Some inequalities for the operator norm and numerical radius of

sums of bounded linear operators in Hilbert spaces are given. Applications for

the Cartesian decomposition of an operator are also provided

Dragomir, Sever S

Banach Journal of Mathematical Analysis

English

Victoria University Eprints Repository

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 22, No. 1 (2007), pp. 61–75NORM AND NUMERICAL RADIUS INEQUALITIES FORSUMS OF BOUNDED LINEAR OPERATORS IN HILBERTSPACESSever S. DragomirAbstract. Some inequalities for the operator norm and numerical radius of sumsof bounded linear operators in Hilbert spaces are given. Applications for theCartesian decomposition of an operator are also provided.1. IntroductionLet B (H) denote the C∗−algebra of all bounded linear operators on acomplex Hilbert space H with inner product 〈·, ·〉. For A ∈ B (H) , let ‖A‖denote the the usual operator norm of A.It is well known that, the following generalized triangle inequality holdstrue ∥∥∥∥ n∑j=1Aj∥∥∥∥ ≤ n∑j=1‖Aj‖for any A1, . . . , An ∈ B (H) and n a natural number.In [1] we obtained several inequalities that provide alternative upperbounds for the norm of the sum∑nj=1Aj :∥∥∥∥ n∑j=1Aj∥∥∥∥2(1.1)Received September 13, 2006.2000 Mathematics Subject Classification. Primary 47A12, 47A05; Secondary 26D15.6162 S.S. Dragomir≤n∑j=1‖Aj‖2 ;n1/p(n∑j=1‖Aj‖2q)1/qif p > 1, 1/p+ 1/q = 1;n max1≤j≤n‖Aj‖2 ;+∑1≤j 6=k≤n‖AjA∗k‖ ;[n (n− 1)]1/r( ∑1≤j 6=k≤n‖AjA∗k‖s)1/sif r > 1, 1/r + 1/s = 1;n (n− 1) max1≤j 6=k≤n‖AjA∗k‖ ;where (1.1) should be seen as all 9 possible configurations.Out of these inequalities, one can remark the following results of interest∥∥∥∥ n∑j=1Aj∥∥∥∥ ≤ ( n∑j,k=1‖AjA∗k‖)1/2=( n∑j=1‖Aj‖2 +∑1≤j 6=k≤n‖AjA∗k‖)1/2(1.2)≤n∑j=1‖Aj‖ , a refinement of the triangle inequality;∥∥∥∥ n∑j=1Aj∥∥∥∥ ≤ √n [ max1≤j≤n ‖Aj‖2 + (n− 1) max1≤j 6=k≤n ‖AjA∗k‖]1/2;(1.3)∥∥∥∥ n∑j=1Aj∥∥∥∥ ≤ √n[max1≤j≤n‖Aj‖2 +( ∑1≤j 6=k≤n‖AjA∗k‖2)1/2]1/2(1.4)and ∥∥∥∥ n∑j=1Aj∥∥∥∥(1.5)≤ n1/2p[( n∑j=1‖Aj‖2q)1/q+ (n− 1)1/p( ∑1≤j 6=k≤n‖AjA∗k‖q)1/q]1/2for p, q > 1 with 1/p+ 1/q = 1, which, for p = q = 2 provides∥∥∥∥ n∑j=1Aj∥∥∥∥(1.6)≤ n1/4[( n∑j=1‖Aj‖4)1/2+ (n− 1)1/2( ∑1≤j 6=k≤n‖AjA∗k‖2)1/2]1/2.Norm and Numerical Radius Inequalities . . . 63A different approach employed in [2] has lead to the following inequalitiesfor iterated sums: ∥∥∥∥ n∑j=1Aj∥∥∥∥2 ≤αβγ(1.7)whereα :=n∑j,k=1‖AjA∗k‖ ;n1/r[n∑j=1(n∑k=1‖AjA∗k‖)s]1/sif r > 1, 1/r + 1/s = 1;n max1≤j≤n(n∑k=1‖AjA∗k‖);β :=n1/pn∑j=1(n∑k=1‖AjA∗k‖q)1/q;n1/t+1/p[n∑j=1(n∑k=1‖AjA∗k‖q)u/q]1/uif u > 1, 1/u+ 1/t = 1;n1+1/p max1≤j≤n(n∑k=1‖AjA∗k‖q)1/q;where p > 1, 1/p+ 1/q = 1, andγ :=nn∑j=1(max1≤k≤n‖AjA∗k‖);n1/m+1[n∑j=1(max1≤k≤n‖AjA∗k‖`)]1/`if m > 1, 1/m+ 1/` = 1;n2 max1≤j,k≤n‖AjA∗k‖ .Note that the choice p = t = 2 (therefore u = q = 2) will produce from theβ−branch the inequality∥∥∥∥ n∑j=1Aj∥∥∥∥2 ≤ n[n∑j=1‖Aj‖4 +∑1≤j 6=k≤n‖AjA∗k‖2]1/2≤ nn∑j=1‖Aj‖2 ,(1.8)which is a refinement of the Cauchy-Bunyakovsky-Schwarz inequality.64 S.S. DragomirThe aim of this paper is to establish various new inequalities for theoperator norm and numerical radius of sums of bounded linear operatorsin