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

Abstract

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.

1. Introduction

In classical analysis a trapezoidal type inequality is an inequality that provides upper and/or lower bounds for the quantity 𝑓(𝑎)+𝑓(𝑏)2(𝑏−𝑎)−^𝑏_𝑎𝑓(𝑡)𝑑𝑡, (1.1)

that is, the error in approximating the integral by a trapezoidal rule, for various classes of integrable functions 𝑓 defined 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 [1] (see also [1, p. 68]):

Theorem 1.1. Let 𝑓∶[𝑎,𝑏]→ℂ be a function of bounded variation. One has the inequality ^𝑏_𝑎𝑓(𝑡)𝑓(𝑎)(𝑏)𝑑𝑡−+𝑓2(≤1𝑏−𝑎)2(𝑏−𝑎)^𝑏_𝑎𝑓, (1.2) where ∨^𝑏_𝑎(𝑓) denotes the total variation of 𝑓 on the interval [𝑎,𝑏]. The constant 1/2 is the best possible one.

This result may be improved if one assumes the monotonicity of 𝑓 as follows (see [1, p. 76]).

Theorem 1.2. Let 𝑓∶[𝑎,𝑏]→ℝ be a monotonic nondecreasing function on [𝑎,𝑏]. Then one has the inequalities ^𝑏_𝑎𝑓(𝑡)𝑓(𝑎)(𝑏)𝑑𝑡−+𝑓2(≤1𝑏−𝑎)2(𝑏−𝑎)𝑓(𝑏)−𝑓(𝑎)−^𝑏_𝑎sgn𝑡−𝑎+𝑏2𝑓(𝑡)≤1𝑑𝑡2(𝑏−𝑎)𝑓(𝑏)−𝑓(𝑎). (1.3) The above inequalities are sharp.

If the mapping is Lipschitzian, then the following result holds as well [3] (see also [1, p. 82]).

Theorem 1.3. Let 𝑓∶[𝑎,𝑏]→ℂ be an 𝐿-Lipschitzian function on [𝑎,𝑏], that is, 𝑓 satisfies the condition 𝑓(𝑠)−𝑓(𝑡)≤𝐿|𝑠−𝑡|forany𝑠,𝑡∈[𝑎,𝑏]𝐿>0isgiven. (L) Then one has the inequality ^𝑏_𝑎𝑓(𝑡)𝑓(𝑎)(𝑏)𝑑𝑡−+𝑓2(≤1𝑏−𝑎)4(𝑏−𝑎)²𝐿. (1.4) The constant 1/4 is best in (1.4).

If we would assume absolute continuity for the function 𝑓, then the following estimates in terms of the Lebesgue norms of the derivative 𝑓^′ hold ([1, p. 93]).

Theorem 1.4. Let 𝑓∶[𝑎,𝑏]→ℂ be an absolutely continuous function on [𝑎,𝑏]. Then one has ^𝑏_𝑎𝑓(𝑡)𝑓(𝑎)(𝑏)𝑑𝑡−+𝑓2(≤1𝑏−𝑎)4(𝑏−𝑎)²𝑓^′_∞if𝑓^′∈𝐿_∞[],1𝑎,𝑏2𝑞+1^1/𝑞(𝑏−𝑎)^1+1/𝑞𝑓^′_𝑝if𝑓^′∈𝐿_𝑝[]1𝑎,𝑏,𝑝>1,𝑝+1𝑞1=1,2(𝑏−𝑎)𝑓^′₁, (1.5) where ‖⋅‖_𝑝(𝑝∈[1,∞]) are the Lebesgue norms, that is, 𝑓^′_∞=esssup_{[]𝑠∈𝑎,𝑏}𝑓^′(𝑠,𝑓_′𝑝)∶=∫^𝑏_𝑎𝑓^′(𝑠)𝑑𝑠^1/𝑝,𝑝≥1. (1.6)

The case of convex functions is as follows [4].

