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Operator P-class functions

  • 1,
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  • 2Email author and
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Journal of Inequalities and Applications20142014:451

https://doi.org/10.1186/1029-242X-2014-451

  • Received: 27 May 2014
  • Accepted: 23 October 2014
  • Published:

Abstract

We introduce and investigate the notion of an operator P-class function. We show that every nonnegative operator convex function is of operator P-class, but the converse is not true in general. We present some Jensen type operator inequalities involving P-class functions and some Hermite-Hadamard inequalities for operator P-class functions.

MSC:47A63, 47A60, 26D15.

Keywords

  • P-class function
  • Jensen operator inequality
  • positive linear map
  • Hermite-Hadamard inequality

1 Introduction and preliminaries

Let B ( H ) denote the C -algebra of all bounded linear operators on a complex Hilbert space with its identity denoted by I. When dim H = n , we identify B ( H ) with the matrix algebra M n of all n × n matrices with entries in the complex field . We denote by σ ( J ) the set of all self-adjoint operators on whose spectra are contained in an interval J. An operator A B ( H ) is called positive (positive semidefinite for a matrix) if A x , x 0 for all x H and in such a case we write A 0 . For self-adjoint operators A , B B ( H ) , we write B A if B A 0 . The Gelfand map f f ( A ) is an isometrical -isomorphism between the C -algebra C ( σ ( A ) ) of a complex-valued continuous functions on the spectrum σ ( A ) of a self-adjoint operator A and the C -algebra generated by I and A. If f , g C ( σ ( A ) ) , then f ( t ) g ( t ) ( t σ ( A ) ) implies that f ( A ) g ( A ) . A real-valued continuous function f on an interval J is called operator increasing (operator decreasing, resp.) if A B implies f ( A ) f ( B ) ( f ( B ) f ( A ) , resp.) for all A , B σ ( J ) . We recall that a real-valued continuous function f defined on an interval J is operator convex if f ( λ A + ( 1 λ ) B ) λ f ( A ) + ( 1 λ ) f ( B ) for all A , B σ ( J ) and all λ [ 0 , 1 ] .

A function f : J R is said to be of P-class on J or is a P-class function on J if
f ( λ x + ( 1 λ ) y ) f ( x ) + f ( y ) ,
(1)
where x , y J and λ [ 0 , 1 ] ; see [1]. Many properties of P-class functions can be found in [14]. Note that the set of all P-class functions contains all convex functions and also all nonnegative monotone functions. Every non-zero P-class function is nonnegative valued. In fact, choose λ = 1 and fix x 0 J . It follows from (1) that
f ( x 0 ) f ( x 0 ) + f ( y ) ,

where y J . Thus 0 f ( y ) for all y J .

For a P-class function f on an interval [ a , b ] ,
f ( a + b 2 ) 2 0 1 f ( t a + ( 1 t ) b ) d t 2 ( f ( a ) + f ( b ) ) ,

which is known as the Hermite-Hadamard inequality for the P-class continuous functions; see [3].

In this paper, we introduce and investigate the notion of an operator P-class function and give several examples. We show that if f is a P-class function on ( 0 , ) such that lim t f ( t ) = 0 , then it is operator decreasing. We also prove that if f is an operator P-class function on an interval J, then
f ( C A C ) 2 C f ( A ) C ,

where A σ ( J ) and C B ( H ) is an isometry. In addition, we present a Hermite-Hadamard inequality for operator P-class functions.

2 Operator P-class functions

In this section, we investigate operator P-class functions and study some relations between the operator P-class functions and the operator monotone functions.

We start our work with the following definition.

Definition 1 Let f be a real-valued continuous function defined on an interval J. We say that f is of operator P-class on J if
f ( λ A + ( 1 λ ) B ) f ( A ) + f ( B )

for all A , B σ ( J ) and all λ [ 0 , 1 ] .

Clearly every nonnegative operator convex function is of operator P-class.

Example 1 Let f ( t ) = t r ( 0 r 1 ) be defined on ( 1 , ) . It follows from the operator concavity of t r ( 0 r 1 ) [5] and the arithmetic-harmonic mean inequality that
( λ A + ( 1 λ ) B ) r ( λ A r + ( 1 λ ) B r ) 1 ( by the concavity of  t r ) λ A r + ( 1 λ ) B r ( by the arithmetic-harmonic mean inequality ) A r + B r ,

where A , B σ ( ( 1 , ) ) and λ [ 0 , 1 ] . Thus f is an operator P-class function on ( 1 , ) .

