Vector Inequalities for Powers of Some Operators in Hilbert Spaces

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OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Vector inequalities for powers of some operators in Hilbert spaces with applications for operator norm, numerical radius, commutators and self-commutators are given.

1. _Introduction

Let (H;h·,·i) be a complex Hilbert space. Thenumerical range of an operator

T is the subset of the complex numbersCgiven by [13, p. 1]: W(T) ={hT x, xi, x∈H, kxk= 1}.

Thenumerical radius w(T) of an operatorT onH is given by [13, p. 8]: (1.1) w(T) = sup{|λ|, λ∈W(T)}= sup{|hT x, xi|,kxk= 1}.

It is well known thatw(·) is a norm on the Banach algebraB(H) of all bounded linear operatorsT :H→H.This norm is equivalent to the operator norm. In fact, the following more precise result holds [13, p. 9]:

(1.2) w(T)≤ kTk ≤2w(T),

for anyT ∈B(H)

For more results on numerical radii, see [14], Chapter 11.

For other results and historical comments on the above see [13, p. 39–41]. For recent inequalities involving the numerical radius, see [2]-[10], [15], [19]-[21] and [22].

The Schwarz inequality for positive operators asserts that if T is a positive operator inB(H),then

(1.3) |hT x, yi|2≤ hT x, xi hT y, yi for allx, y∈H.

For an arbitrary operatorT inB(H) the following ”mixed Schwarz” inequality has been established by Kato in [18] (see also [12] and [14, p. 265]):

(1.4) |hT x, yi|2≤ h(T∗T)αx, xiD(T T∗)1−αy, yE for allx, y∈H

and forα∈[0,1].

An important consequence of Kato’s inequality (1.4) is the famous Heinz in-equality (see [1], [16], [17], [18]) which says that if T, A and B are operators in

B(H) such thatAandBare positive andkT xk ≤ kAxkandkT∗yk ≤ kBykfor all

x, yinH then

|hT x, yi| ≤ kAαxk B1−αy

Date: September 02, 2008.

1991Mathematics Subject Classification. 47A12 (47A30, 47A63, 47B15).

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for allx, y∈H and forα∈[0,1].

In this paper we establish some vector inequalities for powers of various operators in Hilbert spaces. Applications for norm and numerical radius inequalities are provided. Particular cases for commutators and self-commutators are also given.

2. Vector Inequalities for Two Operators The first results concerning powers of two operators is incorporated in:

Theorem 1. For anyA, B∈B(H)andr≥1 we have the vector inequality:

(2.1) |hAx, Byi|r≤1

2[h(A ∗_A₎r

x, xi+h(B∗B)ry, yi],

wherex, y∈H,kxk=kyk= 1.

In particular, we have the norm inequality

(2.2) kB∗Akr≤ 1

2(k(A ∗_A₎r

k+k(B∗B)rk)

and the numerical radius inequality

(2.3) wr(B∗A)≤ 1

2k(A ∗_A₎r

+ (B∗B)rk,

respectively.

The constant 1₂ is best possible in all inequalities (2.1), (2.2) and (2.3).

Proof. By the Schwarz inequality in the Hilbert space (H;h., .i) we have:

|hB∗Ax, yi|=|hAx, Byi| ≤ kAxk · kByk

(2.4)

=hA∗Ax, xi1/2· hB∗By, yi1/2, x, y∈H.

Utilising the arithmetic mean - geometric mean inequality and then the convexity of the functionf(t) =tr, r≥1,we have successively,

hA∗Ax, xi1/2· hB∗By, yi1/2≤ hA

∗_{Ax, x}_i₊_h_B∗_{By, y}_i 2

(2.5)

≤

_h_A∗_{Ax, x}_ir

+hB∗By, yir

2

1r

for anyx, y∈H.

It is known that ifP is a positive operator then for any r≥1 andz∈H with

kzk= 1 we have the inequality (see for instance [20]) (2.6) hP z, zir≤ hPrz, zi.

Applying this property to the positive operatorsA∗AandB∗B,we deduce that

(2.7)

_h_A∗_{Ax, x}_ir

+hB∗By, yir

2

r1

≤

_h₍_A∗_A₎r

x, xi+h(B∗B)ry, yi

2

1r

for anyx, y∈H,kxk=kyk= 1.