Theorem 1.5. Let 𝑓∶[𝑎,𝑏]→ℝ be a convex function on [𝑎,𝑏]. Then one has the inequalities 18(𝑏−𝑎)²𝑓^′₊𝑎+𝑏2−𝑓^′₋𝑎+𝑏2≤𝑓(𝑎)+𝑓(𝑏)2(𝑏−𝑎)−^𝑏_𝑎𝑓(𝑡)≤1𝑑𝑡8(𝑏−𝑎)²𝑓^′₋(𝑏)−𝑓^′₊(𝑎). (1.7) The constant 1/8 is sharp in both sides of (1.7).

For other scalar trapezoidal type inequalities, see [2].

2. 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;⟨⋅,⋅⟩). The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set 𝐶(Sp(𝐴)) of all continuous functions defined 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 [5, page 3]):

For any 𝑓,𝑔∈𝐶(Sp(𝐴)) and any 𝛼,𝛽∈ℂ we have(i)Φ(𝛼𝑓+𝛽𝑔)=𝛼Φ(𝑓)+𝛽Φ(𝑔); (ii)Φ(𝑓𝑔)=Φ(𝑓)Φ(𝑔) and Φ(𝑓)=Φ(𝑓)^∗;(iii)‖Φ(𝑓)‖=‖𝑓‖∶=sup_{𝑡∈Sp(𝐴)}|𝑓(𝑡)|; (iv)Φ(𝑓₀)=1_𝐻 and Φ(𝑓₁)=𝐴, where 𝑓₀(𝑡)=1 and 𝑓₁(𝑡)=𝑡, for 𝑡∈Sp(𝐴).

With this notation we define 𝑓(𝐴)∶=Φ𝑓∀𝑓∈𝐶Sp(𝐴) (2.1)

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∈Sp(𝐴) implies that 𝑓(𝐴)≥0, for example 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:𝑓(𝑡)≥𝑔(𝑡)forany𝑡∈Sp(𝐴)impliesthat𝑓(𝐴)≥𝑔(𝐴) (P)

in the operator order of 𝐵(𝐻).

For a recent monograph devoted to various inequalities for continuous functions of selfadjoint operators, see [5] and the references therein.

For other recent results see [6–12].

Let 𝑈 be a selfadjoint operator on the complex Hilbert space (𝐻,⟨⋅,⋅⟩) with the spectrum Sp(𝑈) included in the interval [𝑚,𝑀] for some real numbers 𝑚<𝑀 and let {𝐸_𝜆}_𝜆 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:𝑓(𝑈)𝑥,𝑦=^𝑀_𝑚−0𝑓(𝜆)𝑑𝐸_𝜆𝑥,𝑦, (2.2) for any 𝑥,𝑦∈𝐻. The function 𝑔_𝑥,𝑦(𝜆)∶=⟨𝐸_𝜆𝑥,𝑦⟩ is of bounded variation on the interval [𝑚,𝑀] and 𝑔_𝑥,𝑦(𝑚−0)=0,𝑔_𝑥,𝑦(𝑀)=𝑥,𝑦 (2.3)

for any 𝑥,𝑦∈𝐻. It is also well-known that 𝑔_𝑥(𝜆)∶=⟨𝐸_𝜆𝑥,𝑥⟩ is monotonic nondecreasing and right continuous on [𝑚,𝑀].

With the notations introduced above, we consider in this paper the problem of bounding the error 𝑓(𝑀)+𝑓(𝑚)2⋅𝑥,𝑦−𝑓(𝐴)𝑥,𝑦 (2.4)

in approximating ⟨𝑓(𝐴)𝑥,𝑦⟩ by the trapezoidal type formula ((𝑓(𝑀)+𝑓(𝑚))/2)⋅⟨𝑥,𝑦⟩, 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.

3. Some Trapezoidal Vector Inequalities

The following result holds.

Theorem 3.1. 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 {𝐸_𝜆}_𝜆 be its spectral family. If 𝑓∶[𝑚,𝑀]→ℂ is a continuous function of bounded variation on [𝑚,𝑀], then one has the inequality 𝑓(𝑀)+𝑓(𝑚)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2𝑀_𝑚𝑓≤12‖𝑥‖𝑦^𝑀_𝑚𝑓 (3.1) for any 𝑥,𝑦∈𝐻.

