The Horn conjecture for compact selfadjoint operators
aa r X i v : . [ m a t h . F A ] S e p THE HORN CONJECTURE FOR SUMS OF COMPACTSELFADJOINT OPERATORSH. BERCOVICI, W.S. LI AND D. TIMOTINAbstra t. We determine the possible eigenvalues of ompa t selfadjoint op-erators
A, B (1) , B (2) , . . . , B ( m ) with the property that A = B (1) + B (2) + · · · + B ( m ) . When all these operators are positive, the eigenvalues were known tobe subje t to ertain inequalities whi h extend Horn's inequalities from the(cid:28)nite-dimensional ase when m = 2 . We (cid:28)nd the proper extension of the Horninequalities and show that they, along with their reverse analogues, provide a omplete hara terization. Our results also allow us to dis uss the more gen-eral situation where only some of the eigenvalues of the operators A and B ( k ) are spe i(cid:28)ed. A spe ial ase is the requirement that B (1) + B (2) + · · · + B ( k ) be positive of rank at most ρ ≥ .1. Introdu tionGiven an N × N omplex Hermitian matrix A , we denote by Λ( A ) the sequen e λ ( A ) ≥ λ ( A ) ≥ · · · ≥ λ N ( A ) of its eigenvalues, in de reasing order and listed a ording to their multipli ities.A. Horn [9℄ onje tured a hara terization of the set of triples ( α, β, γ ) ∈ ( R N ) with the property that there exist Hermitian matri es A, B, C su h that Λ( A ) = α, Λ( B ) = β , Λ( C ) = γ , and A = B + C . More pre isely, for every integer r ∈{ , , . . . , N − } he introdu ed a olle tion T Nr of triples ( I, J, K ) of subsets of { , , . . . , N } su h that | I | = | J | = | K | = r ; here we use | I | to denote the ardinalityof I . We will all Horn triples the elements ( I, J, K ) of T Nr . Horn onje tured thatthe triples (Λ( A ) , Λ( B ) , Λ( C )) (with A = B + C ) are pre isely those triples ( α, β, γ ) of de reasing sequen es satisfying the tra e identity N X i =1 α i = N X j =1 β j + N X k =1 γ k , and the Horn inequalities(1.1) X i ∈ I α i ≤ X j ∈ J β j + X k ∈ K γ k for all Horn triples ( I, J, K ) ∈ T Nr with r < N . The study and eventual proof ofthis onje ture are hroni led by W. Fulton in his ex ellent survey [6℄.Assume now that A is a positive ompa t operator on a omplex Hilbert spa e.In this ase we denote by Λ + ( A ) = { λ ( A ) ≥ λ ( A ) ≥ · · · } The (cid:28)rst author was supported in part by grants from the National S ien e Foundation. These ond and third authors thank Indiana University for its hospitality while this paper was written.1HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 2the sequen e of its eigenvalues, in de reasing order and repeated a ording to theirmultipli ities. Note that λ j ( A ) > for all j when A has in(cid:28)nite rank, so that thepossible eigenvalue is not listed in this ase. The problem of hara terizing thetriples (Λ + ( B + C ) , Λ + ( B ) , Λ + ( C )) when B and C are positive ompa t operatorswas studied by S. Friedland [8℄ and Fulton [7℄. They observed that, if ( α, β, γ ) =(Λ + ( B + C ) , Λ + ( B ) , Λ + ( C )) , the Horn inequalities (1.1) must be satis(cid:28)ed for all ( I, J, K ) ∈ T Nr , for all N ≥ and all r < N . Conversely, if ( α, β, γ ) is a triple ofde reasing sequen es tending to zero, and they satisfy all the Horn inequalities, thenthere exist positive ompa t operators A, B, C su h that Λ + ( A ) = α , Λ + ( B ) = β , Λ + ( C ) = γ , and A ≤ B + C . If in addition ∞ X i =1 α i = ∞ X j =1 β j + ∞ X k =1 γ k < ∞ , we must have A = B + C . If we only know that P ∞ j =1 β j < ∞ , repla ing this tra eidentity by the onditions γ ≤ α and ∞ X j =1 β j = ∞ X k =1 ( α k − γ k ) leads to analogous results; this is shown in [2℄. When all of these sums are in(cid:28)nitethe tra e identities are useless. The results of [2℄ do provide a hara terization evenin this ase, based on an extension of the Littlewood-Ri hardson rule.We will show that the tra e identity an be repla ed in general by a system ofinequalities. To des ribe these inequalities we need some notation. Given a (cid:28)niteset I ⊂ N = { , , . . . } , we denote by I c = N \ I the omplement of I , and by I cp the set onsisting of the p smallest elements of I c . The following statement is aparti ular ase of Theorem 6.2. It will be onvenient to onsider that ( ∅ , ∅ , ∅ ) isa Horn triple.Theorem 1.1. Let α, β, γ be three de reasing sequen es with limit zero. The fol-lowing onditions are equivalent:(1) There exist positive ompa t operators B and C su h that Λ + ( B + C ) = α , Λ + ( B ) = β and λ + ( C ) = γ .(2) For every Horn triple ( I, J, K ) , and for all positive integers p, q we havethe Horn inequality X i ∈ I α i ≤ X j ∈ J β j + X k ∈ K γ k , and the extended reverse Horn inequality X i ∈ I cp + q α i ≥ X j ∈ J cp β j + X k ∈ K cq γ k . The extended reverse Horn inequalities may at (cid:28)rst seem to be altogether wrong,sin e the index sets are of di(cid:27)erent ardinalities. It is important to rememberthough that the problem we onsider is not invariant under addition of onstantmultiples of the identity operator to B and C , whi h is the ase in (cid:28)nite dimensions.These ardinalities are in fa t equalized in the more general situation of ompa tselfadjoint operators. We will onsider more than two summands, as is also donein [7, 8℄. Thus, we produ e a hara terization of