Operator maps of Jensen-type
aa r X i v : . [ m a t h . F A ] O c t OPERATOR MAPS OF JENSEN-TYPE
FRANK HANSEN , MOHAMMAD SAL MOSLEHIAN and HAMED NAJAFI Abstract.
Let B J ( H ) denote the set of self-adjoint operators act-ing on a Hilbert space H with spectra contained in an open interval J . A map Φ : B J ( H ) → B ( H ) sa is said to be of Jensen-type ifΦ( C ∗ AC + D ∗ BD ) ≤ C ∗ Φ( A ) C + D ∗ Φ( B ) D for all A, B ∈ B J ( H ) and bounded linear operators C, D acting on H with C ∗ C + D ∗ D = I , where I denotes the identity operator.We show that a Jensen-type map on a infinite dimensional Hilbertspace is of the form Φ( A ) = f ( A ) for some operator convex function f defined in J. Introduction
We recall that a function f : J → R defined in a real interval J issaid to be n -convex if the inequality(1) f ( λA + (1 − λ ) B ) ≤ λf ( A ) + (1 − λ ) f ( B )holds for all λ ∈ [0 ,
1] and operators
A, B ∈ B J ( H ), when dim H = n. More generally, we call f operator convex if the inequality (1) holds forall natural numbers n . It is known that the inequality in this case alsoholds for operators on an infinite dimensional Hilbert space. Hansenand Pedersen [4, 5] obtained the following characterization of operatorconvexity. Theorem 1.1.
Let f : J → R be a continuous function defined in aninterval J, and let H be an infinite dimensional Hilbert space. Thefollowing conditions are equivalent:(i) f is operator convex.(ii) For each natural number k the inequality f k X i =1 C ∗ i A i C i ! ≤ k X i =1 C ∗ i f ( A i ) C i Mathematics Subject Classification.
Primary 47A63; Secondary 47B10,47A30.
Key words and phrases.
Jensen’s operator inequality; convex operator function. holds for all A , . . . , A k ∈ B J ( H ) and arbitrary operators C , . . . , C k on H with C ∗ C + · · · + C ∗ k C k = I. (iii) For each natural number k the inequality f k X i =1 P i A i P i ! ≤ k X i =1 P i f ( A i ) P i holds for all A , . . . , A k ∈ B J ( H ) and projections P , . . . , P k on H withsum P + · · · + P k = I. One should note that the above result only holds on a Hilbert spaceof infinite dimensions. If H is of finite dimension n, then f should be2 n -convex for statement ( ii ) to hold with k = 2.There are other equivalent conditions of operator convexity, cf. [3, 7].We want to study operator maps Φ given on the form Φ( A ) = f ( A )in a more abstract setting, where f ( A ) is defined by the functionalcalculus, and determine which general properties of Φ that entail thisparticular form. A related problem is to place maps of the said form,Φ( A ) = f ( A ) , in the context of other more general types of operatormaps. To this end we first introduce the notion of a Jensen-type map. Definition 1.1.
