# New Čebyšev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

Authors: Mohammad W. Alomari
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

Several new error bounds for the Čebyšev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well.


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## Body

## 1. Introduction

In recent years the approximation problem of the Riemann-Stieltjes integral∫ab‍f(t)du(t) via the famous Čebyšev functional
(1)T(f,g)=1b-a∫ab‍f(t)g(t)dt-1b-a∫ab‍f(t)dt·1b-a∫ab‍g(t)dt
increasingly became essential. In 1882, Čebyšev [1] derived an interesting result involving two absolutely continuous functions whose first derivatives are continuous and bounded and is given by
(2)|T(f,g)|≤112(b-a)2∥f′∥∞∥g′∥∞,
and the constant 1/12 is the best possible.In 1935, Grüss [2] proved another result for two integrable mappings f, g such that ϕ≤f(x)≤Φ and γ≤f(x)≤Γ; the inequality
(3)|T(f,g)|≤14(Φ-ϕ)(Γ-γ)
holds, and the constant 1/4 is the best possible.In [3, p 302] Beesack et al. have proved the following Čebyšev inequality for absolutely continuous functions whose first derivatives belong to Lp spaces:
(4)|T(f,g)|≤b-a4(2p-1p(p+1))1/p(2q-1q(q+1))1/q∥f′∥p∥g′∥q,
where ∥h∥p:=(∫ab‍|h(t)|pdt)1/p, ∀p>1, and (1/p)+(1/q)=1.For the constant(5)ω(p,q):=14(2p-1p(p+1))1/p(2q-1q(q+1))1/q
we have
(6)18≤ω(p,q)≤14
for all q=p/(p-1), p>1. Furthermore, we have the following particular cases in (4).(1)
Ifp=q=2, we have
(7)|T(f,g)|≤b-a8∥f′∥2∥g′∥2.(2)
Ifq→∞, we have
(8)|T(f,g)|≤b-a4∥f′∥1∥g′∥∞.In 1970, Ostrowski [4] has proved the following combination of the Čebyšev and Grüss results:
(9)|T(f,g)|≤18(b-a)(M-m)∥g′∥∞,
where g is absolutely continuous with g′∈L∞[a,b] and f is Lebesgue integrable on [a,b] and satisfying m≤f(t)≤M, for all t∈[a,b]. The constant 1/8 is the best possible.In 1973, Lupaş [5] has improved Beesack et al. inequality (7), as follows:
(10)|T(f,g)|≤(b-a)π2∥f′∥2∥g′∥2,
provided that f, g are two absolutely continuous functions on [a,b] with f′,g′∈L2[a,b], where ∥h′∥2:=(∫ab‍|h′(t)|2dt)1/2. The constant 1/π2 is the best possible.More recently, and using the identity ([3], page 246),
(11)T(f,g)=1b-a∫ab‍[f(t)-f(a)+f(b)2]mmmmm×[g(t)-1b-a∫abg(s)ds]dt.
Dragomir [6] has proved the following inequality.Theorem 1.
Letf:[a,b]→C be of bounded variation on [a,b] and g:[a,b]→C a Lebesgue integrable function on [a,b]; then
(12)|T(f,g)|≤12⋁ab(f)·1b-a∫ab|g(t)-1b-a∫ab‍g(s)ds|dt,