the set of all ( m + 1) -tuples (Λ ( A ) , Λ ( B (1) ) , Λ ( B (2) ) , . . . , Λ ( B ( m ) )) , where the B ( j ) are ompa t selfadjointHE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 3operators and A = P mj =1 B ( j ) . The eigenvalue sequen e Λ ( A ) will be des ribed inSe tion 4 when A is a ompa t selfadjoint operator. This question an naturallybe put in a more symmetri form by setting A (0) = − B , so that P mj =0 B ( j ) = 0 .The (cid:28)nite-dimensional version of these problems is dis ussed in Fulton [6℄, and itwill be des ribed in Se tion 2 below. A more general problem was onsidered byBu h [4℄, and we will also extend his results to the ompa t selfadjoint ase. Infa t, we will (cid:28)nd ne essary and su(cid:30) ient onditions for the existen e of ompa tselfadjoint operators B ( k ) and A = P mk =1 B ( k ) when their eigenvalues are onlypartially spe i(cid:28)ed. This is a new result even in the matrix ase, and it answers aquestion posed in [6℄.The remainder of the paper is organized as follows. In Se tion 2 we des ribein detail the Horn onje ture in (cid:28)nite dimensions, and we dedu e some importantproperties of Horn triples. In Se tion 3 we prove a (cid:28)nite-dimensional interpolationresult whose in(cid:28)nite-dimensional analogue yields the Horn onje ture in Se tion 4.Se tion 5 ontains the dis ussion of matri es and operators with partially spe i(cid:28)edeigenvalues. The ase of positive operators is dis ussed in Se tion 6, where theextension to m ≥ of Theorem 1.1 is proved. We also di uss brie(cid:29)y the hiveformulation of the Littlewood-Ri hardson rule.2. Horn Inequalities in Finite DimensionsIt will o asionally be onvenient to view a set I = { i < i < · · · < i r } of natural numbers as a fun tion I : { , , . . . , r } → N , i.e., I ( ℓ ) = i ℓ . Thus,if I ′ ⊂ { , , . . . , r } , there is a well-de(cid:28)ned subset I ◦ I ′ ⊂ I . We will also set [ n ] = { , , . . . , n } for n ∈ N .Fix an integer m ≥ . Given integers N, r su h that N ≥ and ≤ r ≤ N , wede(cid:28)ne a olle tion T Nr ( m + 1) of ( m + 1) -tuples ( I, J (1) , J (2) , . . . , J ( m ) ) of subsetsof [ N ] su h that | I | = | J (1) | = · · · = | J ( m ) | = r , ≤ r ≤ N ; when m = 2 , these willbe pre isely the Horn triples. We pro eed by indu tion on N . When N = 0 , thereis only one set to de(cid:28)ne: T ( m + 1) onsists of the ( m + 1) -tuple ( ∅ , ∅ , . . . , ∅ ) .Assume that the sets T Ms have been de(cid:28)ned for all M < N and ≤ s ≤ M .The ( m + 1) -tuples ( I, J (1) , J (2) , . . . , J ( m ) ) in T Nr ( m + 1) are then subje t to therequirements(2.1) r X ℓ =1 ( I ( ℓ ) − ℓ ) = m X k =1 r X ℓ =1 ( J ( k ) ( ℓ ) − ℓ ) , and(2.2) s X ℓ =1 ( I ◦ I ′ ( ℓ ) − ℓ ) ≥ m X k =1 s X ℓ =1 ( J ( k ) ◦ J ′ ( k ) ( ℓ ) − ℓ ) for every s < r and every ( I ′ , J ′ (1) , . . . , J ′ ( m ) ) ∈ T rs ( m + 1) . Observe that the set T NN ( m + 1) onsists of the ( m + 1) -tuple de(cid:28)ned by I = J (1) = · · · = J ( m ) = [ N ] .We will also onsider the larger set T Nr ( m + 1) of ( m + 1) -tuples for whi h theidentity (2.1) is repla ed by r X ℓ =1 ( I ( ℓ ) − ℓ ) ≥ m X k =1 r X ℓ =1 ( J ( k ) ( ℓ ) − ℓ ) . HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 4The following result (in an equivalent form) is proved in [7℄. For a set I ⊂ [ N ] we denote I sym = { N + 1 − i : i ∈ I } .Theorem 2.1. Let α, β (1) , β (2) . . . , β ( m ) ∈ R N be de reasing ve tors. The following onditions are equivalent:(1) There exist Hermitian N × N -matri es A, B (1) , B (2) , . . . , B ( m ) su h that Λ( A ) = α , Λ( B ( k ) ) = β ( k ) for k = 1 , , . . . , m , and A ≤ B (1) + B (2) + · · · + B ( m ) .(2) For every r ≤ N and every ( I, J (1) , J (2) , . . . , J ( m ) ) ∈ T Nr ( m + 1) we havethe Horn inequality r X ℓ =1 α I ( ℓ ) ≤ m X k =1 r X ℓ =1 β J ( k ) ( ℓ ) . (3) For every r ≤ N and every ( I, J (1) , J (2) , . . . , J ( m ) ) ∈ T Nr ( m + 1) we havethe inequality X i/ ∈ I sym α i ≤ m X k =1 X j / ∈ I ( k )sym β j . Note that the requirement (2) (for r = N ) or (3) (for r = 0 ) yields N X ℓ =1 α ℓ ≤ m X k =1 N X ℓ =1 β ℓ . Repla ing this requirement by the tra e ondition N X ℓ =1 α ℓ = m X k =1 N X ℓ =1 β ℓ implies the equality A = B (1) + · · · + B ( m ) . It an also be shown that onditions (2)and (3) are not merely equivalent: they are pre isely the same. We will not needthis stronger assertion, so we do not in lude a proof. The interested reader will beable to supply an indu tive argument based on the de(cid:28)nion of the sets T Nr .An immediate onsequen e of this result is an alternate hara terization of theelements of T Nr ( m + 1) . We an asso iate with ea h subset I ⊂ N of ardinality r an integer partition π ( I ) of length r as follows: π ( I ) = { I ( r ) − r ≥ I ( r − − ( r − ≥ · · · ≥ I (1) − } . Corollary 2.2. Let