Let J be an open real interval, and let H be a Hilbertspace. A (not necessarily linear) map Φ : B J ( H ) → B ( H ) sa is said tobe of Jensen-type if (2) Φ( C ∗ AC + D ∗ BD ) ≤ C ∗ Φ( A ) C + D ∗ Φ( B ) D for all A, B ∈ B J ( H ) and operators C, D on H with C ∗ C + D ∗ D = I . Note that a Jensen-type map is convex. It is also unitarily invariant.Indeed, by choosing C as a unitary U and setting D = 0, we obtainthe inequalityΦ( U ∗ AU ) ≤ U ∗ Φ( A ) U = U ∗ Φ( U U ∗ AU U ∗ ) U ≤ Φ( U ∗ AU ) , implying that Φ( U ∗ AU ) = Φ( U ). We later realize that there existunitarily invariant convex operator maps that are not of Jensen-type.Robertson and Smith [8] showed that if E is an operator system (i.e.a closed ∗ -subspace of a unital C ∗ -algebra containing the identity), B is a C ∗ -algebra and a linear map Ψ : E ⊗ M n → B ⊗ M n satisfiesΨ( U ∗ XU ) = U ∗ Ψ( X ) U for all X ∈ E ⊗ M n and all unitaries U ∈ M n ,then there exist φ, λ : E → B such thatΨ( X ) = ( φ ⊗ id n )( X ) + λ (Tr X ) ⊗ I n for all X ∈ E ⊗ M n , where I n is the identity in the C ∗ -algebra M n of allcomplex n × n matrices. In addition, Bhat [1] proved that any bounded PERATOR MAPS OF JENSEN-TYPE 3 unitarily invariant linear map α : B ( H ) → B ( H ) is of the form α ( X ) = cX + d Tr X · I for some c, d ∈ C if H is finite dimensional, and of the form α ( X ) = cX for some c ∈ C if H is infinite dimensional.2. Unitarily invariant convex operator maps
Let Φ : B J ( H ) → B ( H ) be a unitarily invariant (not necessarilylinear) map. Lemma 2.1.
If operators X ∈ B J ( H ) and Y ∈ B ( H ) commute, thenso do Φ( X ) and Y. Proof.
For any unitary operator U commuting with X we haveΦ( X ) = Φ( U ∗ XU ) = U ∗ Φ( X ) U. Therefore, U Φ( X ) = Φ( X ) U and Φ( X ) is thus contained in the abeliandouble commutant { X } ′′ . Since Y ∈ { X } ′ it follows that Φ( X ) and Y commute. (cid:3) In particular, if X = t · I is a multiple of the identity operator forsome t ∈ J, then we deduce that Φ( t · I ) commutes with every operatorin B ( H ) . It is therefore of the form(3) Φ( t · I ) = f ( t ) · I, t ∈ J for some function f : J → R . We realize that f is convex if Φ is convex. Lemma 2.2.
Let P , . . . , P k be projections with P + · · · + P k = I andput U = θP + θ P + · · · + θ k − P k − + P k , where θ = exp(2 πi/k ) is a k th root of unity. Then U is unitary and k X j =1 P j XP j = 1 k k X j =1 U − j XU j for any X ∈ B ( H ) . Proof.
Take r, s = 1 , . . . , k.
By computation we obtain P r k X j =1 U − j XU j ! P s = k X j =1 θ − jr P r Xθ js P s = P r XP s k X j =1 θ j ( s − r ) . F. HANSEN, H. NAJAFI, M.S. MOSLEHIAN
The sum k X j =1 θ j ( s − r ) = k X j =1 exp (cid:16) j ( s − r ) 2 πik (cid:17) = k for s = r. For s = r we set ω = exp (cid:0) ( s − r )2 πi/k (cid:1) and obtain k X j =1 θ j ( s − r ) = k X j =1 ω j = ω k +1 − ωω − , since ω = 1 and ω k = 1 . The assertion now follows. (cid:3)
The following result is well-known for spectral functions, but it holdsunder the weaker conditions of only unitary invariance and convexity.
Proposition 2.1.
Let H be a Hilbert space and Φ : B J ( H ) → B ( H ) sa a unitarily invariant convex map. Then Φ k X j =1 P j XP j ! ≤ k X j =1 P j Φ( X ) P j for positive integers k , operators X ∈ B ( H ) , and projections P , . . . , P k on H with P + · · · + P k = I. Proof.
By repeated application of Lemma 2.2 we obtainΦ k X j =1 P j XP j ! = Φ k k X j =1 U − j XU j ! ≤ k k X j =1 Φ( U − j XU j )= 1 k k X j =1 U − j Φ( X ) U j = k X j =1 P j Φ( X ) P j , where we used the unitary invariance and convexity of Φ . (cid:3) Proposition 2.2.