Now, on making use of the inequalities (2.4), (2.5) and (2.7), we get the inequal-ity:

(2.8) |h(B∗A)x, yi|r≤1

2[h(A ∗_A₎r

x, xi+h(B∗B)ry, yi]

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Taking the supremum over x, y ∈ H, kxk = kyk = 1 in (2.8) and since the operators (A∗A)r and (B∗B)r are self-adjoint, we deduce the desired inequality (2.2).

Now, if we takey=xin (2.1), then we get

(2.9) |h(B∗A)x, xi|r≤1

2[h[(A ∗_A₎r

+ (B∗B)r]x, xi]

for anyx∈H,kxk= 1.Taking the supremum overx∈H,kxk= 1 in (2.9) we get (2.3).

The sharpness of the constant follows by takingr= 1 andB=Ain all inequal-ities (2.1), (2.2) and (2.3). The details are omitted.

Corollary 1. For anyA∈B(H)andr≥1 we have the vector inequalities:

(2.10) |hAx, yi|r≤1

2[h(A ∗_A₎r

x, xi+ 1],

and

(2.11)

A2x, y

r ≤ 1

2[h(A ∗_A₎r

x, xi+h(AA∗)ry, yi],

wherex, y∈H,kxk=kyk= 1.

In particular, we have the norm inequalities

(2.12) kAkr≤1

2(k(A ∗_A₎r

k+ 1)

and

(2.13) A2

r ≤ 1

2(k(A ∗_A₎r

k+k(AA∗)rk),

respectively.

We also have the numerical radius inequalities

(2.14) wr(A)≤1

2k(A ∗_A₎r

+Ik

and

(2.15) wr A2

≤1

2k(A ∗_A₎r

+ (AA∗)rk,

respectively.

A different approach is considered in the following result:

Theorem 2. For any A, B∈B(H), anyα∈(0,1) andr≥1, we have the vector inequality:

(2.16) |hAx, Byi|2r≤αD(A∗A)αr _{x, x}

E

+ (1−α)D(B∗B)1−rα_{y, y}

E

for any x, y∈H withkxk=kyk= 1.

In particular, we have the norm inequality

(2.17) kB∗Ak2r≤α (A

∗_A₎αr

+ (1−α) (B

∗_B₎1−rα

and the numerical radius inequality

(2.18) w2r(B∗A)≤ α(A

∗_A₎αr _{+ (1}₋_α_{) (}_B∗_B₎ r

1−α

,

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Proof. By Schwarz’s inequality, we have:

|h(B∗A)x, yi|2≤ h(A∗A)x, xi · h(B∗B)y, yi

(2.19)

=Dh(A∗A)1α

iα

x, xE·

_h

(B∗B)1−1α

i1−α

y, y

,

for anyx, y∈H.

It is well known that (see for instance [20]) ifPis a positive operator andq∈(0,1] then for anyu∈H,kuk= 1,we have

(2.20) hPqu, ui ≤ hP u, uiq.

Applying this property to the positive operators (A∗A)α1 _{and (}_B∗_B₎

1

1−α ₍_α∈₍₀_,₁₎₎_, we have

(2.21) Dh(A∗A)1α

iα

x, xE·

_h

(B∗B)1−1α

i1−α

y, y

≤D(A∗A)α1 _{x, x}

Eα

·D(B∗B)1−1α_{y, y}

E1−α

,

Now, utilising the weighted arithmetic mean - geometric mean inequality, i.e.,

aα_b1−α_≤_αa_{+ (1}₋_α₎_{b, α}_∈₍₀_,₁₎_{, a, b}_≥₀_, _{we get}

(2.22) D(A∗A)α1 _{x, x}

Eα

·D(B∗B)1−1α_{y, y}

E1−α

≤αD(A∗A)α1 _{x, x}

E

+ (1−α)D(B∗B)1−1α_{y, y}

E

Moreover, by the elementary inequality following from the convexity of the func-tionf(t) =tr_{, r}_≥₁_,_namely

αa+ (1−α)b≤(αar+ (1−α)br)1r_, _α_∈₍₀_,₁₎_{, a, b}_≥₀_, we deduce that

αD(A∗A)α1 _{x, x}

E

+ (1−α)D(B∗B)1−1α_{y, y}

E

(2.23)

≤hαD(A∗A)α1 _{x, x}

Er

+ (1−α)D(B∗B)1−1α_{y, y}

Eri1r

≤hαD(A∗A)αr _{x, x}

E

+ (1−α)D(B∗B)1−rα_{y, y}

Ei1_r

,

for any x, y ∈ H, kxk = kyk = 1, where, for the last inequality we used the inequality (2.6) for the positive operators (A∗A)α1 _{and (}_B∗_B₎1−1α_.