I, J (1) , . . . , J ( m ) ⊂ { , , . . . , N } be su h that | I | = | J (1) | = · · · = | J ( m ) | = r . The following onditions are equivalent:(1) ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) ( respe tively, ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m +1)) .(2) There exist Hermitian r × r -matri es A, B (1) , B (2) , . . . , B ( m ) su h that Λ( A ) = π ( I ) , Λ( B ( k ) ) = π ( J ( k ) ) for k = 1 , , . . . , m , and A = B (1) + B (2) + · · · + B ( m ) ( respe tively, A ≥ B (1) + B (2) + · · · + B ( m ) ) .Proof. The tra e identity s X ℓ =1 ( I ′ ( ℓ ) − ℓ ) = m X k =1 s X ℓ =1 ( J ′ ( k ) ( ℓ ) − ℓ ) HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 5shows that ondition (2.2) an be rewritten as s X ℓ =1 ( I ◦ I ′ ( ℓ ) − I ′ ( ℓ )) ≥ m X k =1 s X ℓ =1 ( J ( k ) ◦ J ′ ( k ) ( ℓ ) − J ′ ( k ) ( ℓ )) . Therefore the r -tuples − π ( I ) , − π ( J (1) ) , . . . , − π ( J ( m ) ) satisfy ondition (2) of The-orem 2.1. Moreover, (2.1) is pre isely the tra e identity for these tuples. The orollary follows easily from these observations. (cid:3) Corollary 2.3. Assume that ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) and I ⊂ [ n ] for some n < N . Then we also have J ( k ) ⊂ [ n ] for j = 1 , , . . . , m .It should be noted that the results in [6℄ are a tually formulated in symmetri form, i.e., the operator A is repla ed by B (0) = − A , so that the ondition on theseoperators is P mk =0 B ( k ) = 0 (or ≤ ). The passage from one formulation to theother is straightforward. Denote indeed α = Λ( A ) , β (0) = Λ( B (0) ) , and observethat for every subset I ⊂ { , , . . . , N } we have X i ∈ I α i = − X i ∈ I sym β (0) i . If | I | = r , we also have r X ℓ =1 ( I ( ℓ ) − ℓ ) + r X ℓ =1 ( I sym ( ℓ ) − ℓ )) = r ( N − r ) . This allows rewriting the de(cid:28)nition of T Nr ( m + 1) in terms of the sets J (0) = I sym , J (1) , . . . , J ( m ) . We prefer however the less symmetri version of these results.One reason is that we have T Nr ( m + 1) ⊂ T N +1 r ( m + 1) , and therefore we obtainHorn inequalities whi h are valid for matri es of arbitrary size. In fa t, T Nr ( m + 1) onsists pre isely of those ( m + 1) -uples in T N +1 r ( m + 1) whi h are ontained in [ N ] . Another reason is that we an assume that all the operators are positive.The elements of T Nr ( m + 1) and T Nr ( m + 1) also have a ohomologi al inter-pretation, related with the ring stru ture of the ohomology of the Grassmannian G ( N, r ) of r -dimensional subspa es of C N . Sin e this onne tion was ru ial inthe proof of the Horn onje ture, we des ribe it in more detail. Fix subspa es X ⊂ X ⊂ · · · ⊂ X N = C N with dim X j = j , and a subset I = { i < i < · · · < i r } of [ N ] . The asso iated S hubert ell S = { M ∈ G ( N, r ) : dim( M ∩ X i j ) ≥ j for j = 1 , , . . . , r } determines a homology lass η I ∈ H ∗ ( G ( N, r )) whi h is independent of the hoi eof X j . Moreover, H ∗ ( G ( N, r )) is the free abelian group generated by the lasses η I as I runs over all subsets I ⊂ [ N ] with | I | = r . The parti ular set I = [ r ] orresponds with the lass of one point. The ohomology ring H ∗ ( G ( N, r )) has adual basis ω I indexed by I ⊂ N , | I | = r , i.e., h η I , ω J i = δ IJ . The set T Nr ( m + 1) onsists of those ( m + 1) -tuples ( I, J (1) , . . . , J ( m ) ) su h that the produ t(2.3) ω I ω J (1) sym · · · ω J ( m ) symis not equal to zero, while T Nr ( m +1) is hara terized by the fa t that this produ t isa nonzero multiple of the lass ω [ r ] of one point. One an also identify a smaller lass ˙ T Nr ( m + 1) ⊂ T Nr ( m + 1) orresponding with produ ts (2.3) whi h are exa tly equalHE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 6to ω [ r ] . As seen in [1, 6℄, the Horn inequalities orresponding with ˙ T Nr ( m + 1) , r =1 , , . . . , N − , are independent, and they imply (for de reasing sequen es) all theHorn inequalities orresponding with T Nr ( m +1) , or even T Nr ( m +1) . We will alwaysformulate our results in terms of the sets T , but generally one an use either T (ifone wants to dedu e more inequalities) or ˙ T (if one wants minimal hypotheses).We on lude this se tion with a few useful properties of the sets T and T .Lemma 2.4. Fix ( I, J (1) , J (2) , . . . , J ( m ) ) ∈ T Nr ( m +1) , integers ≤ q, q , . . . , q m ≤ r su h that q = P mk =1 q k ≤ r , and an additional M ∈ N . Denote by I ′ the setobtained by in reasing the largest q elements by M , i.e., I ′ ( ℓ ) = ( I ( ℓ ) if ℓ ≤ r − q,I ( ℓ ) + M if ℓ > r − q. Analogously, de(cid:28)ne J ′ ( k ) by J ′ ( k ) ( ℓ ) = ( J ( k ) ( ℓ ) if ℓ ≤ r − q k ,J ( k ) ( ℓ ) + M if ℓ > r − q k . for k = 1 , , . . . , m . Then ( I ′ , J ′ (1) , J ′ (2) , . . . , J ′ ( m ) ) ∈ T N + Mr ( m + 1) .Proof. By indu tion, it su(cid:30) es to onsider the ase q = 1 . For simpli ity, assumethat q = 1 and q k = 0 for k > . Let us set α = π ( I ) , α ′ = π ( I ′ ) , β k = π ( J ( k ) ) ,and β ′ k = π ( J ′ ( k ) ) for k = 1 , , . . . r . By Corollary 2.2 (or by the de(cid:28)nition of T Nr ) ,the sequen es ( α, β (1) , . . . , β ( k ) ) satisfy ondition (1) of Theorem 2.1, plus the tra eidentity. Moreover, the di(cid:27)eren es ( α ′ − α, β ′ (1) − β (1) , . . . , β ′ ( k ) − β ( k ) ) also satisfy ondition (1) of Theorem 2.1 and the tra e identity. Indeed, α ′ − α = β ′ (1) − β (1) =( M, , . . . , , while β ′ ( k ) − β ( k ) = 0 for k > . We dedu e that ( α ′ , β ′ (1) , . . . , β ′ ( k ) ) satisfy ondition (1) of Theorem 2.1 as well. The on lusion follows from Corollary2.2. (cid:3) Proposition 2.5. Fix integers x < r < N and y < N − r .(1) If ( I, J (1) , . . . J ( m ) ) ∈ T Nr ( m + 1) and ( I ′ , J ′ (1) , . . . , J ′ ( m ) ) ∈ T rx ( m + 1) then ( I ◦ I ′ , J (1) ◦ J ′ (1) , . . . J ( m ) ◦ J ′ ( m ) ) ∈ T Nx ( m + 1) .