In [P. Renaud, "A matrix formulation of Gr\"uss inequality", Linear Algebra
Appl. 335 (2001), 95--100] it was proved an operator inequality involving the
usual trace functional. In this article, we give a refinement of such result
and we answer positively the Renaud's open problem

Bottazzi, Tamara

Conde, Cristian

Annals of Functional Analysis

English

arXiv

Revista con referatoFil: Bottazzi, Tamara Paula. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina.Fil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática Alberto Calderón; Argentina.Fil: Conde, Cristian Marcelo. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.En 2001, Renaud obtuvo una desigualdad de operador de tipo Grüss que involucraba la funcional de traza habitual. En este artículo, refinamos dicho resultado y respondemos positivamente al problema abierto de Renaud.In 2001, Renaud obtaines a Grüss type operator inequality involving the usual trace functional. In this article, we give a refinement of that result, and we answer positively Renaud´s open problem.Em 2001, Renaud obtém uma desigualdade de operador do tipo Grüss envolvendo o funcional trace usual. Neste artigo, damos um refinamento desse resultado e respondemos positivamente ao problema aberto de Renaud

Bottazzi, Tamara Paula

Conde, Cristian Marcelo

Repositorio Institucional UNGS (Universidad Nacional de General Sarmiento)

A Grüss type operator inequality

T. Bottazzi

C. Conde

Crossref

In 2001, Renaud obtaines a Grüss type operator inequality involving the usual trace functional. In this article, we give a refinement of that result, and we answer positively Renaud´s open problem.Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin

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Fil: Bottazzi, Tamara P. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15; ArgentinaFil: Bottazzi, Tamara P. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Conde, Cristian M. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15; ArgentinaFil: Conde, Cristian M. Instituto Argentino de Matemática Alberto Calderón; Argentina.Fil: Conde, Cristian M. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaIn [P. Renaud,A matrix formulation of Gr ̈uss inequality,Linear Al-gebra Appl.335(2001), 95–100] it was proved an operator inequality involvingthe usual trace functional.  In this article, we give a refinement of such result andwe answer positively the Renaud’s open problem

Conde, Cristian M.

