aa r X i v : . [ m a t h . F A ] D ec EPIGRAPH OF OPERATOR FUNCTIONS
MOHSEN KIAN
Abstract.
It is known that a real function f is convex if and only if the setE( f ) = { ( x, y ) ∈ R × R ; f ( x ) ≤ y } , the epigraph of f is a convex set in R . We state an extension of this result for operatorconvex functions and C ∗ -convex sets as well as operator log-convex functions and C ∗ -log-convex sets. Moreover, the C ∗ -convex hull of a Hermitian matrix has been representedin terms of its eigenvalues. Introduction and preliminaries
Throughout the paper, assume that B ( H ) is the C ∗ -algebra of all bounded linear operatorson a Hilbert space H . If dim ( H ) = n , then we identify B ( H ) with M n , the algebra of all n × n matrices. We denote by H n the set of all Hermitian matrices in M n . An operator X ∈ B ( H ) is called positive (denoted by X ≥
0) if h Xa, a i ≥ a ∈ H . If inaddition X is invertible, then it is called strictly positive(denoted by X > B ( H ) + the set of all strictly positive operators on H .For a real interval J , we mean by sp ( J ) the set of all self-adjoint operators on H whosespectra are contained in J . A continuous function f : J → R is said to be operator convexif f (cid:0) X + Y (cid:1) ≤ f ( X )+ f ( Y )2 for all X, Y ∈ sp ( J ). It is well known that [6, 7] f is operatorconvex if and only if the Jensen operator inequality f ( C ∗ XC ) ≤ C ∗ f ( X ) C (1)holds for every isometry C and every X ∈ sp ( J ). A continuous function f : (0 , ∞ ) → (0 , ∞ ) is called operator log-convex [1, 8] if f (cid:0) X + Y (cid:1) ≤ f ( X ) ♯f ( Y ) for all strictly positiveoperators X, Y , where the geometric mean ♯ is defined by X♯Y = X (cid:16) X − Y X − (cid:17) X for all strictly positive operators X and Y , see for example [6]. In the case where f isoperator log-convex a sharper inequality than (1) is valid as f n X i =1 C ∗ i X i C i ! ≤ n X i =1 C ∗ i f ( X i ) − C i ! − (2)for all X , · · · , X n > C , · · · , C n ∈ B ( H ) with P ni =1 C ∗ i C i = I , see [8, Corollary3.13]. Mathematics Subject Classification.
Primary 46L89 ; Secondary 52A01, 46L08.
Key words and phrases.
Epigraph, convex hull, C ∗ -convexity, operator convex, operator log-convex. A set
K ⊆ B ( H ) is called C ∗ -convex if X , · · · , X n ∈ K and C , · · · , C n ∈ B ( H ) with P ni =1 C ∗ i C i = I implies that P ni =1 C ∗ i X i C i ∈ K . This kind of convexity has been intro-duced by Loebl and Paulsen [9] as a non-commutative generalization of linear convexityand has been studied by many authors, see e.g. [5, 11, 12] and references therein. Typicalexamples of C ∗ -convex sets are { T ∈ B ( H ) : 0 ≤ T ≤ I } and { T ∈ B ( H ); k T k ≤ M } for a fix scalar M >
0. It is evident that the C ∗ -convexity of a set K in B ( H ) implies itsconvexity in the usual sense. For if X, Y ∈ K and λ ∈ [0 , C = √ λI and C = √ − λI we have C ∗ C + C ∗ C = I and λX + (1 − λ ) Y = C ∗ XC + C ∗ Y C ∈ K . But the converse is not true in general. For example if A ≥
0, then[0 , A ] = { X ∈ B ( H ); 0 ≤ X ≤ A } is convex but not C ∗ -convex [9]. The concept of C ∗ -convexity can be generalized to thesets which have a B ( H )-module structure. Assume that M is a B ( H )-module. We saythat a subset K of M is C ∗ -convex whenever X , · · · , X n ∈ K , C , · · · , C n ∈ B ( H ) and P ni =1 C ∗ i C i = I implies that P ni =1 C ∗ i X i C i ∈ K . For example, let M = { ( X , · · · , X k ); X j ∈ B ( H ) , j = 1 , · · · , k } . Then M is a B ( H )-module under M × B ( H ) → M (( X , · · · , X k ) , T ) ( X T, · · · , X k T ) B ( H ) × M → M ( S, ( X , · · · , X k )) ( SX , · · · , SX k ) . Now,
K ⊆ M is called C ∗ -convex if X i = ( X i , · · · , X ik ) ∈ K , C i ∈ B ( H ) ( i = 1 , · · · , n )and P ni =1 C ∗ i C i = I implies that n X i =1 C ∗ i X i C i = n X i =1 C ∗ i X i C i , · · · , n X i =1 C ∗ i X ik C i ! ∈ K . As an example, it is easy to see that K = { ( X , · · · , X k ) ∈ M ; 0 ≤ X j ≤ I, j = 1 , · · · , k } is C ∗ -convex.The epigraph of a real function f is defined to be the setE( f ) = { ( x, y ) ∈ R × R ; f ( x ) ≤ y } . It is known that f is a convex function if and only if E( f ) is a convex set in R . The mainpurpose of this paper is to present this result for operator functions. In particular, we givethe connection between operator convex functions and C ∗ -convex sets as well as operatorlog-convex functions and C ∗ -log-convex sets. It is also shown that the C ∗ -convex hull ofa Hermitian matrix can be represented in terms of its eigenvalues. PIGRAPH OF OPERATOR FUNCTIONS 3 Main Result
For a continuous real function f : J → R , we define the operator epigraph of f byOE( f ) := { ( X, Y ) ∈ sp ( J ) × B ( H ); f ( X ) ≤ Y } . The next result gives the connections between operator convex functions and C ∗ -convexsets. Theorem 2.1.