Proof. If 𝑓,𝑢∶[𝑚,𝑀]→ℂ are such that the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑢(𝑡) exists, then a simple integration by parts reveals the identity ^𝑏_𝑎𝑓(𝑡)(𝑡)=𝑓(𝑎)(𝑏)𝑑𝑢+𝑓2[𝑢(𝑏)−𝑢(𝑎)]−^𝑏_𝑎𝑢(𝑡)−𝑢(𝑎)+𝑢(𝑏)2𝑑𝑓(𝑡). (3.2) If we write the identity (3.2) for 𝑢(𝜆)=⟨𝐸_𝜆𝑥,𝑦⟩, then we get ^𝑀_𝑚−0𝑓(𝜆)𝑑𝐸_𝜆=𝑓(𝑚)(𝑀)𝑥,𝑦+𝑓2⋅𝑥,𝑦−^𝑀_𝑚−0𝐸_𝜆−1𝑥,𝑦2𝑥,𝑦𝑑𝑓(𝜆) (3.3) which, by (2.2), gives the following identity of interest in itself 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)=1𝑥,𝑦𝑥,𝑦2^𝑀_𝑚−0𝐸_𝜆𝑥,𝑦+𝐸_𝜆−1_𝐻𝑥,𝑦𝑑𝑓(𝜆), (3.4) for any 𝑥,𝑦∈𝐻.
It is well-known that if 𝑝∶[𝑎,𝑏]→ℂ is a continuous function and 𝑣∶[𝑎,𝑏]→ℂ is of bounded variation, then the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡) exists and the following inequality holds: ^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡)≤max_{[]𝑡∈𝑎,𝑏}𝑝(𝑡)^𝑏_𝑎(𝑣), (3.5) where ∨^𝑏_𝑎(𝑣) denotes the total variation of 𝑣 on [𝑎,𝑏].
Utilizing the property (3.5), we have from (3.4) that 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑦+𝐸_𝜆−1_𝐻𝑥,𝑦^𝑀_𝑚𝑓≤12max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑦+1_𝐻−𝐸_𝜆𝑥,𝑦^𝑀_𝑚𝑓. (3.6) If 𝑃 is a nonnegative operator on 𝐻, that is, ⟨𝑃𝑥,𝑥⟩≥0 for any 𝑥∈𝐻, then the following inequality is a generalization of the Schwarz inequality in the Hilbert space 𝐻: 𝑃𝑥,𝑦²≤⟨𝑃𝑥,𝑥⟩𝑃𝑦,𝑦, (3.7) for any 𝑥,𝑦∈𝐻.
On applying the inequality (3.7) we have 𝐸_𝜆𝑥,𝑦≤𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2,1_𝐻−𝐸_𝜆𝑥,𝑦≤1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2, (3.8) which, together with the elementary inequality for 𝑎,𝑏,𝑐,𝑑≥0𝑎𝑏+𝑐𝑑≤𝑎²+𝑐^21/2𝑏²+𝑑^21/2 (3.9) produce the inequalities 𝐸_𝜆𝑥,𝑦+1_𝐻−𝐸_𝜆≤𝐸𝑥,𝑦_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2≤𝐸_𝜆𝑥,𝑥+1_𝐻−𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦+1_𝐻−𝐸_𝜆𝑦,𝑦^1/2=‖𝑥‖𝑦 (3.10) for any 𝑥,𝑦∈𝐻.
On utilizing (3.6) and taking the maximum in (3.10) we deduce the desired result (3.1).

The case of Lipschitzian functions may be useful for applications.

Theorem 3.2. 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 {𝐸_𝜆}_𝜆 be its spectral family. If 𝑓∶[𝑚,𝑀]→ℂ is Lipschitzian with the constant 𝐿>0 on [𝑚,𝑀], then one has the inequality 𝑓(𝑀)+𝑓(𝑚)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2𝐿^𝑀_𝑚−0𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2≤1𝑑𝜆2(𝑀−𝑚)𝐿‖𝑥‖𝑦 (3.11) for any 𝑥,𝑦∈𝐻.