Let H be a Hilbert space, and let Φ : B J ( H ) → B ( H ) sa be a unitarily invariant map. Then Φ k X j =1 P j XP j ! = k X j =1 P j Φ k X i =1 P i XP i ! P j for X ∈ B J ( H ) sa and projections P , . . . , P k with P + · · · + P k = I. Proof.
The projections P , . . . , P k are necessarily mutually orthogonaland the sum ˜ X = k X i =1 P i XP i commutes with P i for i = 1 , . . . , k. It then follows by Lemma 2.1 thatalso Y = Φ( ˜ X ) commutes with every P i and the assertion follows. (cid:3) PERATOR MAPS OF JENSEN-TYPE 5 The structure of Jensen-type maps
Take an open real interval J, and let Φ : B J ( H ) → B ( H ) sa be aJensen-type map. Since Φ is unitarily invariant we learned in (3) thatΦ( t · I ) = f ( t ) · I, t ∈ J, for a function f : J → R . The convexity of Φimplies that f is convex and thus continuous since J is open. Lemma 3.1.
Let
Φ : B J ( H ) → B ( H ) sa be a Jensen-type map. Thenthe following statements are true.(i) Let P be a projection on H . The equality P Φ (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) P = f ( t ) P, t ∈ J holds for any Y ∈ B J ( H ) . (ii) If λ is an eigenvalue of an operator X ∈ B J ( H ) with correspondingeigenprojection P , then P Φ( X ) P = P Φ (cid:0) λP + ( I − P ) X ( I − P ) (cid:1) P = f ( λ ) P. Proof.
Since Φ is of Jensen-type we obtainΦ (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) ≤ P Φ( t ) P + ( I − P )Φ( Y )( I − P )= f ( t ) P + ( I − P )Φ( Y )( I − P ) . Furthermore, f ( t ) = Φ (cid:0) P (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) P + ( I − P ) t ( I − P ) (cid:1) ≤ P Φ (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) P + ( I − P )Φ( t )( I − P ) ≤ P (cid:0) f ( t ) P + ( I − P )Φ( Y )( I − P ) (cid:1) P + f ( t )( I − P )= f ( t ) P + f ( t )( I − P ) = f ( t ) . We therefore have the equality f ( t ) = P Φ (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) P + f ( t )( I − P )and thus P Φ (cid:0) tP + ( I − P ) Y ( I − P ) (cid:1) P = f ( t ) P independent of Y, which proves ( i ) . Statement ( ii ) follows from thespectral theorem and ( i ) . (cid:3) Theorem 3.1.
Let J be an open real interval, and let H be a Hilbertspace of finite dimension n. If Φ : B J ( H ) → B ( H ) sa is of Jensen-type,then Φ( A ) = f ( A ) A ∈ B J ( H ) , where f is the function defined in (3). Furthermore, f is n -convex. F. HANSEN, H. NAJAFI, M.S. MOSLEHIAN
Proof.
Let P , . . . , P k be the spectral projections of X. By the spectraltheorem and Lemma 3.1 ( ii ) we obtainΦ( X ) = k X i =1 P i Φ( X ) = k X i =1 P i Φ( X ) P i = k X i =1 f ( λ i ) P i = f ( X ) , where the second equality follows from Lemma 2.1. Since Φ is convex,it follows that f is an n -convex function. (cid:3) Note that to obtain Theorem 3.1 we only used that Φ is unitarilyinvariant together with the inequality in (2) for projections C = P and D = I − P. However, to conclude that a map of the form Φ( X ) = f ( X )is of Jensen-type, we need that f is 2 n -convex, where n is the dimensionof the underlying Hilbert space.Note also that even if when the underlying Hilbert space is infinitedimensional the proof of the preceding theorem implies that Φ( A ) = f ( A ) for any finite rank operator A ∈ B J ( H ) . Lemma 3.2.
Let
A, Y be self-adjoint operators on a Hilbert space with α < A ≤ Y for some constant α. Then there exist operators C and D such that A = C ∗ Y C + αD ∗ D and C ∗ C + D ∗ D = I. Proof.