where ⋁ab(f) denotes the total variation of f on the interval [a,b]. The constant 1/2 is best possible in (12).Another result when both functions are of bounded variation was considered in the same paper [6], as follows.Theorem 2.
Iff,g:[a,b]→C are of bounded variation on [a,b], then
(13)|T(f,g)|≤14⋁ab(f)·⋁ab(g).
The constant 1/4 is best possible in (13).Many authors have studied the functional (1) and, therefore, several bounds under various assumptions have been obtained; for more new results and generalizations the reader may refer to [6–21].On other hand and in order to study the difference between two Riemann integral means, Barnett et al. [22] have proved the following estimates.Theorem 3.
Letf:[a,b]→R be an absolutely continuous function with the property that f′∈L∞[a,b]; that is,
(14)∥f′∥∞:=esssup⁡t∈[a,b]|f′(t)|.
Then for a≤c<d≤b, we have the inequality
(15)|1b-a∫ab‍f(t)dt-1d-c∫cd‍f(s)ds|≤[14+((a+b)/2-(c+d)/2(b-a)-(d-c))2]×[(b-a)-(d-c)]∥f′∥∞≤12[(b-a)-(d-c)]∥f′∥∞.
The constant1/4 in the first inequality and 1/2 in the second inequality are the best possible.After that, Cerone and Dragomir [23] have obtained the following three results as well.Theorem 4.
Letf:[a,b]→R be an absolutely continuous mapping. Then for a≤c<d≤b, we have the inequalities
(16)|1b-a∫ab‍f(t)dt-1d-c∫cd‍f(s)ds|≤{(b-a)(q+1)1/q[1+(ρ1+ρ)q]1/q×[vq+1+λq+1]1/q∥f′∥p,f′∈Lp[a,b],1≤p<∞,1p+1q=1;12[1-ρ+|v-λ|]∥f′∥1,f′∈L1[a,b],
where (b-a)v=c-a, (b-a)ρ=d-c and (b-a)λ=b-d. Both inequalities in (16) are sharp.Theorem 5.
Assume that the mappingf:[a,b]→R is of r-H-Hölder type on [a,b]. Then for a≤c<d≤b, we have the inequality
(17)|1b-a∫ab‍f(t)dt-1d-c∫cd‍f(s)ds|≤H(c-a)r+1+(b-d)r+1(r+1)[(b-a)-(d-c)].
Inequality (17) is best possible in the sense that we cannot put in the right-hand side a constant less than 1.Theorem 6.
Letf:[a,b]→R be a mapping of bounded variation on [a,b]. The following bounds hold:
(18)|1b-a∫ab‍f(t)dt-1d-c∫cd‍f(s)ds|≤{[b-a-(d-c)2+|c+d2-a+b2|]∨ab(f)b-a;L(c-a)2+(b-d)22[(b-a)-(d-c)];iffisL-Lipschitzian(b-db-a)f(b)-(c-ab-a)f(a)+[c+d-(a+b)b-a]f(s0);iffismonotonicnondecreasing,
where s0=(cb-ad)/((b-a)-(d-c))∈[c,d].In this paper by utilising amongst others the inequalities from Theorems3–6, several new bounds for theČebyšev functionalT(f,g) are provided.The inequalities (15)–(18) are used in an essential way to obtain new error bounds for the T(f,g), which gives a significant application for these inequalities. Applications for functions of self-adjoint operators on complex Hilbert Spaces are provided as well.