(2) If ( I, J (1) , . . . J ( m ) ) ∈ T Nr ( m + 1) and ( I ′ , J ′ (1) , . . . , J ′ ( m ) ) ∈ T N − ry ( m + 1) then ( I ′′ , J ′′ (1) , . . . , J ′′ ( m ) ) ∈ T Nr + y ( m + 1) , where I ′′ = I ∪ ( I c ◦ I ′ ) and J ′′ ( k ) = J ( k ) ∪ ( J ( k ) c ◦ J ′′ ( k ) ) for k = 1 , , . . . m .(3) If ( I, J (1) , . . . J ( m ) ) ∈ T ( m +1) Nr ( m + 1) then | I ∩ [ N ] | + X | J ( k ) \ [ mN ] | ≤ r. Proof. Properties (1) and (2) are known; see, for instan e Bu h [4, Lemma 1℄.To prove (3), onsider mutually orthogonal proje tions P (1) , P (2) , . . . , P ( m ) of size ( m + 1) N × ( m + 1) N and of rank N , and set P = P (1) + · · · + P ( m ) . Observe that α = Λ( − P ) = (0 , . . . , | {z } N , − , . . . , − | {z } mN ) , β ( k ) = λ ( − P ( k ) ) = (0 , . . . , | {z } mN , − , . . . , − | {z } N ) and therefore X i ∈ I α i = −| I \ [ N ] | , X j ∈ J ( k ) β ( k ) j = −| J ( k ) \ [ mN ] | . HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 7Thus the Horn inequality applied to the eigenvalues of − P, − P (1) , . . . , − P ( m ) yields −| I \ [ N ] | ≤ − m X k =1 | J ( k ) \ [ mN ] | . The desired inequality is obtained sin e | I \ [ N ] | = r − | I ∩ [ N ] | . (cid:3)
3. An Interpolation ResultGiven three ve tors α, α ′ α ′′ ∈ R N , we will say that α is between α ′ and α ′′ if min { α ′ i , α ′′ i } ≤ α i ≤ max { α ′ i , α ′′ i } , i = 1 , , . . . , N. We denote by α ∗ the de reasing rearrangement of a ve tor α ∈ R N .Lemma 3.1. Let α, α ′ , α ′′ ∈ R N be su h that α ′ and α ′′ are de reasing, and α isbetween α ′ and α ′′ . Then α ∗ is also between α ′ and α ′′ .Proof. Repla ing α ′ i by min { α ′ i , α ′′ i } and α ′′ i by max { α ′ i , α ′′ i } , we may assume that α ′ ≤ α ≤ α ′′ . We have then α ′∗ ≤ α ∗ ≤ α ′′∗ , and learly α ′∗ = α ′ , α ′′∗ = α ′′ . (cid:3) We are now ready to interpolate between N -tuples whi h satisfy the Horn in-equalities and the reverse Horn inequalities. If α ∈ R N , and I = { i < i < · · · . Proposition 2.5(1) implies then the existen e of r × r -matri es A , B (1)0 , . . . , B ( m )0 su h that A = P mk =1 B ( k )0 , Λ( A ) = α ( τ ) ◦ I , and Λ( B ( k )0 ) = β ( k ) ( τ ) ◦ J ( k )0 . We laim that the ve tors α ′ = α ( τ ) ◦ I cN − r , β ′ ( k )1 = β ( k ) ( τ ) ◦ J ( k ) cN − r , α ′′ = α ′′ ◦ I cN − r , β ′′ ( k )1 = β ′′ ◦ J ( k ) cN − r satsify the hypothesis of theproposition, with N − r in pla e of N . Indeed, the required Horn inequalities for α ′ and β ′ ( k )1 follow from Proposition 2.5(2) (the equality (3.1) must be subtra tedfrom the orresponding Horn inequality for α ( τ ) and β ( k ) ( τ ) ). The required Horninequalities for α ′′ and β ′′ ( k )1 follow dire tly from Proposition 2.5(1). The indu tivehypothesis implies the existen e of ( N − r ) × ( N − r ) -matri es A , B ( k )1 su h that A = P mk =1 B ( k )1 , Λ( A ) is between α ′ and α ′′ , and Λ( B ( k )1 ) is between β ′ ( k )1 and β ′′ ( k )1 . The on lusion of the proposition is satis(cid:28)ed by the matri es A = A ⊕ A and B ( k ) = B ( k )0 ⊕ B ( k )1 , k = 1 , , . . . , m . The veri(cid:28) ation of this fa t is an easyappli ation of the pre eding lemma be ause Λ( A ) = (Λ( A ) , Λ( A )) ∗ .Assume now that the entries of α ′ , α ′′ , β ′ ( k ) , β ′′ ( k ) are integers for k = 1 , , . . . , m .Unfortunately, α ( τ ) does not generally have integer entries. The argument willpro eed however in a similar manner. We onstru t for n = 0 , , . . . de reasinginteger ve tors α ( n ) , β ( k ) ( n ) su h that(1) α (0) = α ′ , β ( k ) (0) = β ′ ( k ) , (2) α ( n + 1) is between α ( n ) and α ′′ , and β ( k ) ( n + 1) is between β ( k ) ( n ) and β ′′ ( k ) for k = 1 , , . . . , m, (3) P i ∈ I α ( n ) i ≤ P mk =1 P j ∈ J ( k ) β ( k ) ( n ) j for every ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m +1) , r ≤ N , and(4) P Ni =1 | α ( n + 1) i − α ( n ) i | + P mk =1 P Nj =1 | β ( k ) ( n + 1) j − β ( k ) ( n ) j | = 1 . In other words, only one entry of one of the ve tors is modi(cid:28)ed in passing from n to n + 1 . The onstru tion pro eeds by indu tion until either α ( n ) = α ′′ and β ( k ) ( n ) = β ′′ for all n , or one of the inequalities in (3) is an equality. The remainderof the argument pro eeds as before. (cid:3) In the following statement, the inequality I ′ ≤ I is simply an inequality betweenfun tions, i.e., I ′ ( ℓ ) ≤ I ( ℓ ) for ℓ = 1 , , . . . r .Corollary 3.3. Given r < N and ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) , there exists ( I ′ , J ′ (1) , . . . , J ′ ( m ) ) ∈ T Nr ( m + 1) su h that I ′ ≤ I and J ′ ( k ) ≥ J ( k ) for k =1 , , . . . , m .Proof. Let us de(cid:28)ne partitions of length r as follows: α ′′ = π ( I ) , β ′′ ( k ) = π ( J ( k ) ) ,α ′ = (0 , , . . . , , and β ′ ( k ) = ( N − r, N − r, . . . , N − r ) . We laim that these ve torssatisfy the hypotheses of Proposition 3.2. Indeed, the inequalities for α ′′ , β ′′ ( k ) HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 9follow from the fa t that ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) , and the Horn inequalitiesfor α ′ , β ′ ( k ) are obvious. Proposition 3.2 provides partitions α = Λ( A ) , β ( k ) =Λ( B ( k ) ) whi h satisfy the Horn inequalities and the tra e identity, and su h that α ′ ≤ α ≤ α ′′ and β ′′ ( k ) ≤ β ≤ β ′ ( k ) for k = 1 , , . . . m . To on lude the proof, we de(cid:28)ne I ′ , J ′ ( k ) ⊂ [ N ] su h that π ( I ′ ) = α and π ( J ′ ( k ) ) = β ( k ) for k = 1 , , . . . , m . (cid:3)