Ann. Funct. Anal. 8 (2017), no. 1, 124–132http://dx.doi.org/10.1215/20088752-3764566ISSN: 2008-8752 (electronic)http://projecteuclid.org/afaA GRÜSS TYPE OPERATOR INEQUALITYT. BOTTAZZI1 and C. CONDE1,2*Communicated by T. YamazakiAbstract. In 2001, Renaud obtained a Grüss type operator inequality involv-ing the usual trace functional. In this article, we give a refinement of that result,and we answer positively Renaud’s open problem.1. IntroductionIn 1935, Grüss [6] obtained the following inequality: if f, g are integrable realfunctions on [a, b] and there exist real constants α, β, γ, δ such that α ≤ f(x) ≤β, γ ≤ g(x) ≤ δ for all x ∈ [a, b], then∣∣∣ 1b− a∫ baf(x)g(x) dx− 1(b− a)2∫ baf(x) dx∫ bag(x) dx∣∣∣ ≤ 14(β − α)(δ − γ),and the inequality is sharp in the sense that the constant 1/4 cannot be replacedby a smaller constant. This inequality has been investigated, applied, and gener-alized by many mathematicians, including Banić, Bourin, Matharu, Moslehian,Ilĭsević, Renaud, and Varos̆anec, among others, in different areas of mathematics(see [8] and the references within).In this work,H denotes a (complex, separable) Hilbert space with inner product〈·, ·〉. Let (B(H), ‖ · ‖) be the C∗-algebra of all bounded linear operators actingon (H, 〈·, ·〉) with the uniform norm. We denote by Id the identity operator, andfor any A ∈ B(H), we consider A∗ its adjoint and |A| = (A∗A)1/2 the absoluteCopyright 2017 by the Tusi Mathematical Research Group.Received May 5, 2016; Accepted Jul. 25, 2016.*Corresponding author.2010 Mathematics Subject Classification. Primary 39B05; Secondary 39B42, 47B10, 47A12,47A30.Keywords. Grüss inequality, variance, trace inequality, distance formula.124A GRÜSS TYPE OPERATOR INEQUALITY 125value of A. For A ∈ B(H), we use R(A), N(A), respectively, to denote the rangeand kernel of A.By B(H)+, we denote the cone of positive operators of B(H); that is, B(H)+ :={T ∈ B(H) : 〈Th, h〉 ≥ 0 ∀h ∈ H}. In the case when dimH = n, we identifyB(H) with the full matrix algebra Mn of all n × n matrices with entries in thecomplex field C. For each T ∈ B(H), we denote its spectrum by σ(T ); that is,σ(T ) = {λ ∈ C : T − λ Id is not invertible} and a complex number λ ∈ C is saidto be in the approximate point spectrum of the operator T , and we denote byσap(T ) if there is a sequence {xn} of unit vectors satisfying (T − λ)xn → 0.For each operator T , we considerr(T ) = sup{|λ| : λ ∈ σ(T )}spectral radius of T,W (T ) ={〈Th, h〉 : ‖h‖ = 1}numerical range of T,andw(T ) = sup{|λ| : λ ∈ W (T )}numerical radius of T.Recall, for all T ∈ B(H), that r(T ) ≤ w(T ) ≤ ‖T‖ ≤ 2w(T ), σ(T ) ⊆ W (T ),and by the Toeplitz–Hausdorff theorem, W (T ) is convex.Renaud [11] gave a bounded linear operator analogue of the Grüss inequalityby replacing integrable functions by operators and the integration by a tracefunction as follows. Let A, T ∈ B(H), and suppose that W (A) and W (T ) arecontained in disks of radii RA and RT , respectively. Then, for any positive traceclass operator, P with tr(P ) = 1 holds∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ 4RART , (1.1)and if A and T are normal (i.e., T ∗T = TT ∗), the constant 4 can be replacedby 1. We can easily see that, if A = α Id or T = β Id with α, β ∈ C, then theleft-hand side is equal to zero. In the same article, Renaud proposed the