Further Inequalities for Sequences and Power Series of Operators in Hilbert Spaces Via Hermitian Forms

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This is the Published version of the following publication

Dragomir, Sever S (2016) Further Inequalities for Sequences and Power Series of Operators in Hilbert Spaces Via Hermitian Forms. Moroccan Journal of Pure and Applied Analysis, 2 (1). 47 - 64. ISSN 2351-8227

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Moroccan J. Pure and Appl. Anal.(MJPAA) RESEARCH ARTICLE Volume 2(1), 2016, Pages 47–64

ISSN: 2351-8227

Further Inequalities for Sequences and Power Series of Operators in Hilbert Spaces Via

Hermitian Forms

S. S. Dragomir^a,b

Abstract. By the use of some inequalities for nonnegative Hermitian forms some new inequalities for sequences and power series of bounded linear operators in complex Hilbert spaces are established. Applications for some fundamental functions of interest are also given.

2010 Mathematics Subject Classification. 47A63; 47A99.

Key words and phrases. Hermitian forms, Inner product, Hilbert space, Schwarz inequality, Numerical radius, Operator norm, Functions of operators.

1. Introduction

Let K be the field of real or complex numbers, i.e., K=R or C and X be a linear space over K.

Definition 1.1. A functional (·,·) :X×X →K is said to be a Hermitian form on X if

(H1) (ax+by, z) =a(x, z) +b(y, z) for a, b∈K and x, y, z ∈X;

(H2) (x, y) = (y, x) for all x, y ∈X.

The functional (·,·) is said to be positive semi-definite on a subspace Y of X if (H3) (y, y)≥0 for everyy ∈Y,

Received March 15, 2016 - Accepted May 13, 2016.

The Author(s) 2016. This article is published with open access by Sidi Mohamed Ben Abdallahc University

aMathematics, School of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia.

bSchool of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

e-mail: [email protected]

. 47

DOI 10.7603/s40956-016-0005-1

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and positive definite onY if it is positive semi-definite on Y and (H4) (y, y) = 0, y ∈Y implies y= 0.

The functional (·,·) is said to be definite onY provided that either (·,·) or−(·,·) is positive semi-definite on Y.

When a Hermitian functional (·,·) is positive-definite on the whole space X, then, as usual, we will call it an inner product on X and will denote it byh·,·i.

We use the following notations related to a given Hermitian form (·,·) on X : X₀ :={x∈X|(x, x) = 0}, K :={x∈X|(x, x)<0}

and, for a given z∈X,

X^(z):={x∈X|(x, z) = 0} and L(z) :={az|a∈K}. The following fundamental facts concerning Hermitian forms hold:

Theorem 1.1 (Kurepa, 1968 [28]). Let X and (·,·) be as above.

(1) If e∈X is such that (e, e)6= 0, then we have the decomposition X =L(e)M

X^(e), (1)

where L

denotes the direct sum of the linear subspaces X^(e) and L(e) ;

(2) If the functional (·,·) is positive semi-definite on X^(e) for at least one e ∈ K, then (·,·) is positive semi-definite on X^(f⁾ for each f ∈K;

(3) The functional (·,·) is positive semi-definite on X^(e) with e ∈K if and only if the inequality

|(x, y)|² ≥(x, x) (y, y) (2)

holds for all x∈K and all y ∈X;

(4) The functional(·,·)is semi-definite onX if and only if the Schwarz’s inequality

|(x, y)|² ≤(x, x) (y, y) (3)

holds for all x, y ∈X;

(5) The case of equality holds in (3) for x, y ∈ X and in (2), for x ∈ K, y ∈ X, respectively; if and only if there exists a scalar a ∈K such that

y−ax ∈X₀^(x):=X₀∩X^(x).

LetXbe a linear space over the real or complex number fieldKand let us denote by H(X) the class of all positive semi-definite Hermitian forms on X, or, for simplicity, nonnegative forms on X.

If (·,·) ∈ H(X), then the functional k·k = (·,·)¹² is a semi-norm on X and the following equivalent versions of Schwarz’s inequality hold:

kxk²kyk² ≥ |(x, y)|² or kxk kyk ≥ |(x, y)| (4) for any x, y ∈X.