## 2. The Case Whenf Is of Bounded Variation

We may start with the following result.Theorem 7.
Letf,g:[a,b]→R be such that f is of bounded variation on [a,b] and g is absolutely continuous on [a,b]; then
(19)|T(f,g)|≤⋁ab(f)·{18(b-a)∥g′∥∞,g′∈L∞[a,b];12(1+(1/2q)2(q+1))1/q∥g′∥p,g′∈Lp[a,b];18∥g′∥1,g′∈L1[a,b],
where ∥·∥p are the usual Lebesgue norms; that is,
(20)∥h∥p:=(∫ab‍|h(t)|pdt)1/p,forp≥1,∥h∥∞:=esssup⁡t∈[a,b]|h(t)|.Proof.
Using integration by parts, we have(21)T(f,g)=-1b-a∫ab‍(∫at‍g(s)ds-t-ab-a∫ab‍g(s)ds)df(t).
It is known that for a continuous function p:[a,b]→R and a function ν:[a,b]→R of bounded variation, the Riemann-Stieltjes integral ∫ab‍p(t)dν(t) exists and one has the inequality
(22)|∫ab‍p(t)dν(t)|≤sup⁡t∈[a,b]|p(t)|⋁ab(ν).
As f is of bounded variation on [a,b], by (22) we have
(23)|-1b-a∫ab‍(∫at‍g(s)ds-t-ab-a∫ab‍g(s)ds)df(t)|≤1b-asup⁡|∫at‍g(s)ds-t-ab-a∫ab‍g(s)ds|·⋁ab(f)=1b-asup⁡t∈[a,b]|∫ab‍g(s)ds(t-a)[∫ab‍g(s)ds1t-a∫at‍g(s)dsmmmmmmmmmcmmmm-1b-a∫ab‍g(s)ds∫at‍]|⋁ab(f).
In the inequality (15), setting d=t and c=a, we get
(24)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤12(b-t)∥g′∥∞.
Substituting (24) into (23), we get
(25)|T(f,g)|≤1b-a·12∥g′∥∞sup⁡t∈[a,b]{(t-a)(b-t)}⋁ab(f)=18(b-a)∥g′∥∞⋁ab(f),
since sup⁡t∈[a,b]⁡{(t-a)(b-t)}, occurs at t=(a+b)/2, therefore, sup⁡t∈[a,b]⁡{(t-a)(b-t)}=(1/4)(b-a)2, which proves the first inequality in (19).
In the inequality (16), setting d=t and c=a, we get
(26)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤{(b-t)1+(1/q)(q+1)1/q(b-a)1/q×[1+(t-ab+t-2a)q]1/q∥g′∥p,g′∈Lp[a,b],12b-tb-a∥g′∥1,g′∈L1[a,b].
Substituting (26) into (23), we get
(27)|T(f,g)|≤1b-a⋁ab(f)×sup⁡t∈[a,b](t-a)·{(b-t)1+(1/q)(q+1)1/q(b-a)1/q×[1+(t-ab+t-2a)q]1/q∥g′∥p,g′∈Lp[a,b],12b-tb-a∥g′∥1,g′∈L1[a,b]≤⋁ab(f)×{[1+(1/2)q]1/q21+1/q(1+q)1/q∥g′∥p,g′∈Lp[a,b];18∥g′∥1,g′∈L1[a,b],
where p>1 and (1/p)+(1/q)=1, which proves the second and the third inequalities in (19).Another result wheng is of r-H-Hölder type is as follows.Theorem 8.
Letf,g:[a,b]→R be such that f is of bounded variation on [a,b] and g is of r-H-Hölder type on [a,b]; then
(28)|T(f,g)|≤Hrr(r+1)r+1(b-a)r⋁ab(f).Proof.
Asf is of bounded variation and g is of r-H-Hölder type on [a,b], by (23) and using (17) we have
(29)|T(f,g)|≤1b-asup⁡t∈[a,b]|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmmmmmmmmmm-1b-a∫ab‍g(u)du∫at‍]|⋁ab(f)≤1b-aHr+1sup⁡t∈[a,b](t-a)(b-t)r⋁ab(f)=Hrr(r+1)r+1(b-a)r⋁ab(f),
since sup⁡t∈[a,b]⁡(t-a)(b-t)r=(rr/(r+1)r+1)(b-a)r+1, which completes the proof.Theorem 9.
Letf,g:[a,b]→R be such that f is of bounded variation on [a,b] and g is monotonic nondecreasing on [a,b]; then
(30)|T(f,g)|≤16(b-a)[g(b)-g(a)]⋁ab(f).Proof.
Asf and g is of bounded variation on [a,b] and g is monotonic nondecreasing on [a,b], by (22) and using (23) we have
(31)|T(f,g)|≤1b-a∫ab‍|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmcmmmmmm-1b-a∫ab‍g(u)du∫at‍]|dt⋁ab(f).
In the third part of inequality (18), setting d=t and c=a, we get
(32)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤b-tb-a[g(b)-g(a)].
Substituting (32) into (31), we get
(33)|T(f,g)|≤1b-a·g(b)-g(a)b-a∫ab‍(t-a)(b-t)dt⋁ab(f)=16(b-a)[g(b)-g(a)]⋁ab(f),
and thus the proof is finished.


