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

Authors: S. S. Dragomir
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

In classical analysis atrapezoidal type inequality is an inequality that provides upper and/or lower bounds for the quantity (1.1)f(a)+f(b)2(b-a)-∫abf(t)dt,that is,  the error in approximating the integral by a trapezoidal rule, for various classes of integrable functionsf 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.
Letf:[a,b]→ℂ be a function of bounded variation. One has the inequality
(1.2)|∫abf(t)dt-f(a)+f(b)2(b-a)|≤12(b-a)⋁ab(f),
where ∨ab(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 off as follows (see [1, p. 76]).Theorem 1.2.
Letf:[a,b]→ℝ be a monotonic nondecreasing function on [a,b]. Then one has the inequalities
(1.3)|∫abf(t)dt-f(a)+f(b)2(b-a)|≤12(b-a)[f(b)-f(a)]-∫absgn⁡(t-a+b2)f(t)dt≤12(b-a)[f(b)-f(a)].
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.
Letf:[a,b]→ℂ be an L-Lipschitzian function on [a,b], that is, f satisfies the condition
(L)|f(s)-f(t)|≤L|s-t|foranys,t∈[a,b](L>0isgiven).
Then one has the inequality
(1.4)|∫abf(t)dt-f(a)+f(b)2(b-a)|≤14(b-a)2L.
The constant 1/4 is best in (1.4).If we would assume absolute continuity for the functionf, then the following estimates in terms of the Lebesgue norms of the derivative f′ hold ([1, p. 93]).Theorem 1.4.
Letf:[a,b]→ℂ be an absolutely continuous function on [a,b]. Then one has
(1.5)|∫abf(t)dt-f(a)+f(b)2(b-a)|≤{14(b-a)2‖f′‖∞iff′∈L∞[a,b],12(q+1)1/q(b-a)1+1/q‖f′‖piff′∈Lp[a,b],p>1,1p+1q=1,12(b-a)‖f′‖1,
where ∥·∥p(p∈[1,∞]) are the Lebesgue norms, that is,
(1.6)‖f′‖∞=esssup⁡s∈[a,b]|f′(s)|,‖f′‖p:=(∫ab|f′(s)|ds)1/p,p≥1.The case of convex functions is as follows [4].Theorem 1.5.
Letf:[a,b]→ℝ be a convex function on [a,b]. Then one has the inequalities
(1.7)18(b-a)2[f+′(a+b2)-f-′(a+b2)]≤f(a)+f(b)2(b-a)-∫abf(t)dt≤18(b-a)2[f-′(b)-f+′(a)].
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.LetA 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 1H on H as follows (see for instance [5, page 3]):For anyf,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∥:=supt∈Sp(A)|f(t)|;(iv)
Φ(f0)=1H and Φ(f1)=A, where f0(t)=1 and f1(t)=t, for t∈Sp(A).With this notation we define(2.1)f(A):=Φ(f)∀f∈C(Sp(A))and we call it thecontinuous functional calculus for a selfadjoint operator A.IfA 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:Pf(t)≥g(t)for anyt∈Sp(A)impliesthatf(A)≥g(A)in the operator order ofB(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].LetU 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 {Eλ}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:(2.2)〈f(U)x,y〉=∫m-0Mf(λ)d(〈Eλx,y〉),
for any x,y∈H. The function gx,y(λ):=〈Eλx,y〉 is of bounded variation on the interval [m,M] and (2.3)gx,y(m-0)=0,gx,y(M)=〈x,y〉for anyx,y∈H. It is also well-known that gx(λ):=〈Eλx,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(2.4)f(M)+f(m)2⋅〈x,y〉-〈f(A)x,y〉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.
LetA be a selfadjoint operator in the Hilbert space H with the spectrum Sp(A)⊆[m,M] for some real numbers m<M and let {Eλ}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
(3.1)|f(M)+f(m)2⋅〈x,y〉-〈f(A)x,y〉|≤12max⁡λ∈[m,M][〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]⋁mM(f)≤12‖x‖‖y‖mM(f)
for any x,y∈H.Proof.