4. Eigenvalues of Compa
t Selfadjoint OperatorsLet A be a
ompa
t selfadjoint operator on a
omplex Hilbert spa
e H . One
anrepresent A as Ah = X k µ k h h, e k i e k , h ∈ H , where { e k } is an orthonormal system, and ( µ k ) is a real sequen
e with limit zero.It may no longer be possible to rearrange the eigenvalues µ k in de
reasing order.Instead, we
an de(cid:28)ne numbers λ ± n for n ∈ N su
h that λ n is th n th largest positiveterm in ( µ k ) , while λ − n is the n th smallest negative term in ( µ k ) . Note that λ ≥ λ ≥ · · · ≥ and λ − ≤ λ − ≤ · · · ≤ . If there are only p < ∞ positive terms in ( µ k ) , we set λ n = 0 for n > p , with asimilar
onvention for the negative terms. We will denote by Λ ( A ) the sequen
e λ ≥ λ ≥ · · · ≥ λ − ≥ λ − . Observe that A annot quite be re
onstru
ted, up to unitary equivalen
e, from Λ ( A ) . It may happen that is not an eigenvalue of A , but it (cid:28)gures in(cid:28)nitelymany times in Λ ( A ) . Conversely, may be an eigenvalue of A but λ ± n = 0 for all n . We will also use the notation Λ + ( A ) for the sequen
e { λ n } ∞ n =1 . Thus Λ ( A ) anbe identi(cid:28)ed with the pair (Λ + ( A ) , − Λ + ( − A )) .We will denote by c ↓ ↑ the
olle
tion of all real sequen
es ( α ± n ) ∞ n =1 su
h that α ≥ α ≥ · · · ≥ ≥ · · · ≥ α − ≥ α − and lim n →∞ α ± n = 0 . If α ∈ c ↓ ↑ , we will denote by α the sequen
e ( β ± n ) ∞ n =1 de(cid:28)ned by β k = − α − k for all k = ± n . With this notation, we have Λ ( − A ) = Λ ( A ) for all
ompa
t selfadjoint operators A .We will prove analogues of the Horn inequalities by
ompressing operators to(cid:28)nite-dimensional subspa
es. The following observation shows why this is possible.Lemma 4.1. Let A be a
ompa
t selfadjoint operator on H , let P be an orthogonalproje
tion on H , and set α = Λ ( A ) , β = Λ ( P AP | P H ) . Then we have α n ≥ β n ≥ β − n ≥ α − n , n ∈ N . Proof. This is an immediate
onsequen
e of the usual variational formulas(4.1) α n = inf dim M
1) = p − ) . Similarly, the lower estimates amount to α p ≥ max ( m X k =1 β ( k ) j k : m X k =1 ( N − j k ) = N − p ) . This result was proved by Johnson [10℄. for m = 2 .These onsiderations an be extended to any situation where the ve tors β ( k ) are fully spe i(cid:28)ed, thus answering a question raised in [6℄. Finding a minimal setof inequalities may require some work. We illustrate this in ase α and α arespe i(cid:28)ed. For simpli ity, assume that m = 2 , and set β = β (1) , γ = β (2) . In this ase α min = ( α , α , α , −∞ , . . . , −∞ | {z } N − ) , α max = ( α , α , α , . . . , α | {z } N − ) . For the upper estimates, we need to onsider Horn triples ( I, J, K ) with I ⊂ [3] .When | I | = 1 , we obtain α ≤ β + γ , α ≤ min { β + γ , β + γ , β + γ ) , HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 15when | I | = 3 we obtain the inequality α + 2 α ≤ X j =1 ( β j + γ j ) , and when | I | = 2 we have I = { , } whi h yields(5.1) α + α ≤ β + β + γ + γ ,I = { , } whi h yields α + α ≤ min { β + β + γ + γ , β + β + γ + γ } , and (cid:28)nally I = { , } whi h yields α ≤ β + β + γ + γ β + β + γ + γ β + β + γ + γ . As before, these last three estimates and (5.1) an be dis arded in order to obtaina minimal set of upper estimates. A minimal set of lower estimates is harder toobtain. The inequalities for whi h [ N ] \ I ⊂ [2] or [ N ] \ I ⊂ [2] c involve only one ofthe spe i(cid:28)ed entries, and will be satis(cid:28)ed provided that α ≥ max { β j + γ k : j + k = N + 1 } , α ≥ max { β j + γ k = N + 3 } . In addition, there will be lower bounds for α + ( r − α and α + ( r − α when ≤ r ≤ N . These are provided by the reverse Horn inequalities with [ N ] \ I =[ r +1] \{ } and [ N ] \ I = [ r ] , respe tively. The fa t that one does not need to onsidermore general sets I is dedu ed easily from Corollary 3.3. Indeed, assume that wewrite a reverse Horn inequality su h that P i/ ∈ I α max i = α + ( r − α . Repla ing I by I ′ su h that [ N ] \ I ′ = [ r + 1] \ { } we have P i/ ∈ I α max i = P i/ ∈ I ′ α max i , andthe ( m + 1) -tuple ( I ′ , J (1) , . . . , J ( m ) ) belongs to T NN − r . Corollary 3.3 yields then ( I ′′ , J ′ (1) , . . . , J ′ ( m ) ) in T NN − r su h that I ′′ ≥ I ′ and J ′ ( k ) ≥ J ( k ) . The reverseHorn inequality for this new tuple will be stronger than the one given by theoriginal ( I, J (1) , . . . , J ( m ) ) . One reasons in a similar fashion when P i/ ∈ I α max i =2 α + ( r − α . Some of the inequalities with [ N ] \ I = [ r + 1] \ { } might notbelong to ˙ T NN − r and an therefore be dis arded.The results of [4℄ an also be dedu ed from Proposition 5.1. Namely, one onsid-ers the ve tors β (1) , . . . , β ( m ) to be fully spe i(cid:28)ed, while the only spe i(cid:28)ed entriesof α are α ρ +1 = α ρ +2 = · · · = α N = 0 . This amounts to looking at Hermitian ma-tri es B (1) , . . . , B ( m ) with spe i(cid:28)ed eigenvalues, whose sum is positive and has rankat most ρ . The onditions found in [4℄ form a minimal system of inequalities