Now, on making use of the inequalities (2.19), (2.21), (2.22) and (2.23), we get

(2.24) |h(B∗A)x, yi|2r≤αD(A∗A)rα_{x, x}

E

+ (1−α)D(B∗B)1−rα_{y, y}

E

for anyx, y∈H,kxk=kyk= 1,and the inequality (2.16) is proved.

Taking the supremum over x, y ∈ H, kxk = kyk = 1 in (2.24) produces the desired inequality (2.17).

The numerical radius inequality follows from (2.24) written fory=x.The details are omitted.

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Corollary 2. For any A ∈ B(H) and α ∈ (0,1), r ≥ 1, we have the vector inequalities

(2.25) |hAx, yi|2r≤αD(A∗A)αr _{x, x}

E

+ 1−α,

(2.26)

A2x, y

2r

≤αD(A∗A)αr _{x, x}

E

+ (1−α)D(AA∗)1−rα_{y, y}

E

and

(2.27) |hAx, Ayi|2r≤αD(A∗A)αr _{x, x}

E

+ (1−α)D(A∗A)1−rα_{y, y}

E

,

respectively, wherex, y∈H, kxk=kyk= 1. We have the norm inequalities

(2.28) kAk2r≤α

(A

∗_A₎αr

+ 1−α

and

(2.29) A2

2r ≤α

(A

∗_A₎αr

+ (1−α) (AA

∗₎1−rα

,

respectively.

We have the numerical radius inequalities

(2.30) w2r(A)≤

α(A

∗_A₎αr _{+ (1}₋_α₎_I

and

(2.31) w2r A2≤ α(A

∗_A₎αr _{+ (1}₋_α_{) (}_AA∗₎ r

1−α

,

respectively.

Moreover, we have the norm inequality

(2.32) kAk4r≤ α(A

∗_A₎αr _{+ (1}₋_α_{) (}_A∗_A₎ r

1−α

.

3. Vector Inequalities for the Sum of Two Products The following result concerning four operators may be stated:

Theorem 3. For anyA, B, C, D∈B(H)andr, s≥1we have:

(3.1)

_B∗_A₊_D∗_C 2

x, y

2

≤

(A∗A)r+ (C∗C)r 2

x, x

1r

·

(B∗B)s+ (D∗D)s 2

y, y

1s

for any x, y∈H withkxk=kyk= 1. Moreover, we have the norm inequality

(3.2)

B∗A+D∗C

2

≤

(A∗A)r+ (C∗C)r 2

1

r

·

(B∗B)s+ (D∗D)s 2

1

s

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Proof. By the Schwarz inequality in the Hilbert space (H;h., .i) we have:

|h(B∗A+D∗C)x, yi|2

(3.3)

=|hB∗Ax, yi+hD∗Cx, yi|2 ≤[|hB∗Ax, yi|+|hD∗Cx, yi|]2

≤hhA∗Ax, xi12 _{· h}_B∗_{By, y}_i21 ₊_h_C∗_{Cx, x}_i12 _{· h}_D∗_{Dy, y}_i12 i2

,

for anyx, y∈H.

Now, on utilising the elementary inequality:

(ab+cd)2≤ a2+c2

b2+d2

, a, b, c, d∈R,

we then conclude that:

(3.4) hA∗Ax, xi12 _{· h}_B∗_{By, y}_i 1

2 ₊_h_C∗_{Cx, x}_i 1

2 _{· h}_D∗_{Dy, y}_i 1 2

≤(hA∗Ax, xi+hC∗Cx, xi)·(hB∗By, yi+hD∗Dy, yi),

for anyx, y∈H.