## 3. The Case Whenf Is Lipschitzian

In this section, we give some new bounds whenf is L-Lipschitzian.Theorem 10.
Letf,g:[a,b]→R be such that f is L-Lipschitzian on [a,b] and g is an absolutely continuous on [a,b]; then
(34)|T(f,g)|≤L{112(b-a)2∥g′∥∞,g′∈L∞[a,b];q2(1+(1/2q))1/q(2q+1)(3q+1)(q+1)1/q×(b-a)2∥g′∥p,g′∈Lp[a,b];112(b-a)∥g′∥1,g′∈L1[a,b],
where p>1 and (1/p)+(1/q)=1.Proof.
Using the fact that for a Riemann integrable functionp:[c,d]→R and L-Lipschitzian function ν:[c,d]→R, one has the inequality
(35)|∫cd‍p(t)dν(t)|≤L∫cd‍|p(t)|dt.
As f′ is L-Lipschitzian on [a,b], by (35) we have
(36)|T(f,g)|≤Lb-a∫ab|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)du‍mmmmmcmmmmm-1b-a∫ab‍g(u)du∫at‍]|dt≤12(b-a)L∥g′∥∞∫ab‍(t-a)(b-t)dt=112L(b-a)2∥g′∥∞,
where for the last inequality we used the inequality (15), with d=t and a=c, (see (24)).
In the inequality (16), setting d=t and a=c, we get
(37)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤{(b-t)1+(1/q)(q+1)1/q(b-a)1/q×[1+(t-ab+t-2a)q]1/q∥g′∥pg′∈Lp[a,b],1≤p<∞,1p+1q=1;12b-tb-a∥g′∥1,g′∈L1[a,b].
Substituting (37) into (36), we get
(38)|T(f,g)|≤Lb-a{∥g′∥p(q+1)1/q(b-a)1/q×∫ab‍(t-a)(b-t)1+(1/q)mmmm×[1+(t-ab+t-2a)q]1/qdt,mmmmmcmcmmg′∈Lp[a,b],12∥g′∥1b-a∫ab‍(t-a)(b-t)dt,mmmmmcmcmmg′∈L1[a,b],≤Lb-a{∥g′∥p(q+1)1/q(b-a)1/q×sup⁡t∈[a,b][1+(t-ab+t-2a)q]1/q×∫ab‍(t-a)(b-t)1+(1/q)dt,mmmcmmmmg′∈Lp[a,b],112(b-a)2∥g′∥1,mmmcmmmmg′∈L1[a,b],=L{∥g′∥p(q+1)1/q[1+12q]1/q×(b-a)2q2(2q+1)(3q+1),g′∈Lp[a,b],112(b-a)∥g′∥1,g′∈L1[a,b],
where sup⁡t∈[a,b]⁡[1+((t-a)/(b+t-2a))q]1/q=(1+(1/2q))1/q, and
(39)∫ab‍(t-a)(b-t)1+(1/q)dt=(b-a)3+(1/q)∫01‍(1-t)t1+(1/q)dt=(b-a)3+(1/q)q2(2q+1)(3q+1),
which proves the second and the third inequalities in (34).Theorem 11.
Letf,g:[a,b]→R be such that f is L-Lipschitzian on [a,b] and g is of r-H-Hölder type on [a,b]; then
(40)|T(f,g)|≤LH(b-a)r+1(r+1)2(r+2).Proof.
Asf is L-Lipschitzian and g is of r-H-Hölder type on [a,b], by (36) and using (17) we have
(41)|T(f,g)|≤Lb-a∫ab‍|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmmmmmcmm-1b-a∫ab‍g(u)du∫at‍]|dt≤LH(r+1)(b-a)∫ab‍(t-a)(b-t)rdt=LH(b-a)r+1(r+1)2(r+2),
where, for the last inequality, a simple calculation yields that
(42)∫ab‍(t-a)(b-t)rdt=(b-a)r+2∫01‍(1-t)trdt=(b-a)r+2(r+1)(r+2),
which completes the proof.Corollary 12.
In Theorem11, if g is M-Lipschitzian on [a,b], then we have
(43)|T(f,g)|≤112LM(b-a)2.Theorem 13.
Letf,g:[a,b]→R be such that f is of bounded variation on [a,b] and g is monotonic nondecreasing on [a,b]; then
(44)|T(f,g)|≤16L(b-a)2[g(b)-g(a)].Proof.
Asf is L-Lipschitzian on [a,b] and g is monotonic nondecreasing on [a,b], by (35) and using (36) we have
(45)|T(f,g)|≤Lb-a∫ab‍|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmmmmmm-1b-a∫ab‍g(u)du∫at‍]|dt.
In the third part of inequality (18), setting d=t and c=a, we get
(46)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤b-tb-a[g(b)-g(a)].
Substituting (46) into (45), we get
(47)|T(f,g)|≤Lb-ag(b)-g(a)b-a∫ab‍(t-a)(b-t)dt=16L(b-a)[g(b)-g(a)],
which completes the proof.