equiv-alent to the ones provided by Proposition 5.1. The upper estimates whi h must bekept are those orresponding to ( I, J (1) , . . . , J ( m ) ) ∈ ˙ T Nr su h that I = [ N ] \ [ N − r ] ,as an be seen by an argument using Corollary 3.3. The lower estimates orrespondwith I ⊃ [ ρ ] , and one only needs to onsider those tuples in ˙ T .We onsider now ompa t selfadjoint operators with partially spe i(cid:28)ed eigenval-ues. Let S be a set of nonzero integers. We assume, for simpli ity, that S ∩ N and S ∩ ( − N ) are in(cid:28)nite. A fun tion n α n ∈ R de(cid:28)ned on S will be alled apartially spe i(cid:28)ed element of c ↓ ↑ if there exists β ∈ c ↓ ↑ su h that β n = α n forall n ∈ S . Let α be su h a sequen e, and assume S ∩ N = { n < n < · · · } ,HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 16 S ∩ ( − N ) = { m > m > . . . } . We introdu e two sequen es ( α max ± n ) ∞ n =1 and ( α min ± n ) ∞ n =1 as follows: α max n = ∞ if < n < n α n j if n j ≤ n < n j +1 α m if m ≤ n < α m j +1 if m j +1 ≤ n < m j α min n = α n if < n ≤ n α n j +1 if n j < n ≤ n j +1 −∞ if m < n < α m j if m j +1 < n ≤ m j . The elements β ∈ c ↓ ↑ whi h extend α are pre isely those satisfying the inequalities α min ≤ β ≤ α max . We will use the notation β ⊃ α to indi ate that the sequen e β extends the partially de(cid:28)ned α .In the following result, as in its (cid:28)nte-dimensional ounterpart, an extended Horninequality in whi h an in(cid:28)nite term appears is to be automati ally satis(cid:28)ed. Inorder to see what the di(cid:30) ulty is in extending the proof of Proposition 5.1, observethat there are in(cid:28)nitely many extended Horn inequalities. Thus, when repla ingan in(cid:28)nite entry by a (cid:28)nite onstant, we impose in(cid:28)nitely many onditions on that onstant. We need to show that these onstraints an be met simultaneously.Proposition 5.2. Let α, β (1) , β (2) , . . . , β ( m ) be partially spe i(cid:28)ed elements of c ↓ ↑ .The following onditions are equivalent:(1) There exist ompa t selfadjoint operators A, B (1) , . . . , B ( k ) su h that A = P mk =1 B ( k ) , Λ ( A ) ⊃ α , and Λ ( B ( k ) ) ⊃ β ( k ) for k = 1 , , . . . , m .(2) Both ( α min , β (1) max , . . . , β ( m ) max ) and ( α max , β (1) min , . . . , β ( m ) min ) satisfyall the Horn inequalities.Proof. The impli ation (1) ⇒ (2) follows immediately from the orresponding im-pli ation in Theorem 4.4. Theorem 4.4 also shows that, to prove the oppositeimpli ation, it will su(cid:30) e to (cid:28)nd sequen es α ′ , α ′′ , β ′ ( k ) , β ′′ ( k ) ∈ c ↓ ↑ su h that the ( m + 1) -tuples ( α ′ , β ′ (1) , . . . , β ′ ( k ) ) and ( α ′′ , β ′′ (1) , . . . , β ′′ ( k ) ) satisfy all the Horninequalities, α ′ ≥ α min , α ′′ ≤ α max , β ′ ( k ) ≤ β ( k ) max , and β ′′ ( k ) ≥ β ( k ) min . Bysymmetry, it will su(cid:30) e to onstru t the sequen es α ′ and β ′ ( k ) . We start by on-stru ting α ′ ≥ α min su h that ( α ′ , β (1) max , . . . , β ( m ) max ) satisfy all the extendedHorn inequalities. The ve tors α min and β ( k ) max have (cid:28)nitely many in(cid:28)nite entries.Assume for de(cid:28)niteness that α min i = −∞ for i = − , − , . . . , − u , β max i = + ∞ for i = 1 , , . . . , v k , and all the other entries of these sequen es are (cid:28)nite. We will onstru t indu tively sequen es α (0) = α, α (1) , α (2) , . . . , α ( v ) = α ′ su h that ea h α ( i +1) is obtained from α ( i ) by repla ing one −∞ entry by a large negative number,and ( α ( i ) , β (1) , . . . , β ( m ) ) satisfy all the extended Horn inequalities. It will su(cid:30) eto show that it is possible to onstru t α (1) . Denote by α (1) the sequen e de(cid:28)nedby α (1) i = ( α min i if i = − uτ if i = − u for τ ≤ α min − u − . We verify that ( α (1) , β (1) max , . . . , β ( m ) max ) satis(cid:28)es all the Horninequalities if τ is su(cid:30) iently small. The relevant inequalities whi h are not already overed by the hypothesis of the propositions are r − q X ℓ =1 α (1) I ( ℓ ) + r X ℓ = r − q +1 α ( τ ) I ( ℓ ) − N − ≤ m X k =1 r − q k X ℓ =1 β ( k ) max J ( k ) ( ℓ ) + r X ℓ = r − q k +1 β ( k ) max J ( k ) ( ℓ ) − N − , HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 17where ( I, J (1) , . . . , J ( k ) ) ∈ T Nr ( m + 1) , q = q + · · · + q m < r , I ( r ) − N − − u ,and J ( k ) (1) > v k for k = 1 , , . . . , m . The ( m + 1) -tuple ( J, J, . . . , J ) , with J = { , , . . . , r − } , belongs to T rr − ( m + 1) ; this is an easy onsequen e of Corollary2.2 sin e π ( J ) = 0 . Therefore ( I ◦ J, J (1) ◦ J, . . . , J ( m ) ◦ J ) ∈ T Nr − ( m + 1) byProposition 2.5(1). Corollary 3.3 provides ( I ′ , J ′ (1) , . . . , J ′ ( k ) ) ∈ T Nr − ( m + 1) su hthat I ′ ≤ I ◦ J and J ′ ( k ) ≥ J ( k ) ◦ J for k = 1 , , . . . , m . The hypothesis implies thenthe inequality r − − q ′ X ℓ =1 α min I ′ ( ℓ ) + r − X ℓ = r − q ′ α min I ′ ( ℓ ) − N − ≤ m X k =1 r − − q ′ k X ℓ =1 β ( k ) max J ′ ( k ) ( ℓ ) + r − X ℓ = r − q ′ k β ( k ) max J ′ ( k ) ( ℓ ) − N − , and the relations between I ′ and I ◦ J , along with the fa t that α is de reasing,imply r − − q ′ X ℓ =1 α min I ( ℓ ) + r − X ℓ = r − q ′ α min I ( ℓ ) − N − ≤ m X k =1 r − − q ′ k X ℓ =1 β ( k ) max J ( k ) ( ℓ ) + r − X ℓ = r − q ′ k β ( k ) max J ( k ) ( ℓ ) − N − . With the hoi e q ′ k = q k − and q ′ k = q k for k = k , we obtain r − q X ℓ =1 α min I ( ℓ ) + r − X ℓ = r − q +1 α min I ( ℓ ) − N − ≤ m X k =1 r − q k X ℓ =1 β ( k ) max J ( k ) ( ℓ ) + r − X ℓ = r − q k +1 β ( k ) max J ( k ) ( ℓ ) − N − after also repla ing some negative terms by positive ones in the right-hand side ofthe inequality. The desired inequality for α (1) will then follow provided that τ ≤ m X k =1 β ( k ) max J ( k ) ( r ) − N − , and for this it will su(cid:30) e that τ ≤ m X k =1 β ( k ) max − . The onstru tion of β ′ ( k ) is then done by indu tion on k so that ( α ′ , β ′ (1) , . . . , β ′ ( k ) , β ( k +1) max , . . . , β ( m ) max ) satis(cid:28)es all the extended Horn inequalities. For instan e, we know from Corollary4.7 that ( β (1) max , α ′ , β (2) max , . . . , β ( m ) max ) satis(cid:28)es all the extended Horn inequal-ities, and the pre eding onstru tion (with β (1) max in pla e of α min ) yields thedesired sequen e β ′ (1) . This on ludes the onstru tion of the sequen es β ′ ( k ) andthe proof of the proposition. (cid:3)
6. Cases of Equality, Positive OperatorsIn (cid:28)nite dimensions it is known [9℄ that, when one of the Horn inequalities is anequality, there is a subspa e whi h redu es all the matri es involved. This resultextends easily to ompa t selfadjoint operators.HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 18Proposition 6.1. Let
A, B (1) , . . . , B ( k ) be ompa t selfadjoint operators su h that A = P mk =1 B ( k ) , and set α = Λ ( A ) and β ( k ) = Λ ( B ( k ) ) . Assume that the ( m + 1) -tuple ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) , q , . . . , q m are su h that q = P mk =1 q k ≤ r ,and r − q X ℓ =1 α I ( ℓ ) + r X ℓ = r − q +1 α I ( ℓ ) − N − = m X k =1 r − q k X ℓ =1 β ( k ) J ( k ) ( ℓ ) + r X ℓ = r − q k +1 β ( k ) J ( k ) ( ℓ ) − N − . Then there is a spa e M of dimension N whi h redu es the operators A and B ( k ) , Λ( A | M ) = ( α I (1) , . . . , α I ( r − q ) , α I ( r − q +1) − N − , . . . , α I ( r ) − N − ) , and similarly Λ( B ( k ) | M ) = ( β ( k ) J ( k ) (1) , . . . , β ( k ) J ( k ) ( r − q k ) , β ( k ) J ( k ) ( r − q k +1) − N − , . . . , β ( k ) J ( k ) ( r ) − N − ) for k = 1 , , . . . , r .Proof. Assume that the operators a t on the separable spa e H . Constru t spa es H n su h that H n ⊂ H n +1 , S H n is dense in H , and H n ontains all eigenve tors for A and B ( k ) orresponding with the eigenvalues α I ( ℓ ) , α I ( ℓ ) − N − , β ( k ) J ( k ) ( ℓ ) , β ( k ) J ( k ) ( ℓ ) − N − for ℓ = 1 , , . . . , r , provided that n ≥ r ( m + 1) . The (cid:28)nite-dimensional resultimplies the existen e of spa es M n , redu ing for P H n A | H n and P H n B ( k ) | H n , su hthat Λ( P H n A | M n ) and Λ( P H n B ( k ) | M n ) have the desired eigenvalues for n large. Ifall of these eigenvalues are di(cid:27)erent from zero, it will follow that all the spa es M n are ontained in a (cid:28)xed (cid:28)nite-dimensional spa e, and then we an hoose M to beany limit point of this sequen e. The ase in whi h some of the eigenvalues involvedare zero requires minor modi(cid:28) ations whi h we leave to the interested reader. (cid:3) Corollary 4.5 spe ializes as follows in the ase of positive operators.Theorem 6.2. Let α, β (1) , . . . , β ( m ) be de reasing sequen es with limit zero. thefollowing onditions are equivalent:(1) There exist positive ompa t operators A, B (1) , . . . , B ( k ) su h that A = P mk =1 B ( k ) , Λ + ( A ) = α , and Λ + ( B ( k ) ) = β ( k ) for k = 1 , , . . . , m .(2) For every r < N , every q , . . . , q m ≥ su h that q = q + · · · + q m ≤ r , andevery ( I, J (1) , . . . , J ( m ) ) ∈ T Nr ( m + 1) , we have the Horn inequality X i ∈ I α i ≤ m X k =1 X j ∈ J ( k ) β ( k ) j and the extended reverse Horn inequality X i ∈ I cq α i ≥ m X k =1 X j ∈ J ( k ) cqk β ( k ) j . The la k of balan e in the reverse Horn inequalities omes from the fa t that allnegative terms in Λ ( A ) and Λ ( B ( k ) ) are equal to zero. The proof of the propo-sition is straighforward if we take into a ount the equivalen e between onditions(2) and (3) in Theorem 2.1. Note that the extended Horn inequalities need notbe required in full generality (cid:21) the ordinary Horn inequalities su(cid:30) e in the aseof positive operators. Indeed, applying the Horn inequalities provided by Lemma2.4 one obtains the extended Horn inequalities as M → ∞ . This is not the aseHE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 19for the reverse inequalities. While this observation does redu e the olle tion ofinequalities to be veri(cid:28)ed, the smaller olle tion is still redundant. As noted in[8, 7℄, one an delete any (cid:28)nite olle tion of Horn inequalities from ondition (2)without altering the on lusion of the theorem.We will now illustrate some of our results with a simple example where m =2 . Consider the sequen es β = γ ′ = (1 / n ) ∞ n =1 , and positive ompa t operators B = C su h that Λ + ( B ) = β. Note that α = Λ + ( B + C ) = (1 /n ) ∞ n =1 . Onthe other hand, let C be a positive ompa t operator su h that Λ + ( C ) = γ ′′ =(1 / (2 n − ∞ n =1 . Then α = Λ + ( B ⊕ C ) = Λ + ( B ⊕ ⊕ C ) . It follows that ( α, β, γ ′ ) and ( α, β, γ ′′ ) satisfy