Since A − α > C = ( Y − α ) − / ( A − α ) / andobtain A − α = C ∗ ( Y − α ) C = C ∗ Y C − αC ∗ C. Since C ∗ C ≤ I we may put D = ( I − C ∗ C ) / and obtain A = C ∗ Y C + αD ∗ D and C ∗ C + D ∗ D = I. (cid:3) Theorem 3.2.
Let H be an infinite dimensional Hilbert space, and let Φ : B J ( H ) → B ( H ) sa be a Jensen-type map. Then Φ( A ) = f ( A ) A ∈ B J ( H ) , where f is the function defined in (3). In addition, f is operator convex.Proof. Take A ∈ B J ( H ) and a constant α ∈ J with α < A. We maydetermine an upper sum operator Y n = f n ( A ) with spectrum in J bychoosing f n as an increasing step function defined on the convex hull of PERATOR MAPS OF JENSEN-TYPE 7 the spectrum of A corresponding to a subdivision with fineness ε > f n ( t ) = t in the right hand side of each subinterval. Then α < A ≤ Y n and k Y n − A k ≤ ε such that Y n converges to A in the norm topol-ogy as the fineness of the subdivision tends to zero. Furthermore, byLemma 3.2 we obtain A = C ∗ n Y n C n + αD ∗ n D n for operators C n and D n with C ∗ n C n + D ∗ n D n = I, and thusΦ( A ) ≤ C ∗ n Φ( Y n ) C n + D ∗ n Φ( α ) D n = C ∗ n f ( Y n ) C n + f ( α ) D ∗ n D n , where we first used that Φ is of Jensen-type, and then that Y n is a finiterank operator such that Φ( Y n ) = f ( Y n ) . Notice that C n by the spectraltheorem converges to the identity operator in the norm topology, whenthe fineness of the subdivision tends to zero. In the limit we thus obtainΦ( A ) ≤ f ( A ) . We next choose a constant β ∈ J such that β < Z n ≤ A, where in this case Z n = g n ( A ) is an under sum operator of A withspectrum in J corresponding to a subdivision of J. We now obtain Z n = C ∗ n AC n + βD ∗ n D n for operators C n and D n such that C ∗ n C n + D ∗ n D n = 1 , and C n convergesto the identity operator in the norm topology for n tending to infinity.By using that Y n is finite rank and that Φ is of Jensen-type we obtainthe inequality f ( Z n ) = Φ( Z n ) ≤ C ∗ n Φ( A ) C n + Φ( β ) D ∗ n D n and thus in the limit f ( A ) ≤ Φ( A ) . (cid:3) Note that a Jensen-type map automatically is strongly continuousby the preceding theorem.
Remark 3.1. If H is infinite dimensional and Φ : B J ( H ) → B ( H ) sa is unitarily invariant, we learned that the inequality Φ( P AP + ( I − P ) B ( I − P )) ≤ P Φ( A ) P + ( I − P )Φ( B )( I − P ) for all A, B ∈ B J ( H ) and projections P on H is sufficient to concludethat Φ is of the form Φ( A ) = f ( A ) for some operator convex function f, and it is therefore, by Theorem 1.1, of Jensen-type.If Φ is just unitarily invariant and convex, then the more restrictedinequality Φ( P AP + ( I − P ) A ( I − P )) ≤ P Φ( A ) P + ( I − P )Φ( A )( I − P ) F. HANSEN, H. NAJAFI, M.S. MOSLEHIAN holds for A ∈ B J ( H ) and projections P on H , cf. Proposition 2.1.The difference between these two inequalities (the latter being morerestricted than the former) elucidates the difference between the generalclass of unitarily invariant convex maps and the more restricted subsetof Jensen-type maps.The map Φ( X ) = Tr X · I is unitarily invariant and convex, butit is not of Jensen-type. To realize this, take A = P and B = 0 inDefinition 1.1. References
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