## 4. More Inequalities

In this section we give other related results.Theorem 14.
Letf,g:[a,b]→R be such that f and g are of bounded variation on [a,b]; then
(48)|T(f,g)|≤14⋁ab(g)⋁ab(f).
The constant 1/4 is the best possible.Proof.
Asf and g is of bounded variation on [a,b], by (22) and using (23), we have
(49)|T(f,g)|≤1b-asup⁡t∈[a,b]|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmmmmmmmm-1b-a∫ab‍g(u)du∫at‍]|⋁ab(f).
In the first inequality of (18), setting d=t and c=a, we get
(50)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤b-tb-a⋁ab(g).
Substituting (50) into (49), we get
(51)|T(f,g)|≤1(b-a)2sup⁡t∈[a,b]{(t-a)(b-t)}⋁ab(g)⋁ab(f)=14⋁ab(g)⋁ab(f),
which proves the inequality. The sharpness case trivially holds by taking f(t)=g(t)=sgn⁡(t-((a+b)/2)), which completes the proof.When the integrator is of bounded variation we have the following.Theorem 15.
Letf,g:[a,b]→R be such that f is L-Lipschitzian on [a,b] and g is of bounded variation on [a,b]; then
(52)|T(f,g)|≤16L(b-a)⋁ab(g).Proof.
Asf is L-Lipschitzian on [a,b] and g is of bounded variation on [a,b], by (22) and using (23), we have
(53)|T(f,g)|≤Lb-a∫ab‍|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)dummmmmlllmmmmm-1b-a∫ab‍g(u)du∫at‍]|dt.
In the second inequality of (18), setting d=t and c=a, we get
(54)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤b-tb-a⋁ab(g).
Substituting (54) into (53), we get
(55)|T(f,g)|≤L(b-a)2∫ab‍(t-a)(b-t)dt⋁ab(g)=16L(b-a)⋁ab(g),
and the proof is completed.When both functions are Lipschitzian we have the following.Theorem 16.
Letf,g:[a,b]→R be, respectively, such that f and g are L1- and L2-Lipschitzian on [a,b]; then
(56)|T(f,g)|≤112L1L2(b-a)2.
The constant 1/12 is the best possible.Proof.
Asf and g are L1- and L2-Lipschitzian on [a,b], respectively, by (35) and using (36), we have
(57)|T(f,g)|≤L1b-a∫ab|∫ab‍(t-a)[∫ab‍1t-a∫at‍g(u)du‍mmmmmmmmmmm-1b-a∫ab‍g(u)du∫at‍]|dt.
In the second inequality of (18), setting d=t and c=a, we get
(58)|1t-a∫at‍g(s)ds-1b-a∫ab‍g(s)ds|≤12L2(b-t).
Substituting (58) into (57), we get
(59)|T(f,g)|≤12(b-a)L1L2∫ab‍(t-a)(b-t)dt=112L1L2(b-a)2,
which proves the inequality. The sharpness case trivially holds by taking f(t)=g(t)=t, which completes the proof.Remark 17.
Letg be as in Theorems 7–16. By applying the same techniques used in the corresponding proofs of each theorem, we may obtain several inequalities for monotonic nondecreasing integrator f using the fact that for a monotonic nondecreasing function ν:[a,b]→R and continuous function p:[a,b]→R, one has the inequality
(60)|∫ab‍p(t)dν(t)|≤∫ab‍|p(t)|dν(t).
We leave the details to the interested reader.