Now, let us observe that H(X) is a convex cone in the linear space of all mappings defined on X² with values in K, i.e.,

(e) (·,·)₁,(·,·)₂ ∈ H(X) implies that (·,·)₁+ (·,·)₂ ∈ H(X) ;

(4)

(ee) α≥0 and (·,·)∈ H(X) implies that α(·,·)∈ H(X). We can introduce on H(X) the following binary relation [23]:

(·,·)₂ ≥(·,·)₁ if and only if kxk₂ ≥ kxk₁ for all x∈X. (5) We observe that the following properties hold:

(b) (·,·)₂ ≥(·,·)₁ for all (·,·)∈ H(X) ;

(bb) (·,·)₃ ≥(·,·)₂ and (·,·)₂ ≥(·,·)₁ implies that (·,·)₃ ≥(·,·)₁; (bbb) (·,·)₂ ≥(·,·)₁ and (·,·)₁ ≥(·,·)₂ implies that (·,·)₂ = (·,·)₁;

i.e., the binary relation defined by (5) is an order relation onH(X).

While (b) and (bb) are obvious from the definition, we should remark, for (bbb), that if (·,·)₂ ≥ (·,·)₁ and (·,·)₁ ≥ (·,·)₂, then obviously kxk₂ = kxk₁ for all x ∈ X, which implies, by the following well known identity:

(x, y)_k := 1 4

kx+yk²_k− kx−yk²_k+i kx+iyk²_k− kx−iyk²_k

(6) with x, y ∈X and k ∈ {1,2}, that (x, y)₂ = (x, y)₁ for all x, y ∈X.

2. Inequalities for Hermitian Forms

The following result is of interest in itself as well:

Lemma 2.1. Let X be a linear space over the real or complex number field Kand (·,·) a nonnegative Hermitian form on X. If y ∈X is such that (y, y)6= 0, then

p_y :H×H →K, p_y(x, z) = (x, z)kyk²−(x, y) (y, z) (7) is also a nonnegative Hermitian form on X.

We have the inequalities

kxk²kyk²− |(x, y)|²

kyk²kzk²− |(y, z)|²

(8)

≥

(x, z)kyk²−(x, y) (y, z)

2

and

kx+zk²kyk²− |(x+z, y)|²¹₂

(9)

≤ kxk²kyk²− |(x, y)|²¹₂

+ kyk²kzk²− |(y, z)|²¹₂ for any x, y, z ∈X.

Proof. By Schwarz’s inequality for the nonnegative Hermitian form (·,·) we have p_y(x, x) = (x, x)kyk²−(x, y) (y, x)

= kxk²kyk²− |(x, y)|² ≥0 for any x∈X.

We have

p_y(αx+βu, z) = (αx+βu, z)kyk²−(αx+βu, y) (y, z)

=α(x, z)kyk²−α(x, y) (y, z)

(5)

+β(u, z)kyk²−β(u, y) (y, z)

=α

(x, z)kyk²−(x, y) (y, z) +β

(u, z)kyk²−(u, y) (y, z)

=αpy(x, z) +βpy(u, z) for any α, β ∈K and x, u∈X.

Also, we have

p_y(z, x) = (z, x)kyk²−(z, y) (y, x)

= (z, x)kyk²−(z, y)(y, x)

= (x, z)kyk²−(x, y) (y, z) = p_y(x, z) for any x, z ∈X.

Ify ∈X is such that (y, y) = 0,then the inequalities (8) and (9) are obviously true.

If y∈X is such that (y, y)6= 0, then by Schwarz’s inequality for py(·,·) we have

|p_y(x, z)|² ≤p_y(x, x)p_y(z, z) for any x, z ∈X, which is equivalent to (8).

The inequality (9) follows by the triangle inequality for the nonnegative formpy(·,·). Remark 2.1. The case when (·,·) is an inner product in Lemma 2.1 was obtained in 1985 by S. S. Dragomir, [3].

Remark 2.2. Putting z =λy in (9), we get:

0≤ kx+λyk²kyk²− |(x+λy, y)|² ≤ kxk²kyk²− |(x, y)|² (10) and, in particular,

0≤ kx±yk²kyk²− |(x±y, y)|² ≤ kxk²kyk²− |(x, y)|² (11) for every x, y ∈H.

We note here that the inequality (10) is in fact equivalent to the following statement sup

λ∈K

kx+λyk²kyk²− |(x+λy, y)|²

=kxk²kyk²− |(x, y)|² (12) for each x, y ∈H.