Now, on making use of a similar argument to the one in the proof of Theorem 1, we have forr, s≥1 that

(3.5) (hA∗Ax, xi+hC∗Cx, xi)·(hB∗By, yi+hD∗Dy, yi)

≤4·

(A∗A)r+ (C∗C)r 2

x, x

1r

·

(B∗B)s+ (D∗D)s 2

y, y

1s

for anyx, y∈H withkxk=kyk= 1. Consequently, by (3.3) – (3.5) we have:

(3.6)

_B∗_A₊_D∗_C 2

x, y

2

≤

₍_A∗_A₎r

+ (C∗C)r 2

x, x

1r

·

₍_B∗_B₎s

+ (D∗D)s 2

y, y

1s

for anyx, y∈H withkxk=kyk= 1, which provides the desired result (3.1). Taking the supremum overx, y∈H withkxk=kyk= 1 in (3.6) we deduce the desired inequality (3.2).

Remark 1. If we make y=xin (3.6) and take the supremum over kxk= 1,then we get the inequality

w2

_B∗_A₊_D∗_C 2

≤

(A∗A)r+ (C∗C)r 2

1

r

·

(B∗B)s+ (D∗D)s 2

1

s

,

which is not as good as (3.2) since we always have

w2

_B∗_A₊_D∗_C 2

≤

B∗A+D∗C

2

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Remark 2. If s=r, then the inequality (3.1) becomes : (3.7)

_B∗_A₊_D∗_C 2 x, y 2r ≤

(A∗A)r+ (C∗C)r 2 x, x ·

(B∗B)r+ (D∗D)r 2

y, y

for any x, y∈H withkxk=kyk= 1while (3.2) is equivalent with

(3.8)

B∗A+D∗C

2 2r ≤

(A∗A)r+ (C∗C)r 2 ·

(B∗B)r+ (D∗D)r 2 .

Corollary 3. For anyA, C∈B(H)we have:

(3.9)

_A₊_C

2 x, y 2r ≤

₍_A∗_A₎r

+ (C∗C)r 2

x, x

for any x, y∈H withkxk=kyk= 1.In particular, we have the norm inequality

(3.10)

A+C

2 2r ≤

(A∗_A₎r

+ (C∗_C₎r

2 ,

wherer≥1.

The result is obvious by choosingB=D=Iin Theorem 3.

Corollary 4. For anyA, C∈B(H)we have:

(3.11)

A2+C2

2 x, y 2 ≤

₍_A∗_A₎r

+ (C∗C)r 2

x, x

r1

·

₍_AA∗₎s

+ (CC∗)s 2

y, y

1s

for any x, y∈H withkxk=kyk= 1.Also, we have the norm inequality

(3.12)

A2₊_C2 2 2 ≤

(A∗A)r+ (C∗C)r 2 1 r ·

(AA∗)s+ (CC∗)s 2 1 s

for allr, s≥1.

If s=r,then we have, in particular,

(3.13)

_A2₊_C2 2 x, y 2r ≤

(A∗A)r+ (C∗C)r 2 x, x ·

(AA∗)r+ (CC∗)r 2

y, y

for any x, y∈H withkxk=kyk= 1and the norm inequality

(3.14)

A2+C2

2 2r ≤

(A∗A)r+ (C∗C)r 2 ·

(AA∗)r+ (CC∗)r 2

forr≥1.

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Corollary 5. For anyA, B∈B(H)we have: (3.15)

_B∗_A₊_A∗_B 2 x, y 2 ≤

₍_A∗_A₎r

+ (B∗_B₎r

2

x, x

1r

·

₍_A∗_A₎s

+ (B∗_B₎s

2

y, y

1s

(3.16)

B∗A+A∗B

2 2 ≤

(A∗A)r+ (B∗B)r 2 1 r ·

(A∗A)s+ (B∗B)s 2 1 s

for any r, s≥1.

In particular we have

(3.17)

_B∗_A₊_A∗_B 2 x, y 2r ≤

₍_A∗_A₎r

+ (B∗B)r 2

x, x (A

∗_A₎r

+ (B∗B)r 2

y, y

for any x, y∈H withkxk=kyk= 1and

(3.18)

B∗A+A∗B

2 r ≤

(A∗A)r+ (B∗B)r 2

wherer≥1.