Iff,u:[m,M]→ℂ are such that the Riemann-Stieltjes integral ∫abf(t)du(t) exists, then a simple integration by parts reveals the identity
(3.2)∫abf(t)du(t)=f(a)+f(b)2[u(b)-u(a)]-∫ab[u(t)-u(a)+u(b)2]df(t).
If we write the identity (3.2) for u(λ)=〈Eλx,y〉, then we get
(3.3)∫m-0Mf(λ)d(〈Eλx,y〉)=f(m)+f(M)2⋅〈x,y〉-∫m-0M(〈Eλx,y〉-12〈x,y〉)df(λ)
which, by (2.2), gives the following identity of interest in itself
(3.4)f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉=12∫m-0M[〈Eλx,y〉+〈(Eλ-1H)x,y〉]df(λ),
for any x,y∈H.
It is well-known that ifp:[a,b]→ℂ is a continuous function and v:[a,b]→ℂ is of bounded variation, then the Riemann-Stieltjes integral ∫abp(t)dv(t) exists and the following inequality holds:
(3.5)|∫abp(t)dv(t)|≤max⁡t∈[a,b]|p(t)|⋁ab(v),
where ∨ab(v)  denotes the total variation of v on [a,b].
Utilizing the property (3.5), we have from (3.4) that
(3.6)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤12max⁡λ∈[m,M]⁡|〈Eλx,y〉+〈(Eλ-1H)x,y〉|⋁mM(f)≤12[max⁡λ∈[m,M][|〈Eλx,y〉|+|〈(1H-Eλ)x,y〉|]]⋁mM(f).
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:
(3.7)|〈Px,y〉|2≤〈Px,x〉〈Py,y〉,
for any x,y∈H.
On applying the inequality (3.7) we have
(3.8)|〈Eλx,y〉|≤〈Eλx,x〉1/2〈Eλy,y〉1/2,|〈(1H-Eλ)x,y〉|≤〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2,
which, together with the elementary inequality for a,b,c,d≥0(3.9)ab+cd≤(a2+c2)1/2(b2+d2)1/2
produce the inequalities
(3.10)|〈Eλx,y〉|+|〈(1H-Eλ)x,y〉|≤〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2≤(〈Eλx,x〉+〈(1H-Eλ)x,x〉)1/2(〈Eλy,y〉+〈(1H-Eλ)y,y〉)1/2=‖x‖‖y‖
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.
LetA be a selfadjoint operator in the Hilbert space H with the spectrum Sp(A)⊆[m,M] for some real numbers m<M and let {Eλ}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
(3.11)|f(M)+f(m)2⋅〈x,y〉-〈f(A)x,y〉|≤12L∫m-0M[〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]dλ≤12(M-m)L‖x‖‖y‖
for any x,y∈H.Proof.
It is well-known that ifp:[a,b]→ℂ is a Riemann integrable function and v:[a,b]→ℂ is Lipschitzian with the constant L>0, that is,
(3.12)|v(s)-v(t)|≤L|s-t|foranyt,s∈[a,b],
then the Riemann-Stieltjes integral ∫abp(t)dv(t) exists and the following inequality holds:
(3.13)|∫abp(t)dv(t)|≤L∫ab|p(t)|dt.
Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation (3.4) that
(3.14)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤12L∫m-0M|〈Eλx,y〉+〈(Eλ-1H)x,y〉|dλ,≤12L∫m-0M[|〈Eλx,y〉|+|〈(1H-Eλ)x,y〉|]dλ,
for any x,y∈H.
Further, integrating (3.10) on [m,M] we have
(3.15)∫m-0M[|〈Eλx,y〉|+|〈(1H-Eλ)x,y〉|]dλ≤∫m-0M[〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]dλ≤(M-m)‖x‖‖y‖
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.