A continuous function f : J → R is operator convex if and only if OE( f ) is C ∗ -convex.Proof. Let f be operator convex. Let ( X i , Y i ) ∈ OE( f ) ( i = 1 , · · · , n ) and C i ∈ B ( H )with P ni =1 C ∗ i C i = I . Therefore f ( X i ) ≤ Y i ( i = 1 , · · · , n ) and so we have by the Jensenoperator inequality that f n X i =1 C ∗ i X i C i ! ≤ n X i =1 C ∗ i f ( X i ) C i ≤ n X i =1 C ∗ i Y i C i . In other words, P ni =1 C ∗ i ( X i , Y i ) C i ∈ OE( f ) and so OE( f ) is C ∗ -convex.Conversely, assume that OE( f ) is C ∗ -convex. For all X , · · · , X n ∈ B ( H ), we have fromthe definition of OE( f ) that ( X i , f ( X i )) ∈ OE( f ). If C i ∈ B ( H ) with P ni =1 C ∗ i C i = I ,then P ni =1 C ∗ i ( X i , f ( X i )) C i ∈ OE( f ) by the C ∗ -convexity of OE( f ). It follows that f n X i =1 C ∗ i X i C i ! ≤ n X i =1 C ∗ i f ( X i ) C i , which means that f is operator convex. (cid:3) Let { f α ; α ∈ Γ } be a family of operator convex functions and M α ∈ R for every α ∈ Γ.By Theorem 2.1, the set { X ∈ B ( H ); f α ( X ) ≤ M α , ∀ α ∈ Γ } is C ∗ -convex. For example,consider the family f α where f α ( t ) = t α and α ∈ [1 , { X ∈ B ( H ); X α ≤ M α , ≤ α ≤ } is C ∗ -convex.The Choi–Davis–Jensen inequality for an operator convex function f : J → R assertsthat f (Φ( X )) ≤ Φ( f ( X )) for every unital positive linear mapping Φ on B ( H ) and every X ∈ sp ( J ), see [2, 3, 6]. Motivating by this result, we state a characterization for C ∗ -convex sets in H n using positive linear mappings. Theorem 2.2. If K ⊆ H n , then the followings are equivalent: (1) K is C ∗ -convex; (2) P mi =1 Φ i ( X i ) ∈ K for every X i ∈ K , ( i = 1 , · · · , m ) and every unital family { Φ i ; i = 1 , · · · , m } of positive linear mappings on M n .Proof. Assume that
K ⊆ H n is C ∗ -convex. First note that if X ∈ K and if λ is aneigenvalue of X , then λI ∈ K . Indeed, it follows from the spectral decomposition thatthere exists a unitary U such that U ∗ XU is a matrix such that λ is its k, k entry. Then λI = P ni =1 E ∗ ki U ∗ XU E ki ∈ K , where { E ij } is the system of unit matrices. Now let X i ∈ K , M. KIAN ( i = 1 , · · · , m ) and let { Φ i ; i = 1 , · · · , m } be a unital family of positive linear mappings on M n . Assume that X i = P nj =1 λ ij P ij be the spectral decomposition of X i for i = 1 , · · · , m so that λ ij I ∈ K for all i, j . Therefore m X i =1 Φ i ( X i ) = m X i =1 Φ i n X j =1 λ ij P ij = m X i =1 n X j =1 λ ij Φ i ( P ij ) = m X i =1 n X j =1 C ∗ ij λ ij C ij , where C ij = p Φ i ( P ij ). Taking into account that m X i =1 n X j =1 C ∗ ij C ij = m X i =1 n X j =1 Φ i ( P ij ) = m X i =1 Φ i n X j =1 P ij = n X i =1 Φ i ( I ) = I, we get by the C ∗ -convexity of K that P mi =1 Φ i ( X i ) ∈ K .Conversely, let X , · · · , X m ∈ K and C i ∈ M n with P mi =1 C ∗ i C i = I . Define positivelinear mappings Φ i on M n by Φ i ( X ) = C ∗ i XC i , ( i = 1 , · · · , m ). Then { Φ i ; i = 1 , · · · , m } is a unital family and so m X i =1 C ∗ i X i C i = m X i =1 Φ i ( X i ) ∈ K , i.e., K is C ∗ -convex. (cid:3) The famous Choi–Davis–Jensen inequality which is a characterization of operator convexfunctions, can be derived from Theorem 2.1 and Theorem 2.2.