Hilbert spaces. In particular, two refinements of the generalized trian-gle inequality for operator norm are obtained. Particular cases of interestfor two bounded linear operators and their applications for the Cartesiandecomposition of an operator are also considered.2. Some General ResultsTheorem 2.1. For any sequence of operators A1, . . . , An ∈ B (H) we have:∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 12∥∥∥∥ ∑1≤k 6=j≤nA∗jAk∥∥∥∥2 + 12(2.1) (=∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 12∥∥∥∥ n∑j=1A∗jn∑k=1Ak −n∑k=1A∗kAk∥∥∥∥2+12).Proof. For any x ∈ H, observe that∥∥∥∥ n∑k=1Akx∥∥∥∥2 = Re[ n∑j=1n∑k=1〈Akx,Ajx〉]=n∑j=1n∑k=1Re 〈Akx,Ajx〉(2.2)=n∑k=1‖Akx‖2 +∑1≤k 6=j≤nRe 〈Akx,Ajx〉=〈( n∑k=1A∗kAk)x, x〉+Re〈 ∑1≤k 6=j≤n(A∗jAk)x, x〉.Then∥∥∥∥ n∑k=1Akx∥∥∥∥2 ≤〈( n∑k=1A∗kAk)x, x〉+12[∥∥∥∥ ∑1≤k 6=j≤n(A∗jAk)x∥∥∥∥2+‖x‖2],(2.3)where, for the last inequality we have used the elementary inequality in(H; 〈·, ·〉) :Re 〈z, u〉 ≤ 12[‖z‖2 + ‖u‖2], z, u ∈ H.Taking the supremum in (2.3) over x ∈ H, ‖x‖ = 1, we get∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 12∥∥∥∥ ∑1≤k 6=j≤nA∗jAk∥∥∥∥2 + 12 .(2.4)Norm and Numerical Radius Inequalities . . . 65Since ∑1≤k 6=j≤nA∗jAk =n∑j=1A∗jn∑k=1Ak −n∑k=1A∗kAk,then the last part of (2.1) is also proved.Remark 2.1. For the case of two operators, we can state that‖B + C‖2 ≤ ‖B∗B + C∗C‖+ 12‖B∗C + C∗B‖2 + 12(2.5)for any B,C ∈ B(H). If in this inequality we choose B = A, C = A∗, then we get‖A+A∗‖2 ≤ ‖A∗A+AA∗‖+ 12∥∥∥A2 + (A∗)2∥∥∥2 + 12(2.6)for any A ∈ B(H).Now, if A = B + C with B = (A+A∗)/2, C = (A−A∗)/2, i.e., B and C arethe Cartesian decomposition of A, then applying (2.5) for B and C as above willgive the inequality‖A‖2 ≤ 12‖A∗A+AA∗‖+ 14‖A∗A−AA∗‖2 + 12(2.7)for any A ∈ B(H).The following result may be stated as well.Theorem 2.2. For any A1, . . . , An ∈ B (H) we have:∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 12∥∥∥∥(n− 2) n∑k=1A∗kAk +n∑k=1A∗kn∑k=1Ak∥∥∥∥.(2.8)Proof. Utilising the elementary inequalityRe 〈z, u〉 ≤ 14‖z + u‖2 , z, u ∈ H,we then have:∑1≤k 6=j≤nRe 〈Akx,Ajx〉 ≤ 14∑1≤k 6=j≤n‖(Ak +Aj)x‖2=14∑1≤k 6=j≤n〈(Ak +Aj)∗ (Ak +Aj)x, x〉=14〈 ∑1≤k 6=j≤n(A∗k +A∗j)(Ak +Aj)x, x〉.66 S.S. DragomirNow, on making use of the identity (2.2), we can state that:∥∥∥∥ n∑k=1Akx∥∥∥∥2 ≤〈( n∑k=1A∗kAk)x, x〉(2.9)+14〈 ∑1≤k 6=j≤n(A∗k +A∗j)(Ak +Aj)x, x〉for any x ∈ H.Taking the supremum in (2.9), we get∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 14∥∥∥∥ ∑1≤k 6=j≤n(A∗k +A∗j)(Ak +Aj)∥∥∥∥.(2.10)Since ∑1≤k 6=j≤n(A∗k +A∗j)(Ak +Aj)=n∑k,j=1(A∗kAk +A∗jAk +A∗kAj +A∗jAj)− 4 n∑k=1A∗kAk= 2nn∑k=1A∗kAk + 2n∑k=1A∗kn∑k=1Ak − 4n∑k=1A∗kAk= 2[(n− 2)n∑k=1A∗kAk +n∑k=1A∗kn∑k=1Ak],hence by (2.10) we deduce the desired inequality (2.8).Remark 2.2. Since, by the triangle inequality we have that∥∥∥∥(n− 2) n∑k=1A∗kAk +n∑k=1A∗kn∑k=1Ak∥∥∥∥ ≤ (n− 2) ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ ∥∥∥∥ n∑k=1Ak∥∥∥∥2hence by (2.8) we deduce that∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ n∥∥∥∥ n∑k=1A∗kAk∥∥∥∥for n ≥ 2 and A1, . . . , An ∈ B (H).Norm and Numerical Radius Inequalities . . . 