Proof. It is well-known that if 𝑝∶[𝑎,𝑏]→ℂ is a Riemann integrable function and 𝑣∶[𝑎,𝑏]→ℂ is Lipschitzian with the constant 𝐿>0, that is, |𝑣(𝑠)−𝑣(𝑡)|≤𝐿|𝑠−𝑡|forany𝑡,𝑠∈[𝑎,𝑏], (3.12) then the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡) exists and the following inequality holds: ^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡)≤𝐿^𝑏_𝑎𝑝(𝑡)𝑑𝑡. (3.13)
Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation (3.4) that 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2𝐿^𝑀_𝑚−0𝐸_𝜆𝑥,𝑦+𝐸_𝜆−1_𝐻≤1𝑥,𝑦𝑑𝜆,2𝐿^𝑀_𝑚−0𝐸_𝜆𝑥,𝑦+1_𝐻−𝐸_𝜆𝑥,𝑦𝑑𝜆, (3.14) for any 𝑥,𝑦∈𝐻.
Further, integrating (3.10) on [𝑚,𝑀] we have ^𝑀_𝑚−0𝐸_𝜆𝑥,𝑦+1_𝐻−𝐸_𝜆≤𝑥,𝑦𝑑𝜆_𝑀𝑚−0𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2≤(𝑥𝑦𝑑𝜆𝑀−𝑚)‖‖ (3.15) which together with (3.14) produces the desired result (3.11).

4. Other Trapezoidal Vector Inequalities

The following result provides a different perspective in bounding the error in the trapezoidal approximation.

Theorem 4.1. 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 {𝐸_𝜆}_𝜆 be its spectral family. Assume that 𝑓∶[𝑚,𝑀]→ℂ is a continuous function on [𝑚,𝑀]. Then one has the inequalities 𝑓(𝑀)+𝑓(𝑚)2⋅−𝑓(𝐴)≤𝑥,𝑦𝑥,𝑦max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆1𝑥−2𝑥,𝑦^M_m𝑓𝐿if𝑓isofboundedvariation,_𝑀𝑚−0𝐸_𝜆1𝑥−2𝑥,𝑦𝑑𝜆if𝑓is𝐿Lipschitzian,_𝑀𝑚−0𝐸_𝜆1𝑥−2(𝜆)≤1𝑥,𝑦𝑑𝑓if𝑓isnondecreasing,2‖𝑥‖𝑦^Mm𝑓𝐿()𝑓(𝑀)(𝑚)if𝑓isofboundedvariation,𝑀−𝑚if𝑓is𝐿Lipschitzian,−𝑓if𝑓isnondecreasing (4.1) for any 𝑥,𝑦∈𝐻.

Proof. From (3.6) we have that 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑦+𝐸_𝜆−1_𝐻𝑥,𝑦^𝑀_𝑚𝑓=max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆1𝑥−2𝑥,𝑦^𝑀_𝑚𝑓 (4.2) for any 𝑥,𝑦∈𝐻.
Utilizing the Schwarz inequality in 𝐻 and the fact that 𝐸_𝜆 are projectors we have successively 𝐸_𝜆1𝑥−2𝑥,𝑦≤𝐸_𝜆1𝑥−2𝑥𝑦=𝐸_𝜆𝑥,𝐸_𝜆𝑥−𝐸_𝜆+1𝑥,𝑥4‖𝑥‖^21/2𝑦=12‖𝑥‖𝑦 (4.3) for any 𝑥,𝑦∈𝐻, which proves the first branch in (4.1).
The second inequality follows from (3.14).
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 ∫^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡) and ∫^𝑏_𝑎|𝑝(𝑡)|𝑑𝑣(𝑡) exist and ^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡)≤^𝑏_𝑎𝑝(𝑡)𝑑𝑣(𝑡). (4.4)
From the representation (3.4) we then have 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2^𝑀_𝑚−0𝐸_𝜆𝑥,𝑦+𝐸_𝜆−1_𝐻(𝜆)=𝑥,𝑦𝑑𝑓_𝑀𝑚−0𝐸_𝜆1𝑥−2𝑥,𝑦𝑑𝑓(𝜆) (4.5) for any 𝑥,𝑦∈𝐻, from which we obtain the last branch in (4.1).