Corollary 2.3.
A continuous function f : J → R is operator convex if and only if f m X i =1 Φ i ( X i ) ! ≤ m X i =1 Φ i ( f ( X i )) (3) for every X i ∈ sp ( J ) and every unital family { Φ i } mi =1 of positive linear mappings on M n .Proof. Let f : J → R be operator convex. Then OE( f ) is C ∗ -convex by Theorem 2.1.For every X i ∈ sp ( J ), we have ( X i , f ( X i )) ∈ OE( f ). Theorem 2.2 then implies that P mi =1 Φ i ( X i , f ( X i )) ∈ OE( f ) for every unital family { Φ i } mi =1 of positive linear mappings.Hence (3) holds true. Conversely, assume that (3) is valid. Let ( X i , Y i ) ∈ OE( f ), ( i =1 , · · · , m ) so that f ( X i ) ≤ Y i . If { Φ i } mi =1 is a unital family of positive linear mappings,then f m X i =1 Φ i ( X i ) ! ≤ m X i =1 Φ i ( f ( X i )) (by (3)) ≤ m X i =1 Φ i ( Y i ) (by f ( X i ) ≤ Y i ) . This concludes that P mi =1 Φ i ( X i , Y i ) ∈ OE( f ). Theorem 2.2 now implies that OE( f ) is C ∗ -convex and so f is operator convex by Theorem 2.1. (cid:3) PIGRAPH OF OPERATOR FUNCTIONS 5
Remark . Let
K ⊆ B ( H ) + be C ∗ -convex and 0 ∈ K . If C ∈ B ( H ) is a contraction, then C ∗ XC ∈ K for every X ∈ K . To see this, put D = ( I − C ∗ C ) . Then C ∗ C + D ∗ D = I andso C ∗ XC = C ∗ XC + D ∗ D ∈ K for every X ∈ K . It follows that if P ni =1 X i ∈ K , then X i ∈ K for i = 1 , · · · , n . For if X, Y ∈ K , put C = ( X + Y ) − X and C = ( X + Y ) − Y which are contractions and X = C ∗ ( X + Y ) C Y = C ∗ ( X + Y ) C . Definition 2.5.
We say that a set
K ⊆ B ( H ) is C ∗ -log-convex if (cid:0)P ni =1 C ∗ i X − i C i (cid:1) − ∈ K for all X i ∈ K and C i ∈ B ( H ) with P ni =1 C ∗ i C i = I .If M is a positive scalar, then { X ∈ B ( H ); 0 < X ≤ M } is an obvious example for C ∗ -log-convex sets. Moreover, if K ⊆ B ( H ) is C ∗ -log-convex, then K − = { X − ; X ∈ K} is convex in the usual sense. For if X, Y ∈ K − and λ ∈ [0 , C = √ λI and C = √ − λI we have C ∗ C + C ∗ C = I and so ( C ∗ XC + C ∗ Y C ) − ∈ K . This followsthat λX + (1 − λ ) Y = C ∗ XC + C ∗ Y C ∈ K − . More generally, a set K ⊆ B ( H ) is a C ∗ -log-convex set if and only if K ⊆
Inv( B ( H )) and K − is a C ∗ -convex set, where wemean by Inv( B ( H )) the set of invertible elements in B ( H ). Proposition 2.6. If L and K are C ∗ -log-convex sets, then so is (cid:0) K − + L − (cid:1) − = n(cid:0) X − + Y − (cid:1) − ; X ∈ K , Y ∈ L o . Proof.