LetA be a selfadjoint operator in the Hilbert space H with the spectrum Sp(A)⊆[m,M]  for some real numbers m<M and let {Eλ}λ be its spectral family.  Assume that f:[m,M]→ℂ is a continuous function on [m,M]. Then one has the inequalities
(4.1)|f(M)+f(m)2⋅〈x,y〉-〈f(A)x,y〉|≤{max⁡λ∈[m,M]|〈Eλx-12x,y〉|⋁mM(f)iffisofboundedvariation,L∫m-0M|〈Eλx-12x,y〉|dλiffisLLipschitzian,∫m-0M|〈Eλx-12x,y〉|df(λ)iffisnondecreasing,≤12‖x‖‖y‖{⋁mM(f)iffisofboundedvariation,L(M-m)iffisLLipschitzian,(f(M)-f(m))iffisnondecreasing
for any x,y∈H.Proof.
From (3.6) we have that
(4.2)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤12max⁡λ∈[m,M]|〈Eλx,y〉+〈(Eλ-1H)x,y〉|⋁mM(f)=max⁡λ∈[m,M]|〈Eλx-12x,y〉|⋁mM(f)
for any x,y∈H.
Utilizing the Schwarz inequality inH and the fact that Eλ are projectors we have successively
(4.3)|〈Eλx-12x,y〉|≤‖Eλx-12x‖‖y‖=[〈Eλx,Eλx〉-〈Eλx,x〉+14‖x‖2]1/2‖y‖=12‖x‖‖y‖
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 ifp:[a,b]→ℂ is of bounded variation and v:[a,b]→ℝ is continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals ∫abp(t)dv(t) and ∫ab|p(t)|dv(t) exist and
(4.4)|∫abp(t)dv(t)|≤∫ab|p(t)|dv(t).
From the representation (3.4) we then have
(4.5)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤12∫m-0M|〈Eλx,y〉+〈(Eλ-1H)x,y〉|df(λ)=∫m-0M|〈Eλx-12x,y〉|df(λ)
for any x,y∈H, from which we obtain the last branch in (4.1).We recall that a functionf:[a,b]→ℂ is called r-H-Hölder continuous with fixed r∈(0,1] and H>0 if (4.6)|f(t)-f(s)|≤H|t-s|rforanyt,s∈[a,b].We have the following result concerning this class of functions.Theorem 4.2.
LetA be a selfadjoint operator in the Hilbert space H with the spectrum Sp(A)⊆[m,M] for some real numbers m<M and let {Eλ}λ be its spectral family.  If f:[m,M]→ℂ is r-H-Hölder continuous on [m,M], then one has the inequality
(4.7)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤12rH(M-m)r⋁mM(〈E(⋅)x,y〉)≤12rH(M-m)r‖x‖‖y‖
for any x,y∈H.Proof.
We start with the equality(4.8)f(M)+f(m)2⋅〈x,y〉-〈f(A)x,y〉=∫m-0M[f(M)+f(m)2-f(λ)]d(〈Eλx,y〉)
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
(4.9)|f(m)+f(M)2⋅〈x,y〉-〈f(A)x,y〉|≤max⁡λ∈[m,M]|f(M)+f(m)2-f(λ)|⋁mM(〈E(⋅)x,y〉)
for any x,y∈H.
Asf:[m,M]→ℂ is r-H-Hölder continuous on [m,M], then we have
(4.10)|f(M)+f(m)2-f(λ)|≤12|f(M)-f(λ)|+12|f(λ)-f(m)|≤12H[(M-λ)r+(λ-m)r]
for any λ∈[m,M].
Since, obviously, the functiongr(λ):=(M-λ)r+(λ-m)r,r∈(0,1] has the property that
(4.11)max⁡λ∈[m,M]gr(λ)=gr(m+M2)=21-r(M-m)r,
then by (4.9) we deduce the first part of (4.7).
