Cebyšev’s type inequalities for positive linear maps of selfadjoint operators in Hilbert spaces

. Some inequalities for positive linear maps of continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Applications for power function and logarithm are provided as well.


Introduction
We say that the functions f, g : [a, b] −→ R are synchronous (asynchronous) on the interval [a, b] if they satisfy the following condition: It is obvious that, if f, g are monotonic and have the same monotonicity on the interval [a, b] , then they are synchronous on [a, b] while if they have opposite monotonicity, they are asynchronous.
In the special case p = a ≥ 0, it appears that the inequality (1) has been obtained by Laplace long before Čebyšev (see for example [19, p. 240]).
The inequality (1) was mentioned by Hardy, Littlewood and Pólya in their book [17] in 1934 in the more general setting of synchronous sequences, i.e., if a, b are synchronous (asynchronous), this means that (2) (a i − a j ) (b i − b j ) ≥ (≤) 0 for any i, j ∈ {1, . . ., n} , then (1) holds true as well.
Assume that A is a positive operator on the Hilbert space H and p, q > 0. Then for each x ∈ H with x = 1 we have by (3) the inequality (4) A p+q x, x ≥ A p x, x A q x, x .
If A is positive definite then the inequality (4) also holds for p, q < 0. If A is positive definite and either p > 0, q < 0 or p < 0, q > 0, then the reverse inequality holds in (4).Assume that A is positive definite and p > 0. Then by (3) we have ( 5) for each x ∈ H with x = 1.If p < 0 then the reverse inequality holds in (5).
The following result that is related to the Čebyšev inequality also holds [14] (see also [13, p. 73] or [15, p.
If f, g are asynchronous, then for any x ∈ H with x = 1.
Let A be a selfadjoint operator with Sp (A) ⊆ [m, M ] for some real numbers m < M. If f, g : [m, M ] −→ R are continuous, synchronous and one is convex while the other is concave on [m, M ] , then by Jensen's inequality for convex (concave) functions and by (6) we have If f, g are asynchronous and either both of them are convex or both of them concave on [m, M ], then Assume that A is a positive operator on the Hilbert space H.If p ∈ (0, 1) and q ∈ (1, ∞) , then for each x ∈ H with x = 1 we have the inequality If A is positive definite and p > 1, q < 0, then Assume that A is positive definite and p > 1.Then also Let H be a complex Hilbert space and B (H) , the Banach algebra of bounded linear operators acting on H.We denote by B + (H) the convex cone of all positive operators on H and by B ++ (H) the convex cone of all positive definite operators on H.
Let H, K be complex Hilbert spaces.Following [5] (see also [30, p. 18]) we can introduce the following definition: Definition 1.1.A map Φ : B (H) → B (K) is linear if it is additive and homogeneous, namely for any λ, µ ∈ C and A, B ∈ B (H) .The linear map Φ : We observe that a positive linear map Φ preserves the order relation, In the recent paper [25] the following results of Čebyšev type have been obtained: In particular, we have the Čebyšev type inequality Motivated by the above results, we obtain in this paper some new inequalities for positive linear maps of continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators.Applications for power function and logarithm are provided as well.

Čebyšev Type Inequalities for Positive Maps
The following generalization of Theorem 1.3 may be stated: In particular, we have (13) and Proof.We consider only the case of synchronous functions.In this case we have that ( 17) Using the continuous functional calculus for the operator A we have If we apply to this inequality the positive map Φ then we get ( 18) If we take the inner product in ( 18), then we get Using the functional calculus for the operator B we get from (19) that If we apply to this inequality the positive map Ψ then we get for any x ∈ K with x = 1.
Let y ∈ K with y = 1.By taking the inner product in (21) we deduce the desired result (15).Remark 2.1.If we take in (13) B = A, then we get for any x, y ∈ K with x = y = 1 and in particular, for y = x, in (22) we get the Čebyšev type inequality ( 14) If the map Ψ : ) is normalised and by (23) we get Moreover, if in (24) we take for any v ∈ K.
We also have the following Cauchy-Schwarz' type inequalities: In particular, we have for and, for y = x, we get the following Cauchy-Schwarz inequality Assume that A is a positive operator on the Hilbert space H and p, q > 0. Then for each x ∈ H with x = 1 we have by (23) the inequality (29) Φ A p+q x, x ≥ Φ (A p ) x, x Φ (A q ) x, x .
If A is positive definite then the inequality (29) also holds for p, q < 0. If A is positive definite and either p > 0, q < 0 or p < 0, q > 0, then the reverse inequality holds in (29).Assume that A is positive definite and p > 0. Then by (22) we have for each x ∈ H with x = 1.If p < 0 then the reverse inequality holds in (30).These results generalize the corresponding inequalities from (3)-( 5).Let P j ∈ B (H) , j = 1, ..., k be contractions with Assume that A is a positive operator on the Hilbert space H and p, q > 0. Then for each x ∈ H with x = 1 we have by (32) that (33) for each x ∈ H with x = 1.If A is positive definite and either p > 0, q < 0 or p < 0, q > 0, then the reverse inequality holds in (33).In this case, by taking the supremum over x ∈ H with x = 1, we get the norm inequality (34) where A is positive definite and either p > 0, q < 0 or p < 0, q > 0.
Moreover, by the elementary arithmetic mean-geometric mean inequality, we have and by taking the supremum over x ∈ H with x = 1, we get (35) where A is positive definite and either p > 0, q < 0 or p < 0, q > 0.
Assume that A is positive definite and p > 0. Then by (32) we have (36) for each x ∈ H with x = 1.If p < 0 then the reverse inequality holds in (36).
If we assume that A ≥ 1 H and p < 0, then by (36) we have and by taking the supremum over x ∈ H with x = 1 we get (37) In general, we can state the following norm inequality: In particular, we have Proof.From the inequality (15) we have Taking the supremum in (42) over x, y ∈ K with x = y = 1, we get and We observe that, if P j ∈ B (H) , j = 1, ..., k are contractions satisfying condition (31)
In particular, Proof.From the inequality (46) we have for B = A and y = x that that is equivalent to for any x ∈ K with x = 1.
The case of asynchronous functions goes likewise and the details are omitted.
for any v ∈ K with v = 0, when f, g are synchronous on [m, M ] , and We need the following Jensen's type inequality that has been obtained recently in [16]: We can establish now some refinements of the Čebyšev type inequality (23) when some convexity properties are assumed.
73]): Theorem 1.2.Let A be a selfadjoint operator with Sp (A) ⊆ [m, M ] for some real numbers m < M. If f, g : [m, M ] −→ R are continuous and synchronous on [m, M ] , then be continuous and synchronous (asynchronous) on [m, M ] .If A and B are selfadjoint operators with spectra contained in [m, M ] and Φ ∈ P N [B (H) , B (K)] , then for any x, y ∈ K with x = y = 1 we have