Assume that C i ∈ B ( H ) with P ni =1 C ∗ i C i = I . If Z , · · · , Z n ∈ (cid:0) K − + L − (cid:1) − ,then Z i = (cid:0) X − i + Y − i (cid:1) − for some X i ∈ K and Y i ∈ L , ( i = 1 , · · · , n ). It follows from the C ∗ -log-convexity of K and L that (cid:0)P ni =1 C ∗ i X − i C i (cid:1) − ∈ K and (cid:0)P ni =1 C ∗ i Y − i C i (cid:1) − ∈ L .Therefore, n X i =1 C ∗ i Z − i C i ! − = n X i =1 C ∗ i (cid:0) X − i + Y − i (cid:1) C i ! − = n X i =1 C ∗ i X − i C i + n X i =1 C ∗ i Y − i C i ! − ∈ (cid:0) K − + L − (cid:1) − , which implies that (cid:0) K − + L − (cid:1) − is C ∗ -log-convex. (cid:3) Proposition 2.7.
Let
K ⊆ B ( H ) be inverse closed in the sense that K − ⊆ K . If K is C ∗ -log-convex, then it is C ∗ -convex.Proof. Assume that K is a C ∗ -log-convex set with K − ⊆ K . Let C i ∈ B ( H ) and P ni =1 C ∗ i C i = I . If X i ∈ K , then X − i ∈ K and we have from the C ∗ -log-convexity of K that ( P ni =1 C ∗ i X i C i ) − ∈ K . It follows that P ni =1 C ∗ i X i C i ∈ K and so K is C ∗ -convex. (cid:3) The convex hull of a set K in a vector space X is defined to be the smallest convex setin X containing K . It is known that the convex hull of X is the setCH( K ) = ( m X i =1 t i a i ; a i ∈ K , m ∈ N , m X i =1 t i = 1 ) . (4) M. KIAN
The C ∗ -convex hull [9] of a set K ⊆ B ( H ) is the smallest C ∗ -convex set in B ( H ) whichcontains K . This is the generalization of convex hull in the non-commutative setting. Itis known [9, Corollary 20] that given T ∈ B ( H ), the C ∗ -convex hull of { T } is the set C ∗ − CH( T ) = (X i C ∗ i T C i ; X i C ∗ i C i = I ) . Moreover, we define the C ∗ -log-convex hull of a set K ⊆ B ( H ) to be the smallest C ∗ -log-convex set in B ( H ) which contains K . It is easy to see that if T ∈ B ( H ) and T >
0, thenthe C ∗ -log-convex hull of { T } turns out to be C ∗ − LCH( T ) = X i C ∗ i T − C i ! − ; X i C ∗ i C i = I . When T ∈ H n , we can present the C ∗ -convex hull of { T } in terms of its eigenvalues. Theorem 2.8. If λ , · · · , λ n are eigenvalues of T ∈ H n , then C ∗ − CH( T ) = ( n X i =1 λ i E i ; E i ≥ , i = 1 , · · · , n, n X i =1 E i = I ) . Proof.
Let T = P ni =1 λ i P i be the spectral decomposition of T . PutΩ = ( n X i =1 λ i E i ; E i ≥ , i = 1 , . . . , n, n X i =1 E i = I ) . If X ∈ C ∗ − CH( T ), then X = P i C ∗ i T C i for some C i ∈ M n with P i C ∗ i C i = I . Therefore, X = X i C ∗ i T C i = X i C ∗ i n X j =1 λ j P j C i = n X j =1 λ j X i C ∗ i P j C i . Putting E j = P i C ∗ i P j C i , we have E j ≥
0, ( j = 1 , . . . , n ) and n X j =1 E j = n X j =1 X i C ∗ i P j C i = X i C ∗ i n X j =1 P j C i = X i C ∗ i C i = I. Hence, X = P nj =1 λ j E j and P nj =1 E j = I , i.e., X ∈ Ω.For the converse, note that the C ∗ − CH( T ) is C ∗ -convex and contains all eigenvalues of T . Now if X = P nj =1 λ j E j in which P nj =1 E j = I and E j ≥
0, ( j = 1 , · · · , n ), then X = n X j =1 λ j E j = n X j =1 p E j λ j p E j ∈ C ∗ − CH( T ) , by C ∗ -convexity of C ∗ − CH( T ). (cid:3) Let f : J → R be a continuous function. If T ∈ H n has the spectral decomposition T = P ni =1 λ i P i in which the eigenvalues λ , · · · , λ n are contained in J , then the well PIGRAPH OF OPERATOR FUNCTIONS 7 known functional calculus yields that f ( T ) = P ni =1 f ( λ i ) P i . By use of Theorem 2.8, the C ∗ -convex hull of f ( T ) turns to be C ∗ − CH( f ( T )) = ( n X i =1 f ( λ i ) E i ; E i ≥ , i = 1 , · · · , n, n X i =1 E i = I ) . The next result reveals the reason of naming C ∗ -log-convex sets. First note that, thenotion of C ∗ -log-convexity can be extended to subsets of an algebra with a B ( H )-modulestructure. For example, a set K ⊆ B ( H ) × B ( H ) is called C ∗ -log-convex if ( X i , Y i ) ∈ K , C i ∈ B ( H ) and P ni =1 C ∗ i C i = I implies that n X i =1 C ∗ i ( X i , Y i ) − C i ! − = n X i =1 C ∗ i X − i C i ! − , n X i =1 C ∗ i Y − i C i ! − ∈ K . Theorem 2.9.