The following result holds:

Theorem 2.1. Let X be a linear space over the real or complex number field K and (·,·)a nonnegative Hermitian form onX.For any x, y, z ∈X,the following refinement of the Schwarz inequality holds:

kxk kzk kyk² ≥

(x, z)kyk²−(x, y) (y, z)

+|(x, y) (y, z)| (13)

≥ |(x, z)| kyk².

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Proof. Applying the inequality (8), we can state that kxk²kyk²− |(x, y)|²

(14)

≥

(x, z)kyk²−(x, y) (y, z)

2

any x, y, z ∈X.

Utilising the elementary inequality for real numbers m²−n²

p²−q²

≤(mp−nq)², (15)

we can easily see that

kxk²kyk²− |(x, y)|²

(16)

≤ kxk kyk²kzk − |(x, y) (y, z)|² for any x, y, z ∈X.

Since, by Schwarz’s inequality we have

kxk kyk ≥ |(x, y)| and kyk kzk ≥ |(y, z)|, (17) hence

kxk kyk²kzk − |(x, y) (y, z)| ≥0 for any x, y, z ∈X.

Therefore, by (14) and (16) we deduce kxk kyk²kzk − |(x, y) (y, z)| ≥

(x, z)kyk²−(x, y) (y, z) ,

which proves the first inequality in (13).

Corollary 2.1. For any x, y, z ∈X we have 1

2[kxk kzk+|(x, z)|]kyk² ≥ |(x, y) (y, z)|. (18) Proof. By the modulus property we have

(x, z)kyk²−(x, y) (y, z)

≥ |(x, y) (y, z)| − |(x, z)| kyk² and by the first inequality in (13) we have

kxk kzk kyk² ≥

(x, z)kyk²−(x, y) (y, z)

+|(x, y) (y, z)|

≥ |(x, y) (y, z)| − |(x, z)| kyk²+|(x, y) (y, z)|

for any x, y, z ∈X, which is equivalent to (18).

Remark 2.3. We observe that if (·,·) is an inner product, then (18) reduces to Buzano’s inequality obtained in 1974 [2] in a different way.

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3. Vector Inequalities for n-Tuple of Operators

Let T = (T₁, ..., T_n) ∈ B(H)×...× B(H) := B⁽ⁿ⁾(H) be an n-tuple of bounded linear operators on the Hilbert space (H;h·,·i) andp = (p₁, ..., p_n)∈R^∗n+ ann-tuple of nonnegative weights not all of them equal to zero. For an x∈H, x6= 0 we define

hT,Vi_p,x:=

n

X

j=1

p_jhT_jx, V_jxi=

* _n X

j=1

p_jV_j^∗T_j

! x, x

+

(19) where T= (T₁, ..., T_n),V= (V₁, ..., V_n)∈ B⁽ⁿ⁾(H).

We need the following result:

Lemma 3.1. For any x∈H, x6= 0 and p= (p₁, ..., p_n)∈R^∗n+ we have that h·,·i_p,x is a nonnegative Hermitian form on B⁽ⁿ⁾(H).

Proof. We have that hT,Ti_p,x =

* _n X

j=1

p_jT_j^∗T_j

! x, x

+

=

* _n X

j=1

p_j|T_j|²

! x, x

+

≥0, (20)

for any T = (T₁, ..., T_n)∈ B⁽ⁿ⁾(H), where the operator modulus is defined by |A|² = A^∗A, A∈ B(H).

The functional h·,·i_p,x is linear in the first variable and hV,Ti_p,x =

* _n X

j=1

p_jT_j^∗V_j

! x, x

+

=

* x,

n

X

j=1

p_jT_j^∗V_j

! x

+

(21)

=

* _n X

j=1

p_jT_j^∗V_j

!∗

x, x +

=

* _n X

j=1

p_jV_j^∗T_j

! x, x

+

=hT,Vi_p,x for any T= (T₁, ..., T_n),V= (V₁, ..., V_n)∈ B⁽ⁿ⁾(H).