followingopen problem: to characterize k(A, T ) such that∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ k(A, T )RART (1.2)with 1 ≤ k(A, T ) ≤ 4, in particular, whether it depends on A and T separately,(i.e., whether we can write k(A, T ) = h(A)h(T )), where h(A), h(T ) are suitablydefined constants.In this paper we give a positive answer to the open problem proposed byRenaud, and we obtain an explicit formula for k(A, T ) = h(A)h(T ). Also, wegeneralize the inequality (1.1) for normal to transloid operators.2. PreliminariesLet us begin with the notation and necessary definitions. The set of compactoperators in H is denoted by B0(H). If T ∈ B0(H), then we denote by {sn(T )}the sequence of singular values of T , that is, the eigenvalues of |T | (decreasinglyordered). The notion of unitary invariant norms can be defined also for operatorson Hilbert spaces, a norm ||| · ||| that satisfies the invariance property |||UXV ||| =|||X||| for a pair of unitary operators U, V . Recall that each unitarily invariant normis defined on a natural subclass J| ·| of B0(H) called the norm ideal associated126 T. BOTTAZZI and C. CONDEwith the norm ||| · |||. There is a one-to-one correspondence between symmetricgauge functions defined on sequences of real numbers and unitarily invariantnorms defined on norm ideals of operators. More precisely, if ||| · ||| is a unitarilyinvariant norm, then there is a unique symmetric gauge function g such that|||T ||| = g({sn(T )}) for any T ∈ J| ·| . If dimR(T ) = 1, then |||T ||| = s1(T )g(e1) =g(e1)‖T‖. By convention, we assume that g(e1) = 1. If x, y ∈ H, then we denotex⊗ y the rank 1 operator defined on H by (x⊗ y)(z) = 〈z, y〉x. Then ‖x⊗ y‖ =‖x‖‖y‖ = |||x⊗ y|||.The best-known examples of unitary invariant norms are called Schattenp-norms. For 1 ≤ p < ∞, let‖T‖pp =∑nsn(T )p = tr |T |p,and letBp(H) ={T ∈ H : ‖T‖p < ∞},called the p-Schatten class of B(H). This is the subset of compact operatorswith singular values in lp. The positive operators with trace 1 are called den-sity operators (or states), and we denote this set by S(H). The ideal B2(H) iscalled the Hilbert–Schmidt class, and it is a Hilbert space with the inner prod-uct 〈S, T 〉2 = tr(ST ∗). (For a reference on the theory of norm ideals and theirassociated unitarily invariant norms, see [5].)An operator A ∈ B(H) is called normaloid if r(A) = ‖A‖ = ω(A). If A− µ Idis normaloid for all µ ∈ C, then A is called transloid .Finally, for A, T ∈ B(H) and P ∈ S(H), we introduce the following notation:VP (A, T ) = tr(PAT )− tr(PA) tr(PT ).In the particular case T = A∗, we get the variance of A with respect to P . Moreprecisely, Audenaert in [1] considered the following notion: given A,P ∈ Mn, P ≥0, tr(P ) = 1, the variance of A with respect to the matrix P isVP (A) = tr(|A|2P)−∣∣tr(AP )∣∣2 = VP (A,A∗).Note that VP (A− λ Id) = VP (A). Furthermore, he showed that, if A ∈ Mn, thenmax{tr(|A|2P)−∣∣tr(AP )∣∣2 : P ∈ M+n , tr(P ) = 1} = dist(A,C Id)2, (2.1)and the maximization over P on the left-hand side can be restricted to densitymatrices of rank 1.3. Distance formulas and Renaud’s inequalityLet A and T be linear bounded operators acting on H; the vector-functionA− λT is known as the pencil generated by A and T . Evidently, there is at leastone complex number λ0 such that‖A− λ0T‖ = infλ∈C‖A− λT‖.The