A continuous function f : (0 , ∞ ) → (0 , ∞ ) is operator log-convex if andonly if the set K = { ( X, Y ) ∈ B ( H ) + × B ( H ) + ; f ( X − ) ≤ Y } is a C ∗ -log-convex set.Proof. Let f be operator log-convex, C i ∈ B ( H ) and P ni =1 C ∗ i C i = I . If ( X i , Y i ) ∈ K , then f (cid:0) X − i (cid:1) ≤ Y i . It follows from (2) that f n X i =1 C ∗ i X − i C i ! ≤ n X i =1 C ∗ i f (cid:0) X − i (cid:1) − C i ! − ≤ n X i =1 C ∗ i Y − i C i ! − . Hence (cid:16)(cid:0)P ni =1 C ∗ i X − i C i (cid:1) − , (cid:0)P ni =1 C ∗ i Y − i C i (cid:1) − (cid:17) ∈ K and so K is C ∗ -log-convex.Conversely, assume that K is C ∗ -log-convex, C i ∈ B ( H ) and P ni =1 C ∗ i C i = I . If X i ∈B ( H ) + ( i = 1 , · · · , n ), then ( X − i , f ( X i )) ∈ K . Therefore n X i =1 C ∗ i X i C i ! − , n X i =1 C ∗ i f ( X i ) − C i ! − ∈ K by the C ∗ -log-convexity of K . It follows that f n X i =1 C ∗ i X i C i ! ≤ n X i =1 C ∗ i f ( X i ) − C i ! − and so f is operator log-convex by (2). (cid:3) References [1] T. Ando and F. Hiai,
Operator log-convex functions and operator means , Math. Ann. (2011),611-630.[2] M. D. Choi,
A Schwarz inequality for positive linear maps on C ∗ -algebras , Illinois J. Math. 18 (1974),565-574.[3] C. D`avis, A Schwarz inequality for convex operator functions , Proc. Amer. Math. Soc, 8 (1957), 42-44.[4] E. G. Effros and S. Winkler,
Matrix Convexity: Operator Analogues of the Bipolar and Hahn–BanachTheorems , J. Funct. Anal. (1997), 117–152.[5] D. R. Farenick and H. Zhou,
The structure of C ∗ -extreme points in space of completely positive linearmaps on C ∗ -algebras , Proc. Amer. Math. Soc., (1998), 1467–1477. M. KIAN [6] T. Furuta, H. Mi´ci´c, J. Peˇcari´c and Y. Seo,
Mond–Peˇcari´c Method in Operator Inequalities , Zagreb,Element, 2005.[7] F. Hansen and G.K. Pedersen,
Jensen’s operator inequality , Bull. London Math. Soc. (2003), no.4, 553–564.[8] M. Kian and S. S. Dragomir, f -divergence functional of operator log-convex functions , Linear Multi-linear Algebra, doi:10.1080/03081087.2015.1025686.[9] R. I. Loebl and V. I. Paulsen, Some remarks on C ∗ -convexity , Linear Algebra Appl. (1981), 63–78.[10] B. Magajna, C ∗ -convex sets and completely bounded bimodule homomorphisms , Proc. Roy. Soc. End-inburgh Sect. A. (2000), 375–387.[11] P. B. Morenz, The structure of C ∗ -convex sets , Canad. Math. J. Math. (1994), 1007–1026.[12] C. Webster and S. Winkler, The Krein–Milman Theorem in operator convexity , Trans. Amer. Math.Soc. (1999), 307–322.
Mohsen Kian: Department of Mathematics, Faculty of Basic Sciences, University of Bo-jnord, P. O. Box 1339, Bojnord 94531, IranSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:19395-5746, Tehran, Iran.
E-mail address ::