We have the following result for n-tuples of operators:

Theorem 3.1. For T = (T1, ..., Tn), U = (U1, ..., Un), V = (V1, ..., Vn) ∈ B⁽ⁿ⁾(H)\ {0}, p = (p1, ..., pn)∈R^∗n+ and x∈H, we have

hT,Ti_p,xhU,Ui_p,x−

hT,Ui_p,x

2

(22)

×

hU,Ui_p,xhV,Vi_p,x−

hU,Vi_p,x

2

≥

hT,Vi_p,xhU,Ui_p,x− hT,Ui_p,xhU,Vi_p,x

2

,

hT+V,T+Vi_p,xhU,Ui_p,x−

hT+V,Ui_p,x

21/2

(23)

(8)

≤

hT,Ti_p,xhU,Ui_p,x−

hT,Ui_p,x

21/2

+

hV,Vi_p,xhU,Ui_p,x−

hV,Ui_p,x

21/2

hT,Ti^1/2_p,xhV,Vi^1/2_p,xhU,Ui_p,x (24)

≥

hT,Vi_p,xhU,Ui_p,x− hT,Ui_p,xhU,Vi_p,x +

hT,Ui_p,xhU,Vi_p,x

≥

hT,Vi_p,x

hU,Ui_p,x and

1 2

hhT,Ti^1/2_p,xhV,Vi^1/2_p,x+

hT,Vi_p,x

ihU,Ui_p,x ≥

hT,Ui_p,xhU,Vi_p,x

. (25) The proof follows from the corresponding inequalities above, namely (8), (9), (13) and (18) applied for the nonnegative Hermitian formh·,·i_p,x, x∈H,x6= 0.The details are omitted.

Remark 3.1. The inequality (25) can be written as

1 2





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j|V_j|²

! x, x

+1/2

(26)

+

* _n X

j=1

pjV_j^∗Tj

! x, x

+

# * _n X

j=1

pj|Uj|²

! x, x

+

≥

* _n X

j=1

pjU_j^∗Tj

! x, x

+ * _n X

j=1

pjV_j^∗Uj

! x, x

+ ,

that holds for (T₁, ..., T_n), (U₁, ..., U_n), (V₁, ..., V_n) ∈ B⁽ⁿ⁾(H)\ {0}, p = (p₁, ..., p_n)∈ R^∗n+ and x∈H.

If we take V_j =T_j^∗ for j ∈ {1, ..., n} in (26), then we have 1

2





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

(27)

+

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_j|U_j|²

! x, x

+

≥

* _n X

j=1

p_jU_j^∗T_j

! x, x

+ * _n X

j=1

p_jT_jU_j

! x, x

+ ,

(9)

and since

* _n X

j=1

p_jT_jU_j

! x, x

+

=

* x,

n

X

j=1

p_jT_jU_j

!∗

x +

=

* x,

n

X

j=1

p_jU_j^∗T_j^∗

! x

+

=

* _n X

j=1

p_jU_j^∗T_j^∗

! x, x

+ ,

hence the inequality (27) can also be written as 1

2





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

(28)

+

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_j|U_j|²

! x, x

+

≥

* _n X

j=1

p_jU_j^∗T_j

! x, x

+

* _n X

j=1

p_jU_j^∗T_j^∗

! x, x

+ .

If Tj are normal operators for any j ∈ {1, ..., n}, then we get from (28) that 1

2

"* _n X

j=1

p_j|T_j|²

! x, x

+ +

* _n X

j=1

p_jT_j²

! x, x

+

#

(29)

×

* _n X

j=1

p_j|U_j|²

! x, x

+

≥

* _n X

j=1

p_jU_j^∗T_j

! x, x

+

* _n X

j=1

p_jU_j^∗T_j^∗

! x, x

+ ,

for any (U₁, ..., U_n)∈ B⁽ⁿ⁾(H)\ {0} and x∈H.

If U_j are selfadjoint operators for anyj ∈ {1, ..., n}, then we get from (27) that 1

2





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

(30)

+

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_jU_j²

! x, x

+

≥

* _n X

j=1

p_jU_jT_j

! x, x

+ * _n X

j=1

p_jT_jU_j

! x, x

+ ,

for any (T1, ..., Tn)∈ B⁽ⁿ⁾(H)\ {0} and x∈H.

(10)

Moreover, ifUjTj =TjUj for any j ∈ {1, ..., n},then we get from (30) the inequality 1

2





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

(31)

+

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_jU_j²

! x, x

+

≥

* _n X

j=1

p_jU_jT_j

! x, x

+

2

,

for any (T1, ..., Tn)∈ B⁽ⁿ⁾(H)\ {0}, (U1, ..., Un) selfadjoint operators and x∈H.