The proof is obvious by choosingD=AandC=B in Theorem 3. Another particular case that might be of interest is the following one.

Corollary 6. For anyA, D∈B(H)we have:

(3.19)

_A₊_D

2 x, y 2 ≤

₍_A∗_A₎r

+I 2 x, x 1r ·

₍_DD∗₎s

+I

2

y, y

1s

(3.20)

A+D

2 2 ≤

(A∗A)r+I

2 1 r ·

(DD∗)s+I

2 1 s ,

wherer, s≥1.

(3.21) |hAx, yi|2≤

₍_A∗_A₎r

+I 2 x, x 1r ·

₍_AA∗₎s

+I

2

y, y

1s

(3.22) kAk2≤

(A∗A)r+I

2 1 r ·

(AA∗)s+I

2 1 s .

Moreover, for any r≥1 we have

(3.23) |hAx, yi|2r≤

₍_A∗_A₎r

+I 2 x, x ·

₍_AA∗₎r

+I

2

y, y

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for any x, y∈H withkxk=kyk= 1and

(3.24) kAk2r≤

(A∗A)r+I

2

·

(AA∗)r+I

2

.

The proof of (3.19) is obvious by the Theorem 3 on choosingB=I, C=I and writing the inequality forD∗ instead ofD.The details are omitted.

Remark 3. If T ∈ B(H) and T = A+iC, i.e., A and C are its Cartesian decomposition, then we get from (3.9)

(3.25) |hT x, yi|2r≤22r−1h[(A∗A)r+ (C∗C)r]x, xi

for any x, y∈H withkxk=kyk= 1.In particular, we have the norm inequality

(3.26) kTk2r≤22r−1k(A∗A)r+ (C∗C)rk, wherer≥1.

Now, if we use the inequality (3.19) forT, AandB, then we get:

(3.27) |hT x, yi|2≤22−1r−s1h_[(_A∗_A₎r₊_I_]_{x, x}i

1

r _{· h}_[(_CC∗₎s₊_I_]_{y, y}_i1s

(3.28) kTk2≤22−1r−

1

sk(A∗A)r+Ik

1

r _{· k}₍_CC∗₎s₊_I_k1s_,

wherer, s≥1.In particular, we have

(3.29) |hT x, yi|2r≤22r−2h[(A∗A)r+I]x, xi · h[(CC∗)r+I]y, yi

(3.30) kTk2r≤22r−2k(A∗A)r+Ik · k(CC∗)r+Ik, for any r≥1.

In terms of theEuclidean radius of two operators we(·,·), where, as in [2],

we(T, U) := sup kxk=1

|hT x, xi|2+|hU x, xi|2

1 2

,

we have the following result as well.

Theorem 4. For any A, B, C, D ∈B(H) and p, q > 1 with 1_p +1_q = 1, we have the vector inequality:

(3.31) |hAx, Byi|2+|hCx, Dyi|2

≤ h[(A∗A)p+ (C∗C)p]x, xi1/p· h[(B∗B)q+ (D∗D)q]y, yi1/q

for eachx, y∈H withkxk=kyk= 1.

In particular, we have the inequality for the Euclidean radius:

(3.32) w2_e(B∗A, D∗C)≤ k(A∗A)p+ (C∗C)pk1/p· k(B∗B)q+ (D∗D)qk1/q.

Proof. On utilising the elementary inequality

ac+bd≤(ap+bp)1/p·(cq+dq)1/q, a, b, c, d≥0 andp, q >1 with 1

p+

1

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then for anyx, y∈H,kxk=kyk= 1 we have the inequalities:

|hB∗Ax, yi|2+|hD∗Cx, yi|2

≤ hA∗Ax, xi · hB∗By, yi+hC∗Cx, xi · hD∗Dy, yi

≤(hA∗Ax, xip+hC∗Cx, xip)1/p·(hB∗By, yiq+hD∗Dy, yiq)1/q

≤(h(A∗A)px, xi+h(C∗C)px, xi)1/p·(h(B∗B)qy, yi+h(D∗D)qy, yi)1/q

=h[(A∗A)p+ (C∗C)p]x, xi1/p· h[(B∗B)q+ (D∗D)q]y, yi1/q.