In addition, every operator P-class f on an interval J is of operator Q-class in the sense that
f ( λ A + ( 1 λ ) B ) f ( A ) λ + f ( B ) 1 λ

for all A , B σ ( J ) and λ ( 0 , 1 ) ; see [6]. In the next example, we show the converse is not true, in general.

Example 2 The function f ( t ) = 4 t 2 defined on [ 3 , 3 ] is of operator Q-class; see [[7], Example 2.1]. We put λ = 1 2 , A = ( 3 2 0 0 3 2 ) and B = ( 3 2 0 0 3 2 ) . Then f ( λ A + ( 1 λ ) B ) = ( 4 0 0 4 ) f ( A ) + f ( B ) = ( 14 4 0 0 14 4 ) . Hence f is not of operator P-class.

Example 3 Let α > 0 and f be a continuous function on the interval [ α , 2 α ] into itself. It follows from
f ( λ A + ( 1 λ ) B ) 2 α f ( A ) + f ( B ) ( A , B σ ( [ α , 2 α ] ) , λ [ 0 , 1 ] )

that f is of operator P-class on [ α , 2 α ] .

Example 4 Let g be a nonnegative continuous function on an interval [ a , b ] and α = sup x , y [ a , b ] , t [ x , y ] | g ( t ) g ( x ) g ( y ) | . We put f ( t ) = g ( t ) + α . Then
f ( λ A + ( 1 λ ) B ) = g ( λ A + ( 1 λ ) B ) + α ( g ( A ) + α ) + ( g ( B ) + α ) = f ( A ) + f ( B ) ,

where A , B σ ( [ a , b ] ) and λ [ 0 , 1 ] . Hence f is an operator P-class function.

Next, we explore some relations between operator P-class functions and operator monotone functions. In fact, we have the following.

Theorem 1 If f is an operator P-class function on the interval ( 0 , ) such that lim t f ( t ) = 0 , then f is operator decreasing.

Proof Let 0 < A B . Fix ε > 0 . We put C = B A + ε . Let θ > 0 . It follows from lim t f ( t ) = 0 that there exists M > 0 such that f ( t ) θ for all t M . We may assume that the spectrum of the strictly positive operator C is contained in [ α , β ] for some 0 < α < β . It follows from lim λ 1 λ 1 λ = that there exists δ > 0 such that λ 1 λ M α for all λ ( 1 δ , 1 ) . Hence σ ( λ 1 λ C ) [ M , ) for all λ ( 1 δ , 1 ) . Now, by the functional calculus for the positive operator λ 1 λ C , we have f ( λ 1 λ C ) θ for all λ ( 1 δ , 1 ) . Thus f ( λ 1 λ C ) x , x θ x 2 for all λ ( 1 δ , 1 ) and x H . Since λ ( B + ε ) = λ A + ( 1 λ ) ( λ 1 λ ) C and f is P-class we have
f ( λ ( B + ε ) ) f ( A ) + f ( ( λ 1 λ ) C )
for all λ ( 1 δ , 1 ) . Hence
f ( λ ( B + ε ) ) x , x f ( A ) x , x + f ( λ 1 λ C ) x , x f ( A ) x , x + θ x 2 ,

where λ ( 1 δ , 1 ) and x H . As λ 1 and then θ 0 + we obtain f ( B + ε ) x , x f ( A ) x , x for all x H . As ε 0 + , we conclude that f ( B ) f ( A ) . □

3 Jensen operator inequality for operator P-class functions

In this section, we present a Jensen operator inequality for operator P-class functions. We start with the following result in which we utilized the well-known technique of [8].

Theorem 2 Let f be an operator P-class function on an interval J, A σ ( J ) , and C B ( H ) be an isometry. Then
f ( C A C ) 2 C f ( A ) C .
(2)
Proof Let X = ( A 0 0 B ) B ( H H ) for some B σ ( J ) and let U = ( C D 0 C ) and V = ( C D 0 C ) , where D = 1 H C C . Now we can easily conclude from the two facts C D = 1 H C C C = 0 and D C = C 1 H C C = 0 that U and V are unitary operators in B ( H H ) . Further,
U X U = ( C A C C A D D A C D A D + C B C )
and
V X V = ( C A C C A D D A C D A D + C B C ) .
Using the operator P-class property of f we have
( f ( C A C ) 0 0 f ( D A D + C B C ) ) = f ( C A C 0 0 D A D + C B C ) = f ( U X U + V X V 2 ) f ( U X U ) + f ( V X V ) = 2 ( C f ( A ) C 0 0 D f ( A ) D + C f ( B ) C ) .
Therefore
f ( C A C ) 2 C f ( A ) C .

 □

Applying Theorem 2 we have some inequalities including the subadditivity.