number λ0 is unique if 0 /∈ σap(T ) (or, equivalently, if inf{‖Tx‖ : ‖x‖ = 1} >0). Different authors, following [12], called this unique number the center of massA GRÜSS TYPE OPERATOR INEQUALITY 127of A with respect to T , and we denote it by c(A, T ). When T = Id, we write c(A).Following Paul, for A, T ∈ B(H) such that 0 /∈ σap(T ), we considerMT (A) = sup‖x‖=1[‖Ax‖2 − |〈Ax, Tx〉|2〈Tx, Tx〉]1/2= sup‖x‖=1∥∥∥Ax− 〈Ax, Tx〉〈Tx, Tx〉Tx∥∥∥. (3.1)In [9], Paul proved that MT (A) = dist(A,CT ). The unique minimizer is charac-terized by the following conditions: there exists a sequence of unit vectors {xn}such that∥∥(A− λ0T )xn∥∥ → ‖A− λ0T‖ and 〈(A− λ0T )xn, xn〉 → 0.In [4], Gevorgyan proved thatc(A, T ) = limn→∞〈Ayn, T yn〉〈Tyn, T yn〉, (3.2)where {yn} is a sequence of unit vectors which approximate the supremum in(3.1). In the particular case that T = Id and A is a Hermitian operator, then itis easy to see thatminλ∈C‖A− λ Id ‖ = λmax(A)− λmin(A)2, (3.3)where λmax(A) (resp., λmax(A)) denotes the maximum (resp., minimum) eigen-value of A. Observe that the minimum isc(A) =λmax(A) + λmin(A)2.We recall other formulas that express the distance from A to the one-dimensionalsubspace CT . Thendist(A,CT ) = sup{∣∣〈Ax, y〉∣∣ : ‖x‖ = ‖y‖ = 1, 〈Tx, y〉 = 0} (3.4)if A, T ∈ B(H) and 0 /∈ σap(T ). In the particular case where T = Id, we getdist(A,C Id) =12sup{‖AX −XA‖ : X ∈ B(H), ‖X‖ = 1}= sup{∥∥(Id−Q)AQ∥∥ : Q is a rank one projection}= sup{∥∥(Id−Q)AQ∥∥ : Q is a projection}. (3.5)In the following statement we present a new proof of the relation between thevariance of A with respect to P and the distance from A to the unidimensionalsubspace C Id.Proposition 3.1. Let A ∈ B(H), and let P ∈ S(H). Thentr(|A|2P)−∣∣tr(AP )∣∣2 = ‖AP 1/2‖22 − ∣∣〈AP 1/2, P 1/2〉2∣∣2=∥∥AP 1/2 − 〈AP 1/2, P 1/2〉2P 1/2∥∥22= minλ∈C‖AP 1/2 − λP 1/2‖22 ≤ minλ∈C‖A− λ Id ‖.128 T. BOTTAZZI and C. CONDEProof. These inequalities are simple consequences from the following generalstatement for any Hilbert space H: let x, y ∈ H with y 6= 0; theninfλ∈C‖x− λy‖2 = ‖x‖2‖y‖2 − |〈x, y〉|2‖y‖2. The following statement is an extension of Audenaert’s formula to infinitedimension.Remark 3.2. We show that the equality (2.1) holds in the infinite-dimensionalcontext; that is, for A ∈ B(H), we havesup{[tr(|A|2P)−∣∣tr(AP )∣∣2]1/2 : P ∈ S(H)} = dist(A,C Id). (3.6)First, we obtain this equality from Prasanna’s result in [10]. Indeed, note thatdist(A,C Id)2 = sup‖x‖=1‖Ax‖2 −∣∣〈Ax, x〉∣∣2≤ sup{tr(|A|2P)−∣∣tr(AP )∣∣2 : P ∈ S(H)}≤ dist(A,C Id)2.On the other hand, another way to prove (3.6) is to reduce the problem to a finitedimension and use the classical Audenaert formula. Now we give the idea of thisproof.For the sake of clarity, we denotem := minλ∈C‖A− λ Id ‖andM := sup{[tr(|A|2P)−∣∣tr(AP )∣∣2]1/2 : P ∈ S(H)}.By Proposition 3.1, we have that M ≤ m. Suppose by contradiction that M < m.Then there exists ε > 0 such thatM < ‖A− λ Id ‖ − ε (3.7)for any λ ∈ C. By the equality (3.2), we have that c(A) ∈ W (A), and then|c(A)| ≤ w(A). As any closed ball in the complex plane is a compact set, we canfind λ1, . . . , λm ∈ H such thatB(0, ω(A))⊆m⋃j=1{λ ∈ C : |λ− λj| <ε2}.Now, we choose unit vectors h1, . . . , hm ∈ H with the following property: ‖(A−λj Id)hj‖ > ‖A − λj Id ‖ − ε2 . Let H′ = span{h1, . . . , hm, Ah1, . . . , Ahm} andn = dimH′. Applying (2.1) to