We recall that a function 𝑓∶[𝑎,𝑏]→ℂ is called 𝑟-𝐻-Hölder continuous with fixed 𝑟∈(0,1] and 𝐻>0 if 𝑓(𝑡)−𝑓(𝑠)≤𝐻|𝑡−𝑠|^𝑟forany𝑡,𝑠∈[𝑎,𝑏]. (4.6)

We have the following result concerning this class of functions.

Theorem 4.2. 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 {𝐸_𝜆}_𝜆 be its spectral family. If 𝑓∶[𝑚,𝑀]→ℂ is 𝑟−𝐻-Hölder continuous on [𝑚,𝑀], then one has the inequality 𝑓(𝑚)+𝑓(𝑀)2⋅−𝑓(𝐴)≤1𝑥,𝑦𝑥,𝑦2^𝑟𝐻(𝑀−𝑚)^𝑟𝑀_𝑚𝐸_(⋅)≤1𝑥,𝑦2^𝑟𝐻(𝑀−𝑚)^𝑟‖𝑥‖𝑦 (4.7) for any 𝑥,𝑦∈𝐻.

Proof. We start with the equality 𝑓(𝑀)+𝑓(𝑚)2⋅𝑥,𝑦−𝑓(𝐴)𝑥,𝑦=^𝑀_𝑚−0𝑓(𝑀)+𝑓(𝑚)2−𝑓(𝜆)𝑑𝐸_𝜆𝑥,𝑦 (4.8) for any 𝑥,𝑦∈𝐻, that follows from the spectral representation (2.2).
Since the function ⟨𝐸_(⋅)𝑥,𝑦⟩ is of bounded variation for any vector 𝑥,𝑦∈𝐻, by applying the inequality (3.5) we conclude that 𝑓(𝑚)+𝑓(𝑀)2⋅𝑥,𝑦−𝑓(𝐴)𝑥,𝑦≤max_{[]𝜆∈𝑚,𝑀}𝑓(𝑀)+𝑓(𝑚)2−𝑓(𝜆)^𝑀_𝑚𝐸_(⋅)𝑥,𝑦 (4.9) for any 𝑥,𝑦∈𝐻.
As 𝑓∶[𝑚,𝑀]→ℂ is 𝑟-𝐻-Hölder continuous on [𝑚,𝑀], then we have 𝑓(𝑀)+𝑓(𝑚)2(𝜆≤1−𝑓)2𝑓(𝑀)(𝜆+1−𝑓)2𝑓(𝜆)(𝑚≤1−𝑓)2𝐻(𝑀−𝜆)^𝑟+(𝜆−𝑚)^𝑟 (4.10) for any 𝜆∈[𝑚,𝑀].
Since, obviously, the function 𝑔_𝑟(𝜆)∶=(𝑀−𝜆)^𝑟+(𝜆−𝑚)^𝑟,𝑟∈(0,1] has the property that max_{[]𝜆∈𝑚,𝑀}𝑔_𝑟(𝜆)=𝑔_𝑟𝑚+𝑀2=2^1−𝑟(𝑀−𝑚)^𝑟, (4.11) then by (4.9) we deduce the first part of (4.7).
Now, if 𝑑∶𝑚=𝑡₀<𝑡₁<⋯<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 ^𝑀_𝑚𝐸_(⋅)𝑥,𝑦=sup_𝑑^𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑥,𝑦≤sup_𝑑^𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑥,𝑥^1/2𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑦,𝑦^1/2∶=𝐼. (4.12) By the Cauchy-Buniakovski-Schwarz inequality for sequences of real numbers we also have that 𝐼≤sup_𝑑^𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑥,𝑥^1/2𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑦,𝑦^1/2≤sup_𝑑^𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑥,𝑥^1/2sup_𝑑^𝑛−1_𝑖=0𝐸_𝑡𝑖+1−𝐸_𝑡𝑖𝑦,𝑦^1/2=^𝑀_𝑚𝐸_(⋅)𝑥,𝑥^1/2𝑀_𝑚𝐸_(⋅)𝑦,𝑦^1/2=‖𝑥‖𝑦 (4.13) for any 𝑥,𝑦∈𝐻. These prove the last part of (4.7).

5. 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,∞)→ℝ, 𝑓(𝑡)=ln𝑡, then we can state the following result.