## 5. Applications for Self-Adjoint Operators

We denote byB(H) the Banach algebra of all bounded linear operators on a complex Hilbert space (H;〈·,·〉). Let A∈B(H) be self-adjoint and let φλ be defined for all λ∈R as follows:
(61)φλ(s):={1,for-∞<s≤λ,0,forλ<s<+∞.
Then for every λ∈R the operator
(62)Eλ:=φλ(A)
is a projection which reduces A.The properties of these projections are collected in the following fundamental result concerning the spectral representation of bounded self-adjoint operators in Hilbert spaces; see for instance [24, page 256].Theorem 18 (Spectral Representation Theorem).
LetA be a bonded self-adjoint operator on the Hilbert space H and let m=min⁡{λ∣λ∈Sp(A)}=:minSp(A) and M=max⁡{λ∣λ∈Sp(A)}=:maxSp(A). Then there exists a family of projections {Eλ}λ∈R, called the spectral family of A, with the following properties:(a)
Eλ≤Eλ′ for λ≤λ′;(b)
Em-0=0, EM=I and Eλ+0=Eλ for all λ∈R;(c)
we have the representation(63)A=∫m-0M‍λdEλ.
More generally, for every continuous complex-valued functionφ defined on R and for every ɛ>0 there exists a δ>0 such that
(64)∥φ(A)-∑k=1n‍φ(λk′)(Eλk-Eλk-1)∥≤ɛ
whenever
(65)λ0<m=λ1<⋯<λn-1<λn=M,λk-λk-1≤δfor1≤k≤n,λk′∈[λk-1,λk]for1≤k≤n
this means that
(66)φ(A)=∫m-0M‍φ(λ)dEλ,
where the integral is of Riemann-Stieltjes type.Corollary 19.
With the assumptions of Theorem18 for A, Eλ, and φ we have the representations
(67)φ(A)x=∫m-0M‍φ(λ)dEλx∀x∈H,〈φ(A)x,y〉=∫m-0M‍φ(λ)d〈Eλx,y〉∀x,y∈H.
In particular,
(68)〈φ(A)x,x〉=∫m-0M‍φ(λ)d〈Eλx,x〉∀x∈H.
Moreover, we have the equality
(69)∥φ(A)x∥2=∫m-0M‍|φ(λ)|2d∥Eλx∥2∀x∈H.We recall the following result (see [25]) that provides an upper bound for the total variation of the function R∋λ↦〈Eλx,y〉∈C on an interval [α,β].Theorem 20.
Let{Eλ}λ∈R be the spectral family of the bounded self-adjoint operator A and let m=minSp(A) and M=maxSp(A). Then for any x,y∈H and α<β we have the inequality
(70)[⋁αβ(〈E(·)x,y〉)]2≤〈(Eβ-Eα)x,x〉〈(Eβ-Eα)y,y〉,
where ⋁αβ(〈E(·)x,y〉) denotes the total variation of the function 〈E(·)x,y〉 on [α,β].Remark 21.
Forα=m-ɛ with ɛ>0 and β=M we get from (70) the inequality
(71)⋁m-ɛM(〈E(·)x,y〉)≤〈(1H-Em-ɛ)x,x〉1/2〈(1H-Em-ɛ)y,y〉1/2
for any x,y∈H.
This implies, for anyx,y∈H, that
(72)⋁m-0M(〈E(·)x,y〉)≤∥x∥∥y∥,