Corollary 1 Let f be operator P-class on an interval J, A j σ ( J ) ( 1 j n ), and C j B ( H ) ( 1 j n ), where j = 1 n C j C j = 1 . Then
f ( j = 1 n C j A j C j ) 2 j = 1 n C j f ( A j ) C j .
Proof Let
A ˜ = A ˜ = ( A 1 A 2 A n ) B ( H H ) , C ˜ = ( C 1 C 2 C n ) B ( H H ) .
It follows from C ˜ C ˜ = 1 and (2) that
f ( j = 1 n C j A j C j ) = f ( C ˜ A ˜ C ˜ ) 2 C ˜ f ( A ˜ ) C ˜ = 2 j = 1 n C j f ( A j ) C j .

 □

Corollary 2 Let f be operator P-class on [ 0 , ) such that f ( 0 ) = 0 , A σ ( [ 0 , ) ) , and C B ( H ) be a contraction. Then
f ( C A C ) 2 C f ( A ) C .
Proof For every contraction C B ( H ) , we put D = 1 H C C . It follows from C C + D D = 1 H and (2) that
f ( C A C ) = f ( C A C + D 0 D ) 2 f ( C A C ) + 2 f ( D 0 D ) = 2 C f ( A ) C .

 □

Corollary 3 Let f be operator P-class on [ 0 , ) such that f ( 0 ) = 0 and A , B σ ( ( 0 , ) ) such that A B . Then
A 1 f ( A ) 2 B 1 f ( B ) .
Proof Let A , B σ ( ( 0 , ) ) such that 0 < A B . We put C = B 1 / 2 A 1 / 2 . Then C C = B 1 / 2 A B 1 / 2 1 H , so C is a contraction. It follows from (2) that
f ( A ) = f ( C B C ) 2 C f ( B ) C = 2 A 1 / 2 B 1 / 2 f ( B ) B 1 / 2 A 1 / 2 .
Therefore
A 1 f ( A ) 2 B 1 f ( B ) .

 □

In the following theorem, we obtain the Choi-Davis-Jensen type inequality for operator P-class functions.

Theorem 3 Let Φ be a unital positive linear map on B ( H ) , A σ ( J ) and f be operator P-class on an interval J. Then
f ( Φ ( A ) ) 2 Φ ( f ( A ) ) .
(3)
Proof Let A σ ( J ) . We put Ψ the restriction of Φ to the C -algebra C ( A , I ) generated by I and A. Then Ψ is a unital completely positive map on C ( A , I ) . The celebrated Stinespring dilation theorem [[9], Theorem 1] states that there exist an isometry V : H H and a unital -homomorphism π : C ( A , I ) B ( H ) such that Ψ ( A ) = V π ( A ) V . Hence
f ( Φ ( A ) ) = f ( Ψ ( A ) ) = f ( V π ( A ) V ) 2 V f ( π ( A ) ) V ( by  (2) ) = 2 V π ( f ( A ) ) V = 2 Ψ ( f ( A ) ) = 2 Φ ( f ( A ) ) .

 □

We will show that the constant 2 is the best possible such one in the following example.

Example 5 Let f ( t ) = 2 t 2 for t [ 1 , 1 ] . Then 1 f ( t ) 2 and
f ( λ A + ( 1 λ ) B ) = 2 ( λ A + ( 1 λ ) B ) 2 2 2 A 2 + 2 B 2 = f ( A ) + f ( B ) ,

where A , B σ ( [ 1 , 1 ] ) . Hence f is of operator P-class on [ 1 , 1 ] . Now, consider that the unital positive map Φ : M 2 M 2 is defined by Φ ( A ) = tr ( A ) 2 I . Then for the Hermitian matrix A = ( 1 0 0 1 ) we have Φ ( A ) = 0 , f ( Φ ( A ) ) = 2 , f ( A ) = I , and Φ ( f ( A ) ) = I . Therefore f ( Φ ( A ) ) = 2 Φ ( f ( A ) ) . This shows that the coefficient 2 in (2) and (3) is the best.

Example 6 Consider (the nonnegative increasing function and so) P-class function f ( t ) = t where t ( 0 , ) . Let the unital positive map Ψ : M 2 ( C ) C be defined by Ψ ( A ) = a 22 with A = ( a i j ) 1 i , j 2 and let A = ( 2 2 2 1 ) 2 . Then Ψ ( f ( A ) ) = 1 and f ( Ψ ( A ) ) = 8 . Hence f ( Ψ ( A ) ) 2 Ψ ( f ( A ) ) . It follows from (3) that f is not of operator P-class.

We present a Hermite-Hadamard inequality for operator P-class functions in the next theorem.