Proposition 5.1. 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 {𝐸_𝜆}_𝜆 be its spectral family. Then for any 𝑥,𝑦∈𝐻 one has 𝑥,𝑦ln√𝑀𝑚𝑀−ln𝐴𝑥,𝑦≤ln𝑚×12max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆1𝑥−2≤1𝑥,𝑦2‖𝑥𝑦𝑀‖ln𝑚,𝑥,𝑦ln√≤1𝑚𝑀−ln𝐴𝑥,𝑦𝑚×12^𝑀_𝑚−0𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2𝑑𝜆_𝑀𝑚−0𝐸_𝜆1𝑥−2≤1𝑥,𝑦𝑑𝜆2‖𝑥𝑦𝑀‖𝑚,√−1𝑥,𝑦ln𝑚𝑀−ln𝐴𝑥,𝑦≤^𝑀_𝑚−0𝐸_𝜆1𝑥−2𝑥,𝑦𝜆⁻¹≤1𝑑𝜆2‖𝑥𝑦𝑀‖ln𝑚. (5.1)

The proof is obvious from Theorems 3.1, 3.2, and 4.1 applied for the logarithmic function. The details are omitted.

(2) Consider now the power function 𝑓∶(0,∞)→ℝ, 𝑓(𝑡)=𝑡^𝑝 with 𝑝∈(−∞,0)∪(0,∞). In the case when 𝑝∈(0,1), the function is 𝑝-𝐻-Hölder continuous with 𝐻=1 on any subinterval [𝑚,𝑀] of [0,∞). By making use of Theorem 4.2 we can state the following result.

Proposition 5.2. 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 {𝐸_𝜆}_𝜆 be its spectral family. Then for 𝑝∈(0,1) one has 𝑚^𝑝+𝑀^𝑝2⋅𝑥,𝑦−𝐴^𝑝≤1𝑥,𝑦2^𝑝(𝑀−𝑚)^p𝑀_𝑚𝐸_(⋅)≤1𝑥,𝑦2^𝑝(𝑀−𝑚)^𝑝‖𝑥‖𝑦, (5.2) for any 𝑥,𝑦∈𝐻.

The case of powers 𝑝≥1 is embodied in the following.

Proposition 5.3. 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 {𝐸_𝜆}_𝜆 be its spectral family. Then for 𝑝≥1  and for any 𝑥,𝑦∈𝐻 one has𝑚^𝑝+𝑀^𝑝2⋅𝑥,𝑦−𝐴^𝑝≤𝑀𝑥,𝑦_𝑝−𝑚^𝑝×12max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2max_{[]𝜆∈𝑚,𝑀}𝐸_𝜆1𝑥−2≤1𝑥,𝑦2‖𝑥‖𝑦𝑀^𝑝−𝑚^𝑝, (5.3)𝑚^𝑝+𝑀^𝑝2⋅𝑥,𝑦−𝐴^𝑝𝑥,𝑦≤𝑝𝑀^𝑝−1×12^𝑀_𝑚−0𝐸_𝜆𝑥,𝑥^1/2𝐸_𝜆𝑦,𝑦^1/2+1_𝐻−𝐸_𝜆𝑥,𝑥^1/21_𝐻−𝐸_𝜆𝑦,𝑦^1/2𝑑𝜆_𝑀𝑚−0𝐸_𝜆1𝑥−2≤1𝑥,𝑦𝑑𝜆2𝑝‖𝑥‖𝑦𝑀^𝑝−1, (5.4)𝑚^𝑝+𝑀^𝑝2⋅𝑥,𝑦−𝐴^𝑝𝑥,𝑦≤𝑝^𝑀_𝑚−0𝐸_𝜆1𝑥−2𝑥,𝑦𝜆^𝑝−1≤1𝑑𝜆2‖𝑥‖𝑦𝑀^𝑝−𝑚^𝑝. (5.5)

The proof is obvious from Theorems 3.1, 3.2, and 4.1 applied for the power function 𝑓∶(0,∞)→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.

Acknowledgment

The author would like to thank the anonymous referees for their valuable comments that have been implemented in the final version of the paper.