In particular, if (T1, ..., Tn) are normal operators and (U1, ..., Un) are selfadjoint operators such thatUjTj =TjUj for anyj ∈ {1, ..., n}, then we get from (31) the simpler inequality

1 2

"* _n X

j=1

p_j|T_j|²

! x, x

+ +

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_jU_j²

! x, x

+

(32)

≥

* _n X

j=1

p_jU_jT_j

! x, x

+

2

, for any x∈H.

Remark 3.2. We notice that (32) is an operator version of de Bruijn inequality obtained in 1960 in[1], which provides the following refinement of the Cauchy-Bunyakovsky- Schwarz inequality:

n

X

i=1

aizi

2

≤ 1 2

n

X

i=1

a²_i

" _n X

i=1

|zi|²+

n

X

i=1

z_i²

#

, (33)

provided that a_i are real numbers while z_i are complex for each i∈ {1, ..., n}.

For some inequalities in inner product spaces and operators on Hilbert spaces see [4]-[26] and the references therein.

4. Applications for Functions of Normal Operators

Some important examples of power series with nonnegative coefficients are 1

1−λ =

∞

X

n=0

λⁿ, λ∈D(0,1) ; (34)

ln 1 1−λ =

∞

X

n=1

1

nλⁿ, λ∈D(0,1) ;

(11)

exp (λ) =

∞

X

n=0

1

n!λⁿ, λ∈C; sinhλ=

∞

X

n=0

1

(2n+ 1)!λ²ⁿ⁺¹, λ∈C; coshλ=

∞

X

n=0

1

(2n)!λ²ⁿ, λ∈C.

Other important examples of functions as power series representations with nonnegative coefficients are:

1 2ln

1 +λ 1−λ

=

∞

X

n=1

1

2n−1λ²ⁿ⁻¹, λ∈D(0,1) ; (35) sin⁻¹(λ) =

∞

X

n=0

Γ n+¹₂

√π(2n+ 1)n!λ²ⁿ⁺¹, λ∈D(0,1) ;

tanh⁻¹(λ) =

∞

X

n=1

1

2n−1λ²ⁿ⁻¹, λ∈D(0,1) ;

2F₁(α, β, γ, λ) =

∞

X

n=0

Γ (n+α) Γ (n+β) Γ (γ)

n!Γ (α) Γ (β) Γ (n+γ) λⁿ, α, β, γ >0, λ∈D(0,1)

where Γ is Gamma function.

We have the following result:

Theorem 4.1. Let f(z) :=P∞

j=0p_jz^j be a power series with nonnegative coefficients and convergent on the open disk D(0, R), R >0. If T, U and V are normal operators and V^∗T =T V^∗, U^∗T =T U^∗, V^∗U =U V^∗with kTk²,kUk²,kVk² < R, then we have the inequalities

1 2

h

f |T|²

x, x^1/2

f |V|²

x, x^1/2

+|hf(V^∗T)x, xi|i

(36)

×

f |U|² x, x

≥ |hf(U^∗T)x, xi hf(V^∗U)x, xi|

for any x∈H.

Proof. If we use the inequality (26) for powers of operators we have 1

2





* _m X

j=0

p_j T^j

2

! x, x

+1/2* _m X

j=0

p_j V^j

2

! x, x

+1/2

(37)

+

* _m X

j=0

p_j V^j∗

T^j

! x, x

+

# * _m X

j=0

p_j U^j

2

! x, x

+

(12)

≥

* _m X

j=0

p_j U^j∗

T^j

! x, x

+ * _m X

j=0

p_j V^j∗

U^j

! x, x

+ , for any m ≥1 and x∈H.

Since T, U and V are normal operators andV^∗T =T V^∗, U^∗T =T U^∗, V^∗U =U V^∗, hence from (37) we have

1 2





* _m X

j=0

p_j|T|^2j

! x, x

+1/2* _m X

j=0

p_j|V|^2j

! x, x

+1/2

(38)

+

* _m X

j=0

p_j(V^∗T)^j

! x, x

+

# * _m X

j=0

p_j|U|^2j

! x, x

+

≥

* _m X

j=0

p_j(U^∗T)^j

! x, x

+ * _m X

j=0

p_j(V^∗U)^j

! x, x

+ , for any m ≥1 and x∈H.

