Some Trapezoidal Vector Inequalities for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

On utilising the spectral representation of selfadjoint operators in Hilbert spaces, some trapezoidal inequalities for various classes of continuous functions of such operators are given.


Introduction
In Classical Analysis a trapezoidal type inequality is an inequality that provides upper and/or lower bounds for the quantity that is the error in approximating the integral by a trapezoidal rule, for various classes of integrable functions f de…ned on the compact interval [a; b] : In order to introduce the reader to some of the well known results and prepare the background for considering a similar problem for functions of selfadjoint operators in Hilbert spaces, we mention the following inequalities.
The case of functions of bounded variation was obtained in [2] (see also [1, p. 68]): Theorem 1.Let f : [a; b] !C be a function of bounded variation.We have the inequality is the best possible one.
This result may be improved if one assumes the monotonicity of f as follows (see [1, p.The above inequalities are sharp. If the mapping is Lipschitzian, then the following result holds as well [3] (see also [1, p. 82]).
Then we have the inequality: The constant 1 4 is best in (1.3).If we would assume absolute continuity for the function f , then the following estimates in terms of the Lebesgue norms of the derivative f 0 hold [1, p. 93].
where k k p (p 2 [1; 1]) are the Lebesgue norms, i.e., The case of convex functions is as follows [4]: The constant 1 8 is sharp in both sides of (1.5).For other scalar trapezoidal type inequalities, see [1].

Trapezoidal Operator Inequalities
In order to provide some generalizations for functions of selfadjoint operators of the above trapezoidal inequalities, we need some concepts as results as follows.
Let A be a selfadjoint linear operator on a complex Hilbert space (H; h:; :i) : The Gelfand map establishes a -isometrically isomorphism between the set C (Sp (A)) of all continuous functions de…ned on the spectrum of A; denoted Sp (A) ; and the C -algebra C (A) generated by A and the identity operator 1 H on H as follows (see for instance [8, p. 3]): For any f; g 2 C (Sp (A)) and any ; 2 C we have (i) where f 0 (t) = 1 and f 1 (t) = t; for t 2 Sp (A) : With this notation we de…ne and we call it the continuous functional calculus for a selfadjoint operator A: If A is a selfadjoint operator and f is a real valued continuous function on Sp (A), then f (t) 0 for any t 2 Sp (A) implies that f (A) 0; i:e: f (A) is a positive operator on H: Moreover, if both f and g are real valued functions on Sp (A) then the following important property holds: in the operator order of B (H) : For a recent monograph devoted to various inequalities for continuous functions of selfadjoint operators, see [8] and the references therein.
Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U ) included in the interval [m; M ] for some real numbers m < M and let fE g be its spectral family.Then for any continuous function f : [m; M ] !C, it is well known that we have the following spectral representation in terms of the Riemann-Stieltjes integral : With the notations introduced above, we consider in this paper the problem of bounding the error in approximating hf (A) x; yi by the trapezoidal type formula f (M )+f (m) 2 hx; yi ; where x; y are vectors in the Hilbert space H; f is a continuous functions of the selfadjoint operator A with the spectrum in the compact interval of real numbers [m; M ] : Applications for some particular elementary functions are also provided.

Some Trapezoidal Vector Inequalities
The following result holds: Theorem 6.Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers m < M and let fE g be its spectral family.If f : [m; M ] !C is a continuous function of bounded variation on [m; M ], then we have the inequality for any x; y 2 H: exists, then a simple integration by parts reveals the identity If we write the identity (3.2) for u ( ) = hE x; yi ; then we get which, by (2.1), gives the following identity of interest in itself which, together with the elementary inequality for a; b; c; d 0 produce the inequalities = kxk kyk for any x; y 2 H: On utilizing (3.5) and taking the maximum in (3.7) we deduce the desired result (3.1).
The case of Lipschitzian functions may be useful for applications: Theorem 7. Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers m < M and let fE g be its spectral family.If f : [m; M ] !C is Lipschitzian with the constant L > 0 on [m; M ], then we have the inequality for any x; y 2 H: which together with (3.9) produces the desired result (3.8).