where ⋁m-0M(〈E(·)x,y〉) denotes the limit lim⁡ɛ→0+[⋁m-ɛM(〈E(·)x,y〉)].The inequality (72) was also proved in the recent monographs [26, 27] and will be utilized in the following.After these preparations we can state and prove the following trapezoidal type inequality for functions of self-adjoint operators on Hilbert spaces.Theorem 22.
LetA be a bonded self-adjoint operator on the Hilbert space H and let m=min⁡{λ∣λ∈Sp(A)}=:min⁡Sp(A) and M=max⁡{λ∣λ∈Sp(A)}=:max⁡Sp(A). If f:[m,M]→C is such that its derivative f is of bounded variation on [m,M], then we have the inequality
(73)|〈[f(A)-f(m)(M1H-A)+f(M)(A-m1H)M-m]x,y〉|≤14(M-m)⋁mM(f)⋁m-0M(〈E(·)x,y〉)≤14(M-m)⋁mM(f)∥x∥∥y∥
for any x,y∈H.Proof.
Utilising the inequality (48) for the function of bounded variation g(λ)=〈Eλx,y〉 and the continuous function f we have
(74)|∫m-0M‍f(λ)d〈Eλx,y〉-f(m)∫m-0M‍〈Eλx,y〉dλM-mc-f(M)(〈x,y〉-∫m-0M‍〈Eλx,y〉dλM-m)|≤14(M-m)⋁mM(f)⋁m-0M(〈E(·)x,y〉)≤14(M-m)⋁mM(f)∥x∥∥y∥
for any x,y∈H.
By the Spectral Representation Theorem we have(75)∫m-0M‍f(λ)d〈Eλx,y〉=〈f(A)x,y〉,∫m-0M‍〈Eλx,y〉dλ=〈Eλx,y〉λ|m-0M-∫m-0M‍λd〈Eλx,y〉=M〈x,y〉-〈Ax,y〉=〈(M1H-A)x,y〉
for any x,y∈H.
Substituting these values into (74) we deduce after simple calculations the desired result (73).The above inequality (73) can be utilized for different particular functions of interest in Operator Theory, such as the power, logarithmic, and exponential functions.If we takef(t)=tp with p≥1, then for any positive operator A with Sp(A)⊂[m,M]⊂[0,∞) we have the inequality
(76)|〈[Ap-mp(M1H-A)+Mp(A-m1H)M-m]x,y〉|≤14(M-m)(Mp-mp)⋁m-0M(〈E(·)x,y〉)≤14(M-m)(Mp-mp)∥x∥∥y∥
for any x,y∈H.If we take the functionf(t)=ln⁡t, then for any positive definite operator A with Sp(A)⊂[m,M]⊂(0,∞) we have the inequality
(77)|〈[ln⁡A-ln⁡m(M1H-A)+ln⁡M(A-m1H)M-m]x,y〉|≤ln⁡(Mm)(M-m)/4⋁m-0M(〈E(·)x,y〉)≤ln⁡(Mm)(M-m)/4∥x∥∥y∥
for any x,y∈H.Finally, if we takef(t)=exp⁡(t), then we have for any self-adjoint operator A with Sp(A)⊂[m,M]⊂R the inequality
(78)|〈[exp⁡(A)-exp⁡(m)(M1H-A)+exp⁡(M)(A-m1H)M-mmm-exp⁡(m)(M1H-A)+exp⁡(M)(A-m1H)M-m]x,y〉|≤14(M-m)(exp⁡M-exp⁡m)⋁m-0M(〈E(·)x,y〉)≤14(M-m)(exp⁡M-exp⁡m)∥x∥∥y∥
for any x, y∈H.



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*Source: 363050-2014-06-05.xml*