Since all the series whose partial sums are involved in the inequality (38) are con-

vergent hence by letting m→ ∞ in (38) we get (36).

Corollary 4.1. Let f(z) := P∞

j=0p_jz^j be a power series with nonnegative coefficients and convergent on the open disk D(0, R), R >0. If T andU are normal operators and U^∗T =T U^∗, T U =U T with kTk²,kUk² < R, then we have the inequalities

1 2

f |T|² x, x

+

f T² x, x

f |U|² x, x

(39)

≥ |hf(U^∗T)x, xi hf(T U)x, xi|

for any x∈H, x6= 0.

In particular, if T is normal and U is selfadjoint with T U =U T and kTk²,kUk² <

R, then 1 2

f |T|² x, x

+

f T² x, x

f U² x, x

≥ |hf(T U)x, xi|² (40) for any x∈H.

In order to provide various examples of interesting inequalities we use (40) for some fundamental functions.

If T is normal and U is selfadjoint with T U =U T with kTk,kUk<1,then 1

2 hD

1_H − |T|²⁻¹ x, xE

+ D

1_H −T²−1

x, xE

i D

1_H −U²−1

x, xE

(41)

≥

(1_H −T U)⁻¹x, x

2

and

1 2

h lnD

1_H − |T|²⁻¹ x, xE

+ D

ln 1_H −T²⁻¹ x, xE

i

(42)

(13)

×D

ln 1_H −U²−1

x, xE

≥

ln (1_H −T U)⁻¹x, x

2

for any x∈H.

If T is normal and U is selfadjoint with T U =U T, then 1

2

exp |T|² x, x

+

exp T² x, x

exp U² x, x

(43)

≥ |hexp (T U)x, xi|², 1

2

sinh |T|² x, x

+

sinh T² x, x

sinh U² x, x

(44)

≥ |hsinh (T U)x, xi|², and

1 2

cosh |T|² x, x

+

cosh T² x, x

cosh U² x, x

(45)

≥ |hcosh (T U)x, xi|² for any x∈H.

5. Norm and Numerical Radius Inequalities

The numerical radius w(T) of an operator T on H is given by [27, p. 8]:

w(T) = sup{|λ|, λ ∈W(T)}= sup{|hT x, xi|,kxk= 1}. (46) It is well known thatw(·) is a norm on the Banach algebraB(H) of all bounded linear operators T : H → H. This norm is equivalent with the operator norm. In fact, the following more precise result holds [27, p. 9]:

Theorem 5.1 (Equivalent norm). For any T ∈ B(H) one has

w(T)≤ kTk ≤2w(T). (47)

We recall also that if T is normal operator, then w(T) = kTk.

For a survey of recent inequalities for numerical radius, see [21] and the references therein.

Theorem 5.2. Let (T₁, ..., T_n)∈ B⁽ⁿ⁾(H)\ {0}, p= (p₁, ..., p_n)∈R^∗n+ and (U₁, ..., U_n) be selfadjoint operators such that UjTj =TjUj for anyj ∈ {1, ..., n}.Then we have the inequality

w²

n

X

j=1

p_jU_jT_j

!

≤ 1 2

n

X

j=1

p_jU_j²

(48)

×





n

X

j=1

p_j|T_j|²

1/2

n

X

j=1

p_j T_j^∗

2

1/2

+

n

X

j=1

p_jT_j²



.

(14)

Proof. Taking the supremum overkxk= 1 in the inequality (31) and using its properties we have

sup

kxk=1

* _n X

j=1

pjUjTj

! x, x

+

2

(49)

≤ 1 2 sup

kxk=1











* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

+

* _n X

j=1

p_jT_j²

! x, x

+

# * _n X

j=1

p_jU_j²

! x, x

+)

≤ sup

kxk=1





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

+

* _n X

j=1

p_jT_j²

! x, x

+

# sup

kxk=1

* _n X

j=1

p_jU_j²

! x, x

+ .