Now, ifd:m=t0<t1<⋯<tn-1<tn=M is an arbitrary partition of the interval [m,M], then we have by the Schwarz inequality for nonnegative operators that
(4.12)⋁mM(〈E(⋅)x,y〉)=sup⁡d{∑i=0n-1|〈(Eti+1-Eti)x,y〉|}≤sup⁡d{∑i=0n-1[〈(Eti+1-Eti)x,x〉1/2〈(Eti+1-Eti)y,y〉1/2]}∶=I.
By the Cauchy-Buniakovski-Schwarz inequality for sequences of real numbers we also have that
(4.13)I≤sup⁡d{[∑i=0n-1〈(Eti+1-Eti)x,x〉]1/2[∑i=0n-1〈(Eti+1-Eti)y,y〉]1/2}≤sup⁡d{[∑i=0n-1〈(Eti+1-Eti)x,x〉]1/2sup⁡d[∑i=0n-1〈(Eti+1-Eti)y,y〉]1/2}=[⋁mM(〈E(⋅)x,x〉)]1/2[⋁mM(〈E(⋅)y,y〉)]1/2=‖x‖‖y‖
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 functionf:(0,∞)→ℝ, f(t)=lnt, then we can state the following result.Proposition 5.1.
LetA 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 {Eλ}λ be its spectral family. Then for any  x,y∈H one has
(5.1)|〈x,y〉ln⁡mM-〈ln⁡Ax,y〉|≤ln⁡(Mm)×{12max⁡λ∈[m,M][〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]max⁡λ∈[m,M]|〈Eλx-12x,y〉|≤12‖x‖‖y‖ln⁡(Mm),|〈x,y〉ln⁡mM-〈ln⁡Ax,y〉|≤1m×{12∫m-0M[〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]dλ∫m-0M|〈Eλx-12x,y〉|dλ≤12‖x‖‖y‖(Mm-1),|〈x,y〉ln⁡mM-〈ln⁡Ax,y〉|≤∫m-0M|〈Eλx-12x,y〉|λ-1dλ≤12‖x‖‖y‖ln⁡(Mm).The proof is obvious from Theorems3.1, 3.2, and 4.1 applied for the logarithmic function. The details are omitted.(2) Consider now the power functionf:(0,∞)→ℝ, f(t)=tp 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.
LetA 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 {Eλ}λ be its spectral family. Then for  p∈(0,1) one has
(5.2)|mp+Mp2⋅〈x,y〉-〈Apx,y〉|≤12p(M-m)p⋁mM(〈E(⋅)x,y〉)≤12p(M-m)p‖x‖‖y‖,
for any x,y∈H.The case of powersp≥1 is embodied in the following.Proposition 5.3.
LetA 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 {Eλ}λ be its spectral family. Then for  p≥1  and for any  x,y∈H one has(5.3)|mp+Mp2⋅〈x,y〉-〈Apx,y〉|≤(Mp-mp)×{12max⁡λ∈[m,M][〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]max⁡λ∈[m,M]|〈Eλx-12x,y〉|≤12‖x‖‖y‖(Mp-mp),(5.4)|mp+Mp2⋅〈x,y〉-〈Apx,y〉|≤pMp-1×{12∫m-0M[〈Eλx,x〉1/2〈Eλy,y〉1/2+〈(1H-Eλ)x,x〉1/2〈(1H-Eλ)y,y〉1/2]dλ∫m-0M|〈Eλx-12x,y〉|dλ≤12p‖x‖‖y‖Mp-1,(5.5)|mp+Mp2⋅〈x,y〉-〈Apx,y〉|≤p∫m-0M|〈Eλx-12x,y〉|λp-1dλ≤12‖x‖‖y‖(Mp-mp).The proof is obvious from Theorems3.1, 3.2, and 4.1 applied for the power function f:(0,∞)→ℝ, f(t)=tp with p≥1. The details are omitted.The case of negative powers is similar. The details are left to the interested reader.



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