For the second inequality, let us make the choicey=xto get

|hB∗Ax, xi|2+|hD∗Cx, xi|2

≤ h[(A∗A)p+ (C∗C)p]x, xi1/p· h[(B∗B)q+ (D∗D)q]x, xi1/q,

for anyx∈H,kxk= 1. Taking the supremum overx∈H,kxk = 1 and noticing that the operators (A∗A)p + (C∗C)p and (B∗B)q + (D∗D)q are self-adjoint, we deduce the desired inequality (3.32).

The following particular case is of interest.

Corollary 7. For anyA, C∈B(H)andp, q >1 with 1_p+1_q = 1,we have:

(3.33) |hAx, yi|2+|hCx, yi|2≤21/qh[(A∗A)p+ (C∗C)p]x, xi1/p

for eachx, y∈H,with kxk=kyk= 1. In particular,

w2e(A, C)≤21/qk(A∗A) p

+ (C∗C)pk1/p.

The proof follows from (3.31) and (3.32) forB =D=I.

Corollary 8. For anyA, D∈B(H)andp, q >1 with 1_p+1_q = 1,we have:

(3.34) |hAx, yi|2+|hDx, yi|2≤ h[(A∗A)p+I]x, xi1/p· h[(DD∗)q+I]y, yi1/q

for eachx, y∈H,with kxk=kyk= 1. In particular,

w2_e(A, D)≤ k(A∗A)p+Ik1/p· k(DD∗)q+Ik1/q.

4. _{Inequalities for the Commutator}

The commutator of two bounded linear operatorsT andU is the operatorT U− U T. For the usual norm k·k and for any two operators T and U, by using the triangle inequality and the submultiplicity of the norm, we can state the following inequality:

(4.1) kT U−U Tk ≤2kTk kUk.

In [11], the following result has been obtained as well

(4.2) kT U−U Tk ≤2 min{kTk,kUk}min{kT−Uk,kT+Uk}.

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Proposition 1. For anyT, U ∈B(H)andr, s≥1 we have the vector inequality

(4.3) |h(T U−U T)x, yi|2

≤22−1r−

1

sh[(U∗U)r+ (T∗T)r]x, xi

1

r _{· h}_[(_{U U}∗₎s_{+ (}_{T T}∗₎s_]_{y, y}_i

1

s_,

(4.4) kT U−U Tk2≤22−1r−

1

sk(U∗U)r+ (T∗T)rk

1

r _{· k}₍_{U U}∗₎s_{+ (}_{T T}∗₎s_k

1

s_.

In particular, we have

(4.5) |h(T U−U T)x, yi|2r

≤22r−2h[(U∗U)r+ (T∗T)r]x, xi · h[(U U∗)r+ (T T∗)r]y, yi for any x, y∈H withkxk=kyk= 1and the norm inequality

(4.6) kT U−U Tk2r≤22r−2k(U∗U)r+ (T∗T)rk · k(U U∗)r+ (T T∗)rk, for any r≥1.

Proof. Follows by Theorem 3 on choosingB=T∗, A=U, D=−U∗andC=T.

Now, forU =T∗we can state the following corollary.

Corollary 9. For anyT ∈ B(H)we have the vector inequality for the self com-mutator:

(4.7) |h(T T∗−T∗T)x, yi|2

≤22−1r−

1

sh[(T T∗)r+ (T∗T)r]x, xi

1

r _{· h}_[(_{T T}∗₎s_{+ (}_T∗_T₎s_]_{y, y}_i

1

s

(4.8) kT T∗−T∗Tk2≤22−1r−

1

s k(T T∗)r+ (T∗T)rk

1

r _{· k}₍_{T T}∗₎s_{+ (}_T∗_T₎s_k

1

s_.

(4.9) |h(T T∗−T∗T)x, yi|2r

≤22r−2h[(T T∗)r+ (T∗T)r]x, xi · h[(T T∗)r+ (T∗T)r]y, yi for any x, y∈H withkxk=kyk= 1and the norm inequality

(4.10) kT T∗−T∗Tkr≤2r−1k(T T∗)r+ (T∗T)rk, for any r≥1.

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