However, we have

sup

kxk=1





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

(50)

+

* _n X

j=1

p_jT_j²

! x, x

+

#

≤ sup

kxk=1





* _n X

j=1

p_j|T_j|²

! x, x

+1/2* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2



≤ sup

kxk=1





* _n X

j=1

pj|Tj|²

! x, x

+1/2* _n X

j=1

pj

T_j^∗

2

! x, x

+1/2



+ sup

kxk=1

* _n X

j=1

pjT_j²

! x, x

+

≤ sup

kxk=1

* _n X

j=1

p_j|T_j|²

! x, x

+1/2

sup

kxk=1

* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

+ sup

kxk=1

* _n X

j=1

p_jT_j²

! x, x

+ .

(15)

Since

sup

kxk=1

* _n X

j=1

pjU_j²

! x, x

+

=

n

X

j=1

pjU_j² ,

sup

kxk=1

* _n X

j=1

p_j|T_j|²

! x, x

+1/2

=

n

X

j=1

p_j|T_j|²

1/2

,

sup

kxk=1

* _n X

j=1

p_j T_j^∗

2

! x, x

+1/2

=

n

X

j=1

p_j T_j^∗

2

1/2

and

sup

kxk=1

* _n X

j=1

p_jT_j²

! x, x

+

=

n

X

j=1

p_jT_j² ,

hence by (49) and (50) we get the desired result (48).

Remark 5.1. If we take U_j = a_j1_H with j ∈ {1, ..., n} where a_j ∈ R, j ∈ {1, ..., n}, then we get from (48)

w²

n

X

j=1

p_ja_jT_j

!

≤ 1 2

n

X

j=1

p_ja²_j

(51)

×





n

X

j=1

p_j|T_j|²

1/2

n

X

j=1

p_j T_j^∗

2

1/2

+

n

X

j=1

p_jT_j²





for any (T1, ..., Tn)∈ B⁽ⁿ⁾(H)\ {0} and p = (p1, ..., pn)∈R^∗n+ and, in particular, w²

n

X

j=1

ajTj

!

≤ 1 2

n

X

j=1

a²_j

(52)

×





n

X

j=1

|T_j|²

1/2

n

X

j=1

T_j^∗

2

1/2

+

n

X

j=1

T_j²



. Moreover, if T_j are normal operators for any j ∈ {1, ..., n}, then we have

w²

n

X

j=1

a_jT_j

!

≤ 1 2

n

X

j=1

a²_j

"

n

X

j=1

|T_j|²

+

n

X

j=1

T_j²

#

. (53)

6. The Case for One and Two Operators

If we write the inequality (26) for p_j = 1, j ∈ {1, ..., n}, then we get 1

2





* _n X

j=1

|T_j|²

! x, x

+1/2* _n X

j=1

|V_j|²

! x, x

+1/2

(54)

(16)

+

* _n X

j=1

V_j^∗T_j

! x, x

+

# * _n X

j=1

|U_j|²

! x, x

+

≥

* _n X

j=1

U_j^∗T_j

! x, x

+ * _n X

j=1

V_j^∗U_j

! x, x

+ ,

that holds for (T₁, ..., T_n), (U₁, ..., U_n), (V₁, ..., V_n)∈ B⁽ⁿ⁾(H)\ {0} and x∈H.

If we write this inequality for n= 1 we get 1

2 h

|T|²x, x^1/2

|V|²x, x^1/2

+|hV^∗T x, xi|i

|U|²x, x

(55)

≥ |h(U^∗T)x, xi h(V^∗U)x, xi|, that holds for any T, U, V ∈ B(H) andx∈H.

If we take V =T^∗ in (55), then we get 1

2 h

|T|²x, x^1/2

|T^∗|²x, x^1/2 +

T²x, x

i

|U|²x, x

(56)

≥ |h(U^∗T)x, xi h(T U)x, xi|, that holds for any T, U ∈ B(H) andx∈H.

In particular, if T is normal, then from (56) we have 1

2

|T|²x, x +

T²x, x

|U|²x, x

≥ |h(U^∗T)x, xi h(T U)x, xi|, (57) for any U ∈ B(H) and x∈H.

Also, if U is selfadjoint, then from (56) we have 1

2 h

|T|²x, x^1/2

|T^∗|²x, x^1/2 +

T²x, x

i

U²x, x

(58)

≥ |h(U T)x, xi h(T U)x, xi|, for any T ∈ B(H) and x∈H.

Moreover, if U is selfadjoint and commuting with T ∈ B(H), then we have from (58)

1 2

h

|T|²x, x1/2

|T^∗|²x, x1/2

+

T²x, x

i

U²x, x

≥ |h(T U)x, xi|², (59) for any x∈H.