Theorem 2 . 1 .
Let f, g : [m, M ] −→ R be continuous and synchronous (asynchronous) on [m, M ] .If A and B are selfadjoint operators with spectra contained in [m, M ] and Φ, Ψ ∈ P N [B (H) , B (K)] , then for any x, y ∈ K with x = y = 1 we have

Corollary 2 . 1 .
Let f : [m, M ] −→ R be continuous on [m, M ] .If A and B are selfadjoint operators with spectra contained in [m, M ] and Φ, Ψ ∈ P N [B (H) , B (K)] , then for any x, y ∈ K with x = y = 1 we have P j = 1 H .The map Φ : B (H) → B (H) defined by[30] Φ (A) := k j=1 P * j AP j is a normalized positive linear map on B (H) .If f, g : [m, M ] −→ R are continuous and synchronous (asynchronous) on [m, M ] and A is selfadjoint with Sp (A) ⊆ [m, M ] , then by (23) we have k j=1 P * j f (A) g (A) P j x, x g (A) P j x, x for each x ∈ H with x = 1.
, f, g : [m, M ] −→ R are continuous, asynchronous and nonnegative on [m, M ] , then for any A and B selfadjoint operators with spectra contained in [m, M ] and Φ ∈ P N [B (H) , B (K)] we have the norm inequality (44) k j=1 P * j Φ (f (A) g (A)) P j ≤ k j=1 P * j Φ (f (A)) P j k j=1 P * j Φ (g (A)) P j .

Theorem 3 . 1 .
Let f, g : [m, M ] −→ R be continuous and synchronous (asynchronous) on [m, M ] .If A and B are selfadjoint operators with spectra contained in [m, M ] and Φ, Ψ ∈ P N [B (H) , B (K)] , then for any x, y ∈ K with x = y = 1 we have

Lemma 3 . 1 .
Let f : I → R be a convex function on the interval I and Φ : B (H) → B (K) a normalised positive linear map.Then for any selfadjoint operator A whose spectrum Sp (A) is contained in I we have(56) f ( Φ (A) y, y ) ≤ Φ (f (A)) y, yfor any y ∈ K, y = 1.Proof.For the sake of completeness, we give here a short proof.Let m, M with m < M and such that Sp(A) ⊆ [m, M ] ⊂ I. Then m1 H ≤ A ≤ M 1 H and since Φ ∈ P N [B (H) , B (K)] we have that m1 K ≤ Φ (A) ≤ M 1 K showing that Φ (A) y, y ∈ [m, M ] for any y ∈ K, y = 1.By the gradient inequality for the convex function we have for a = Φ (A) y, y ∈ [m, M ] thatf (t) ≥ f ( Φ (A) y, y ) + (t − Φ (A) y, y ) f + ( Φ (A) y, y )for any t ∈ I, where f + is the right lateral derivative.Using the continuous functional calculus for the operator A we have for a fixed y ∈ K with y = 1 that(57) f (A) ≥ f ( Φ (A) y, y ) 1 H + f + ( Φ (A) y, y ) (A − Φ (A) y, y 1 H ) .Since Φ ∈ P N [B (H) , B (K)] ,then by taking the functional Φ in the inequality (57) we get (58)Φ (f (A)) ≥ f ( Φ (A) y, y ) 1 K + f + ( Φ (A) y, y ) (Φ (A) − Φ (A) y, y 1 K ) for any y ∈ K with y = 1.This inequality is of interest in itself.Taking the inner product in (58) we have for any y ∈ K with y = 1 that Φ (f (A)) y, y ≥ f ( Φ (A) y, y ) y 2 + f + ( Φ (A) y, y ) Φ (A) y, y − Φ (A) y, y y 2 = f ( Φ (A) y, y ) and the inequality (56) is proved.