the compressions of A and Id, respectively, wegetdist(A′,C Idn) = max{[tr(|A′|2P ′)−∣∣tr(A′P ′)∣∣2]1/2 : P ′ ∈ M+n , tr(P ′) = 1}= M ′. (3.8)A GRÜSS TYPE OPERATOR INEQUALITY 129One easily verifies that, if λ ∈ B(0, ω(A)), then there exists j ∈ {1, . . . ,m} suchthat‖A′ − λ Idn ‖ > ‖A′ − λj Idn ‖ −ε2≥∥∥(A′ − λj Idn)hj∥∥− ε2=∥∥(A− λj Id)hj∥∥− ε2> ‖A− λj Id ‖ − ε. (3.9)Thus, combining (3.7) and (3.9), we getminλ∈C‖A′ − λ Idn ‖ > M ≥ M ′, (3.10)and we have here a contradiction with (3.8), and therefore m = M .The following two results give upper bounds for VP (A, T ).Lemma 3.3. Let A, T ∈ B(H), and let P ∈ S(H). Then, for any λ, µ ∈ C holds∣∣VP (A, T )∣∣ ≤ ‖A− λ Id ‖‖T − µ Id ‖ − ∣∣tr(P (A− λ Id)) tr(P (T − µ Id))∣∣.Proof. Define the following semi-inner product for X,Y ∈ B(H) and P ∈ S(H):(X,Y )2,P = 〈P 1/2X,P 1/2Y 〉2.Following the proof given by Dragomir in [3, Theorem 2], we obtain for anyE ∈ B(H) such that (E,E)2,P = 1:∣∣(X,Y )2,P − (X,E)2,P (E, Y )2,P ∣∣ ≤ (X,X)1/22,P (Y, Y )1/22,P − ∣∣(X,E)2,P (E, Y )2,P ∣∣.= (X,X)1/22,P (Y, Y )1/22,P −GE(X,Y ).Since (Id, Id)2,P = 1, we have∣∣VP (A, T )∣∣=∣∣VP (A− λ Id, T − µ Id)∣∣=∣∣(A− λ Id, (T − µ Id)∗)2,P− (A− λ Id, Id)2,P(Id, (T − µ Id)∗)2,P∣∣≤ (A− λ Id, A− λ Id)1/22,P (T∗ − µ̄ Id, T ∗ − µ̄ Id)1/22,P −GId(A− λ Id, T∗ − µ̄ Id)= tr(P∣∣(A− λ Id)∗∣∣2)1/2 tr(P |T − µ Id |2)1/2 −GId(A− λ Id, T ∗ − µ̄ Id)≤∥∥∣∣(A− λ Id)∗∣∣2∥∥1/2∥∥|T − µ Id |2∥∥1/2 − ∣∣tr(P (A− λ Id)) tr(P (T − µ Id))∣∣= ‖A− λ Id ‖‖T − µ Id ‖ −∣∣tr(P (A− λ Id)) tr(P (T − µ Id))∣∣. Proposition 3.4. Let A, T ∈ B(H), and let P ∈ S(H). Then∣∣VP (A, T )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id). (3.11)Proof. By Lemma 3.3, we have∣∣VP (A, T )∣∣ ≤ ‖A− λ Id ‖‖T − µ Id ‖ − ∣∣tr(P (A− λ Id)) tr(P (T − µ Id))∣∣130 T. BOTTAZZI and C. CONDEfor A, T ∈ B(H), P ∈ S(H), and any λ, µ ∈ C. Therefore,∣∣VP (A, T )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id). Remark 3.5. If we define VP : B(H) × B(H) → C, VP (A, T ) := tr(PAT ) −tr(PA) tr(PT ), then VP is a bilinear function, and by (3.11) a continuous mappingwith ‖VP‖ ≤ 1.Now, we give a new proof and a refinement of (1.1).Proposition 3.6. Let A, T ∈ B(H), and we suppose that W (A),W (T ) are con-tained in closed disk D(λ0, RA), D(µ0, RT ), respectively. Then, for any P ∈ S(H),∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id)≤ ‖A− λ0 Id ‖‖T − µ0 Id ‖≤ 4w(A− λ0 Id)w(T − µ0 Id)≤ 4RART . (3.12)In particular, if A and T are normal operators, then we have∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id) = rArT , (3.13)where rS denotes the radius of the unique smallest disc containing σ(S) for anyS ∈ B(H).Proof. The inequalities are consequences of (3.11). In the last inequality, we usethe fact that W (A − λ0 Id) ⊂ D(0, RA) and W (T − µ0 Id) ⊂ D(0, RT ), respec-tively. On the other hand, Björck and Thomée [2] have shown that, for a normaloperator A,dist(A,C Id) = sup‖x‖=1(‖Ax‖2 −∣∣〈Ax, x〉∣∣2)1/2 = rA, (3.14)and this completes the proof. Remark 3.7. From (3.13), if we let A be a positive invertible operator, T = A−1and P = x⊗ x with x ∈ H with ‖x‖ = 1, then∣∣tr(PAT )− tr(PA) tr(PT )∣∣ = ∣∣1− 〈Ax, x〉〈A−1x, x〉∣∣≤ dist(A,C Id) dist(A−1,C Id) = rArA−1 ;that is, we obtain the Kantorovich inequality for an operator A acting on aninfinite-dimensional Hilbert space H with 0 < m ≤ A ≤ M .In 1972, Istratescu [7] generalized the equality (3.14) to the transloid classoperators. Then we have the following statement.A GRÜSS TYPE OPERATOR INEQUALITY 131Proposition 3.8. Let A, T ∈ B(H) with A and T transloid operators. Then∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id) = rArT . (3.15)Proof. It follows from the same arguments in the proof of inequality (3.13). The previous proposition generalizes Renaud’s result for normal operators sincethe classes of transloid and normal operators are related by the inclusion asfollows:normal ⊆ quasinormal ⊆ subnormal ⊆ hyponormal ⊆ transloid,where at least the first inclusion is proper.In the following statement, we obtain a parametric refinement of (1.1).Theorem 3.9. Let A, T ∈ B(H) with A, T /∈ C Id, and suppose that W (A),W (T )are contained in the closed disk D(λ0, RA) and D(µ0, RT ), respectively. Thus, forany P ∈ S(H), we get∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id)≤ hλ(A)hµ(T )ω(A− λ0 Id)ω(T − µ0 Id)≤ hλ(A)hµ(T )RART , (3.16)wherehλ(A) = 2(1− λ) + λ‖A− c(A) Id ‖w(A− λ0 Id), hµ(T ) = 2(1− µ) + µ‖T − c(T ) Id ‖w(T − µ0 Id),and 1 ≤ hλ(A)hµ(T ) ≤ 4 for any λ, µ ∈ [0, 1].Proof. Let λ ∈ [0, 1]. Then∥∥A− c(A) Id∥∥ ≤ λ∥∥A− c(A) Id∥∥+ (1− λ)‖A− λ0 Id ‖≤ λ∥∥A− c(A) Id∥∥+ 2(1− λ)w(A− λ0 Id)= w(A− λ0 Id)(2(1− λ) + λ‖A− c(A) Id ‖w(A− λ0 Id))= w(A− λ0 Id)hλ(A),where 1 ≤ hλ(A) ≤ 2 since ‖A− c(A) Id ‖ ≤ ‖A− λ0 Id ‖ ≤ 2w(A− λ0 Id). Thisinequality completes the proof. Note that the previous result gives a positive answer to Renaud’s open question(1.2).132 T. BOTTAZZI and C. CONDECorollary 3.10. Under the same notation as in Theorem 3.9, if A − λ0 Id andT − µ0 Id are normaloid operators, then, for any λ, µ ∈ [0, 1],∣∣tr(PAT )− tr(PA) tr(PT )∣∣ ≤ supP̃∈S(H)∣∣tr(P̃AT )− tr(P̃A) tr(P̃ T )∣∣≤ dist(A,C Id) dist(T,C Id)≤ (2− λ)(2− µ)ω(A− λ0 Id)ω(T − µ0 Id)≤ (2− λ)(2− µ)RART .References1. K. Audenaert, Variance bounds, with an application to norm bounds for commutators, Lin-ear Algebra Appl. 432 (2010), no. 5, 1126–1143. Zbl 1194.60020. MR2577614. DOI 10.1016/j.laa.2009.10.022. 1262. G. Björck and V. Thomée, A property of bounded normal operators in Hilbert space, Ark.Mat. 4 (1963), 551–555. 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MR3322766. DOI 10.7900/jot.2013nov20.2040. 1249. K. Paul, Translatable radii of an operator in the direction of another operator, Sci. Math.2 (1999), no. 1, 119–122. Zbl 0952.47032. MR1688391. 12710. S. Prasanna, The norm of a derivation and the Björck-Thomeé-Istrăţescu theorem, Math.Japon. 26 (1981), no. 5, 585–588. Zbl 0475.47007. MR0636597. 12811. P. Renaud, A matrix formulation of Grüss inequality, Linear Algebra Appl. 335 (2001),95–100. Zbl 0982.15025. MR1850816. DOI 10.1016/S0024-3795(01)00278-6. 12512. J. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737–747. Zbl 0197.10501.MR0265952. 1261Instituto Argentino de Matemática “Alberto P. Calderón”, Saavedra 15,3◦ piso, (C1083ACA) Buenos Aires, Argentina.E-mail address: tpbottaz@ungs.edu.ar; cconde@ungs.edu.ar2Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Gu-tierrez 1150, (B1613GSX) Los Polvorines, Argentina.E-mail address: cconde@ungs.edu.ar

A Grüss Type Operator Inequality

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A Gr\"uss type operator inequality

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