If we take the supremum over kxk= 1 in (59), then we get w²(T U) = sup

kxk=1

|h(T U)x, xi|²

≤ 1 2 sup

kxk=1

nh

|T|²x, x^1/2

|T^∗|²x, x^1/2 +

T²x, x

i

U²x, xo

≤ 1 2

"

sup

kxk=1

|T|²x, x1/2

sup

kxk=1

|T^∗|²x, x1/2

+ sup

kxk=1

T²x, x

#

(17)

× sup

kxk=1

U²x, x

= 1 2

kTk²+w T² kUk²

since

sup

kxk=1

|T|²x, x^1/2

=

w |T|²^1/2

=

|T|²

1/2 =kTk,

sup

kxk=1

|T^∗|²x, x1/2

=

w |T^∗|²1/2

=

|T^∗|²

1/2 =kT^∗k=kTk and

sup

kxk=1

U²x, x

= U²

=kUk². Therefore we get

w²(T U)≤ 1 2

kTk²+w T²

kUk², (60)

for any T ∈ B(H) and U a selfadjoint operator that commutes with T.

If we take U =I in (60), then we get the sharp inequality w²(T)≤ 1

2

kTk²+w T²

(61) that has been firstly obtained in 2007 in [13].

If we write the inequality (54) for n= 2 we get 1

2 h

|T1|² +|T2|²

x, x^1/2

|V1|²+|V2|²

x, x^1/2i

. (62)

+ |h(V₁^∗T₁+V₂^∗T₂)x, xi|]

|U₁|²+|U₂|² x, x

≥ |h(U₁^∗T₁ +U₂^∗T₂)x, xi h(V₁^∗U₁+V₂^∗U₂)x, xi|, for any (T₁, T₂), (U₁, U₂), (V₁, V₂)∈ B⁽²⁾(H) andx∈H.

If we take T = (A, B) and V = (B^∗,±A^∗) in (62), where A, B ∈ B(H), then we have

1 2

h

|A|² +|B|²

x, x^1/2

|B^∗|²+|A^∗|²

x, x^1/2i

. (63)

+|h(BA±AB)x, xi|]

|U₁|²+|U₂|² x, x

≥ |h(U₁^∗A+U₂^∗B)x, xi h(BU₁±AU₂)x, xi|, for any (U₁, U₂)∈ B⁽²⁾(H) and x∈H.

If we take in this inequality U₁ =B and U₂ =A, then we get 1

2 h

|A|² +|B|²

x, x1/2

|B^∗|²+|A^∗|²

x, x1/2i

. (64)

+|h(BA±AB)x, xi|]

|A|²+|B|² x, x

≥

h(B^∗A+A^∗B)x, xi

B²±A² x, x

,

(18)

for any A, B ∈ B(H). Moreover, if we take in this inequality B =A^∗, then we get 1

2

|A|²+|A^∗|² x, x

. (65)

+|h(A^∗A−AA^∗)x, xi|]

|A|² +|A^∗|² x, x

≥

A²+ (A^∗)²

x, x (A^∗)²−A² x, x

, for any A∈ B(H).

If we take V₂ =T₁ and V₁ =T₂ then we get from (62) that 1

2

|T₁|² +|T₂|² x, x

+|h(T₂^∗T₁+T₁^∗T₂)x, xi| |U₁|²+|U₂|² x, x

(66)

≥ |h(U₁^∗T₁ +U₂^∗T₂)x, xi h(T₂^∗U₁+T₁^∗U₂)x, xi|, for any (T₁, T₂), (U₁, U₂)∈ B⁽²⁾(H) and x∈H.

If we take V₂ =T₁^∗ and V₁ =T₂^∗ then we get from (62) that 1

2 h

|T₁|²+|T₂|²

x, x^1/2

|T₁^∗|²+|T₂^∗|²

x, x^1/2i

. (67)

+

T₁²+T₂² x, x

|U₁|²+|U₂|² x, x

≥ |h(U₁^∗T₁+U₂^∗T₂)x, xi h(T₂U₁+T₁U₂)x, xi|, for any (T₁, T₂), (U₁, U₂)∈ B⁽²⁾(H) and x∈H.

One can state other particular inequalities by taking specific values for (T1, T2), (U1, U2).The details are however omitted.

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

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