 − 1)

(36)

and F (y) = y 2 so that the exact solution is given by ye (x, t) = 10−4 (x2 − 1) exp(−t)

(37)

from which the corresponding values of parameters are λexact = 10−8 τexact = 10−4

(38)

Note that to generate the observed solution we have added a perturbation term β(t, x) to the exact solution yobs = ye (t, x) + β(t, x) (39) such that β L2 (0,T,Ω) 1. We should also note that all computations were performed by means of Matlab language programming. For the optimization computation, the Matlab function ”fmincon”, a sequential quadratic programming routine was used. The Matlab function ”randn”, a random function, was used to generate β(ti, xj ) for computing observations yobs (ti , xj ). The outline of the exact solution and the computing observation is pictured in Figure 1. Results on the identiﬁcation of the pollution parameter are presented in Table 5, Figures 2 and 3. Table 1 gives values of

the identiﬁed

numerical

pollution parameter λN as well as relative errors λN − λexact λexact . One can see that the relative error of the computed pollution parameters decreases as N ∞. This convergence issue is also observed in Figures 2 and 3 where N takes its values between 5 and 50. Figure 4 shows the convergence process of the perturbed state as N ∞.

5

Conclusion

We have presented a new numerical approach of the sentinel method for detecting pollution. The numerical example we have considered conﬁrmed the convergence of the approximation scheme. Note that it is possible to extend the approach developed in this work to parabolic systems over two and three dimensional complex domains. In these cases the Galerkin discontinuous spectral method could be applied to approximate the state system as well as the objective function. It should be also noted that it is easy to adapt the approach outline here to the sentinel problem where the control domain is diﬀerent to the observed ones.

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Fig. 1 Exact and observed solutions with pollution over the observed domain

759

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Fig. 2 The log representation of the relative error norm of the computation of the pollution parameter within the discretized parameter N

Fig. 3 The picture of the computed values of the identiﬁed pollution parameter within the discretized parameter N

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761

Fig. 4 Convergence process of the state system relatively with the computed values of the pollution parameter. Numerical solutions of the state system are presented at diﬀerent parameter values

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N

Computed values of λ

Relative errors

5

0.0245 ×

10−7

0.7691

7

0.0467 × 10−7

0.5642

10

0.0719 ×

10−7

0.3162

12

0.0857 × 10−7

0.2917

14

0.0941 ×

10−7

0.1488

17

0.0980 × 10−7

0.0557

20

10−7

0.0174

0.1020 ×

Table 1 Table of relative errors between values of pollution parameters and its approximated exact value

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