Other Trapezoidal Vector Inequalities
The following result provides a di¤erent perspective in bounding the error in the trapezoidal approximation: for any x; y 2 H: Proof.From (3.5) we have that From the representation (3.3) we then have x; y df ( ) for any x; y 2 H; from which we obtain the last branch in (4.1).
We recall that a function f : We have the following result concerning this class of functions.Proof.We start with the equality for any x; y 2 H; that follows from the spectral representation (2.1).Since the function E ( ) x; y is of bounded variation for any vector x; y 2 H; by applying the inequality (3.4) we conclude that for any x; y 2 H: for any 2 [m; M ] : Since, obviously, the function g r ( ) := (M ) r + ( m) r ; r 2 (0; 1) has the property that then by (4.8) we deduce the …rst part of (4.6).Now, if d : m = t 0 < t 1 < ::: < t n 1 < t n = M is an arbitrary partition of the interval [m; M ] ; then we have by the Schwarz inequality for nonnegative operators that By the Cauchy-Buniakovski-Schwarz inequality for sequences of real numbers we also have that = kxk kyk for any x; y 2 H: These prove the last part of (4.6).

Applications for Some Particular Functions
It is obvious that the results established above can be applied for various particular functions of selfadjoint operators.We restrict ourselves here to only two examples, namely the logarithm and the power functions.
1.If we consider the logarithmic function f : (0; 1) !R, f (t) = ln t; then we can state the following result: Proposition 1.Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers with 0 < m < M and let fE g be its spectral family.Then for any x; y 2 H we have hx; yi ln p mM hln Ax; yi (5.1) and hx; yi ln p mM hln Ax; yi kxk kyk ln M m respectively.
The proof is obvious from Theorems 6, 7 and 8 applied for the logarithmic function.The details are omitted.
2. Consider now the power function f : (0; 1) !R, f (t) = t p with p 2 ( 1; 0) [ (0; 1) : In the case when p 2 (0; 1) ; the function is p H-Hölder continuous with H = 1 on any subinterval [m; M ] of [0; 1): By making use of Theorem 9 we can state the following result: Proposition 2. Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers with 0 m < M and let fE g be its spectral family.Then for p 2 (0; 1) we have  respectively.
The proof is obvious from Theorems 6, 7 and 8 applied for the power function f : (0; 1) !R, f (t) = t p with p 1: The details are omitted.
The case of negative powers is similar.The details are left to the interested reader.
denotes the total variation of f on the interval [a; b].The constant 1 2

Theorem 3 .
Let f : [a; b] !C be an L Lipschitzian function on [a; b] ; i.e., f satis…es the condition:

Theorem 4 .
Let f : [a; b] !C be an absolutely continuous function on [a; b].Then we have Z b a

(2. 1 )
hf (U ) x; yi = Z M m 0 f ( ) d (hE x; yi) ; for any x; y 2 H: The function g x;y ( ) := hE x; yi is of bounded variation on the interval [m; M ] and g x;y (m 0) = 0 and g x;y (M ) = hx; yi for any x; y 2 H: It is also well known that g x ( ) := hE x; xi is monotonic nondecreasing and right continuous on [m; M ].
denotes the total variation of v on [a; b] : Utilising the property (3.4), we have from (3.3) that

Theorem 8 .
Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers m < M and let fE g be its spectral family.Assume that f : [m; M ] !C is a continuous function on [m; M ].Then we have the inequalities

Theorem 9 .
Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers m < M and let fE g be its spectral family.If f : [m; M ] !C is r H-Hölder continuous on [m; M ], then we have the inequality f (m) + f (M ) 2 hx; yi hf (A) (M m) r kxk kyk for any x; y 2 H:

m p + M p 2 hx; yi hA p x; yi 1
M m) p kxk kyk ; for any x; y 2 H:The case of powers p 1 is embodied in the following: Proposition 3. Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m; M ] for some real numbers with 0 m < M and let fE g be its spectral family.Then for p 1 and for any x; y 2 H we have m p + M p 2 hx; yi hA p x; yi (5.5)(M p m p ) From the theory of Riemann-Stieltjes integral is well known that if p : [a; b] !C is of bounded variation and v : [a; b] !R is continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals for any x; y 2 H; which proves the …rst branch in (4.1).The second inequality follows from (3.9).