all the Horn and reverse Horn inequali-ties, and moreover γ ′′ n > γ ′ n for all n . This situation is not possible in the ase of sum-mable sequen es. Indeed, the tra e identity would imply that P ∞ n =1 ( γ ′ n − γ ′′ n ) = 0 . Note moreover that for any sequen e γ su h that γ ′ ≤ γ ≤ γ ′′ , the triple ( α, β, γ ) satis(cid:28)es all the Horn and reverse Horn inequalities, and therefore there exist positive ompa t operators A, B, C su h that Λ + ( A ) = α, Λ + ( B ) = β , Λ + ( C ) = γ , and A = B + C . A parti ular ase is the sequen e γ = (1 , , , . . . , n , . . . ) . Sin e α + α = β + γ , it follows that A, B, C have a ommon redu ing spa e M of dimension 2, su h that Λ( A | M ) = ( α , α ) , Λ( B | M ) = ( β , , and Λ( C | M ) =( γ , . Thus the operators B and C must ne essarily have the eigenvalue zero.Restri ting the operators A, B and C to M ⊥ , we see that the sequen es e α = (1 / ( n +2)) ∞ n =1 , e β = e γ = (1 / n + 1)) ∞ n =1 must satisfy all the Horn and extended reverseHorn inequalities. We will return to these sequen es a little later.It is an amusing exer ise to show that, in the ase of (cid:28)nite tra es, the tra eidentity does follow from the dire t and reverse Horn inequalities.Let us note that the triple ( α = 2 β, β, γ = (1 + ε ) β ) , where β = (1 / n ) ∞ n =1 does not satisfy all the reverse Horn inequalities if ε > . Indeed, among theseinequalities is N X i =1 α i ≥ N X j =1 β j + N X k =1 γ k . The three sums are asymptoti to log N , (1 /
2) log N , and ((1 + ε ) /
2) log N . This an also be seen dire tly from the additivity of the Dixmier tra e applied to the ompa t operators that would have these sequen es as eigenvalues.As pointed out in the introdu tion, a di(cid:27)erent hara terization of the triples (Λ + ( B + C ) , Λ + ( B ) , Λ + ( C )) was given in [2℄ for positive ompa t operators B, C .This involves a ontinuous version of the Littlewood-Ri hardson rule. It was pointedout to us by Cristian Lenart [5℄ that it might be possible to reformulate this rule interms of honey ombs or hives [12, 13, 3℄, and we take the opportunity to do this.For our purposes, a hive will simply be a on ave fun tion f : [0 , ∞ ) → R , whoserestri tion to the triangles T ij = o { ( i − , j ) , ( i − , j − , ( i, j − } , S ij = o { ( i − , j ) , ( i, j ) , ( i, j − } is a(cid:30)ne for all i, j ∈ N ; here we use o ( S ) to denote the onvex hull of S . Su h afun tion is determined, up to an additive onstant, by the points ( x ij , y ij , z ij ) ∈ R HE HORN CONJECTURE FOR SUMS OF COMPACT SELFADJOINT OPERATORS 20de(cid:28)ned by x ij = f ( i, j − − f ( i − , j − , y ij = f ( i − , j − − f ( i − , j ) , z ij = f ( i − , j ) − f ( i, j − for i, j ∈ N . These points form the verti es of an in(cid:28)nite honey omb in the plane x + y + z = 0 ; see [12, 13℄ for a dis ussion of (cid:28)nite honey ombs. De reasingsequen es α, β, and γ onverging to zero and su h that α ≥ β satisfy the ontinuousLittlewood-Ri hardson rule if and only if there exists a hive f su h that x i = α i , y j = − β j , lim j →∞ z ij = − γ i for i, j ∈ N . Generally it is rather di(cid:30) ult to onstru t a hive f realizing this rule. We will onstru t a hive for the parti ular ase of the sequen es e α, e β and e γ onsideredabove. Note that the hive will be determined up to an additive onstant from thevalues of x i , y j and z ij . The reader will verify without di(cid:30) ulty that the followingvalues do yield a hive: z ij = 12 (cid:20) i + j + 1 − i + 1 (cid:21) , i, j ∈ N . Referen es[1℄ P. Belkale, Lo al systems on P − S for S a (cid:28)nite set, Compositio Math. 129(2001), no. 1,67(cid:21)86.[2℄ H. Ber ovi i, W. S. Li, and T. Smotzer, Continuous versions of the Littlewood-Ri hardsonrule, selfadjoint operators, and invariant subspa es, J. Operator Theory 54(2005), no. 1,69(cid:21)92.[3℄ A. S. Bu h,The saturation onje ture (after A. Knutson and T. Tao), With an appendix byWilliam Fulton, Enseign. Math. (2) 46(2000), 43(cid:21)60.[4℄ A. S. Bu h, Eigenvalues of Hermitian matri es with positive sum of bounded rank, LinearAlgebra Appl. 418(2006), no. 2-3, 480(cid:21)488.[5℄ C. Lenart, Personal ommuni ation.[6℄ W. Fulton, Eigenvalues, invariant fa tors, highest weights, and S hubert al ulus, Bull.Amer. Math. So . (N.S.) 37(2000), no. 3, 209(cid:21)249 .[7℄ (cid:22)(cid:22)(cid:22), Eigenvalues of majorized Hermitian matri es and Littlewood-Ri hardson oe(cid:30) ients,Linear Algebra Appl. 319(2000), no. 1-3, 23(cid:21)36.[8℄ S. Friedland, Finite and in(cid:28)nite dimensional generalizations of Klya hko's theorem, LinearAlgebra Appl. 319(2000), no. 1-3, 3(cid:21)22.[9℄ A. Horn, Eigenvalues of sums of Hermitian matri es, Pa i(cid:28) J. Math. 12(1962), 225(cid:21)241.[10℄ S. Johnson, The S hubert al ulus and eigenvalue inequalities for sums of Hermitian matri- es, Ph.D. thesis, University of California. Santa Barbara, 1979.[11℄ A. Klya hko, Stable bundles, representation theory and Hermitian operators, Sele t Math.(N.S.) 4(1998), 419(cid:21)445.[12℄ A. Knutson and T. Tao, The honey omb model of GL n ( C ) tensor produ ts. I. Proof of thesaturation onje ture, J. Amer. Math. So . 12(1999), 1055(cid:21)1090.[13℄ A. Knutson, T. Tao, and C. Woodward, The honey omb model of GL n ( C ))