Theorem 4 Let Φ be a unital positive linear map on B ( H ) and f be operator P-class on J. Then
f ( Φ ( A ) + Φ ( B ) 2 ) 2 0 1 f ( λ Φ ( A ) + ( 1 λ ) Φ ( B ) ) d λ 4 ( Φ ( f ( A ) ) + Φ ( f ( B ) ) ) ,

where A , B σ ( J ) and λ [ 0 , 1 ] .

Proof Let A , B σ ( J ) and λ [ 0 , 1 ] . Then
f ( Φ ( A ) + Φ ( B ) 2 ) = f ( λ Φ ( A ) + ( 1 λ ) Φ ( B ) + ( 1 λ ) Φ ( A ) + λ Φ ( B ) 2 ) f ( λ Φ ( A ) + ( 1 λ ) Φ ( B ) ) + f ( ( 1 λ ) Φ ( A ) + λ Φ ( B ) ) 2 ( f ( Φ ( A ) ) + f ( Φ ( B ) ) ) .
(4)
Integrating both sides of (4) over [ 0 , 1 ] we obtain
f ( Φ ( A ) + Φ ( B ) 2 ) 0 1 f ( λ Φ ( A ) + ( 1 λ ) Φ ( B ) ) d λ + 0 1 f ( ( 1 λ ) Φ ( A ) + λ Φ ( B ) ) d λ = 2 0 1 f ( λ Φ ( A ) + ( 1 λ ) Φ ( B ) ) d λ 2 ( f ( Φ ( A ) ) + f ( Φ ( B ) ) ) 4 ( Φ ( f ( A ) ) + Φ ( f ( B ) ) ) ( by  (3) ) .

 □

4 Some inequalities for P-class functions involving continuous operator fields

Let A be a C -algebra of operators acting on a Hilbert space and let T be a locally compact Hausdorff space. A field ( A t ) t T of operators in A is called a continuous field of operators if the mapping t A t is norm continuous on T. If μ ( t ) is a Radon measure on T and the function t A t is integrable, one can form the Bochner integral T A t d μ ( t ) , which is the unique element in A such that
φ ( T A t d μ ( t ) ) = T φ ( A t ) d μ ( t )

for every linear functional φ in the norm dual A of A .

Let C ( T , A ) denote the set of bounded continuous functions on T with values in A . It is easy to see that the set C ( T , A ) is a C -algebra under the pointwise operations and the norm ( A t ) t T = sup t T A t ; cf. [10].

Assume that there is a field ( Φ t ) t T of positive linear mappings Φ t : A B from A to another C -algebra . We say that such a field is continuous if the mapping t Φ t ( A ) is continuous for every A A . If the C -algebras are unital and the field t Φ t ( I ) is integrable with integral I, we say that ( Φ t ) t T is unital; see [10].

Theorem 5 Let f : J R be an operator P-class function defined on an interval J, and let A and be unital C -algebras. If ( Φ t ) t T is a unital field of positive linear mappings Φ t : A B defined on a locally compact Hausdorff space T with a bounded Radon measure μ, then
f ( T Φ t ( A t ) d μ ( t ) ) 2 T Φ t ( f ( A t ) ) d μ ( t )

holds for every bounded continuous field ( A t ) t T of self-adjoint elements in A with spectra contained in J.

Proof We consider the unital positive linear map Ψ : C ( T , A ) B defined by Ψ ( ( A t ) t T ) = T Φ t ( A t ) d μ ( t ) . Let A ˜ = ( A t ) t T C ( T , A ) . It follows from σ ( A ˜ ) J and (3) that
f ( Ψ ( ( A t ) t T ) ) = f ( Ψ ( A ˜ ) ) 2 Ψ ( f ( A ˜ ) ) = 2 Ψ ( f ( ( A t ) t T ) ) = 2 Ψ ( ( f ( A t ) ) t T ) .

 □

In the discrete case, T = { 1 , , n } in Theorem 5, we get the following result.

Corollary 4 Let f : J R be an operator P-class function defined on an interval J, let A j σ ( J ) ( 1 j n ) and Φ j ( 1 j n ) be unital positive linear maps on B ( H ) . Then
f ( j = 1 n Φ j ( A j ) ) 2 j = 1 n Φ j ( f ( A j ) ) .

Declarations

Acknowledgements

The second and the third authors are supported by the Lebanese University grants program for the Discrete Mathematics and Algebra group.

Authors’ Affiliations

(1)
Department of Mathematics, University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran
(2)
Department of Mathematics, Faculty of Sciences, Lebanese University, Hadath, Beirut, Lebanon
(3)
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAA), Ferdowsi University, Mashhad, Iran

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