67Remark 2.3. The case n = 2 provides the following interesting inequality∥∥∥∥B + C2∥∥∥∥2 ≤ ∥∥∥∥B∗B + C∗C2∥∥∥∥(2.11)for any B,C ∈ B(H). If in this inequality we choose B = A, C = A∗, then we get∥∥∥∥A+A∗2∥∥∥∥2 ≤ ∥∥∥∥A∗A+AA∗2∥∥∥∥(2.12)for each A ∈ B(H).Moreover, if in (2.11) we chose the Cartesian decomposition of an operator A,then we get‖A‖2 ≤ ‖A∗A+AA∗‖(2.13)for any A ∈ B(H).Remark 2.4. Note that, as pointed out in [5], (2.13) follows by the inequality(33) from [6]. We have shown above that it can be also easily deduced from (2.11).A similar approach which provides another inequality for the operatornorm is incorporated in:Theorem 2.3. If A1, . . . , An ∈ B (H), then∥∥∥∥ n∑k=1Ak∥∥∥∥2 + n∑k=1‖Ak‖2 ≤∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ n∑k,j=1∥∥∥∥Ak +Aj2∥∥∥∥2 .(2.14)Proof. Since, by the proof of Theorem 2.2, we have∑1≤k 6=j≤nRe 〈Akx,Ajx〉 ≤ 14∑1≤k 6=j≤n‖(Ak +Aj)x‖2 ,then, by (2.2), we can state that:∥∥∥∥ n∑k=1Akx∥∥∥∥2 ≤〈( n∑k=1A∗kAk)x, x〉+14∑1≤k 6=j≤n‖(Ak +Aj)x‖2(2.15)for any x ∈ H, ‖x‖ = 1.Taking the supremum over x, ‖x‖ = 1, we deduce that∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 14 ∑1≤k 6=j≤n‖Ak +Aj‖2=∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 14n∑k,j=1‖Ak +Aj‖2 −n∑k=1‖Ak‖2 ,which is exactly the desired result (2.14).68 S.S. DragomirRemark 2.5. The case n = 2 will also provide the inequality (2.11) obtainedabove from a different inequality.Finally, we have:Theorem 2.4. If A1, . . . , An ∈ B (H), then∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 14∥∥∥∥ ∑1≤k 6=j≤nA∗jAk + I∥∥∥∥2.(2.16)Proof. It follows by the identity (2.2) on noticing thatRe〈 ∑1≤k 6=j≤n(A∗jAk)x, x〉≤ 14∥∥∥∥ ∑1≤k 6=j≤nA∗jAk + x∥∥∥∥2for any x ∈ H. The details are omitted.Remark 2.6. The case n = 2 provides the following inequality‖B + C‖2 ≤ ‖B∗B + C∗C‖+ 14‖B∗C + C∗B + I‖2(2.17)for any B,C ∈ B(H). If in this inequality we choose B = A and C = A∗, then weget‖A+A∗‖2 ≤ ‖A∗A+AA∗‖+ 14‖A2 + (A∗)2 + I‖2(2.18)for any A ∈ B(H).Finally, the inequality (2.17) provides for the Cartesian decomposition of anoperator A ∈ B(H) the following result as well:‖A‖2 ≤ 12[‖AA∗ +A∗A‖+ 12∥∥∥∥A∗A−AA∗2 + I∥∥∥∥2](2.19)for any A ∈ B(H).3. Some Results for Commuting OperatorsFor a bounded linear operator A, let w(A) denotes its numerical radius[3], [4].We recall the following results concerning the numerical radius of a prod-uct of two operators [3, pp. 37-40] that will be used in deriving variousinequalities for the sums of operators on Hilbert spaces:Norm and Numerical Radius Inequalities . . . 69Theorem 3.1. For any two bounded linear operators A,B we havew (AB) ≤ 4w (A)w (B) .(3.1)If A and B commute, i.e., AB = BA, thenw (AB) ≤ 2w (A)w (B) .(3.2)To get closer to the elusive inequalityw (AB) ≤ w (A)w (B) .(3.3)we recall the following result [3, p. 38]:Theorem 3.2. If the operators A and B double commute, i.e., AB = BAand AB∗ = B∗A, thenw (AB) ≤ w (B) ‖A‖ .(3.4)As a particular case of interest that provides an affirmative answer forthe validity of the inequality (3.3) for some pair of operators (A,B), we canstate [3, p. 39]:Corollary 3.2. Let A be a normal operator, i.e., AA∗ = A∗A. If A com-mutes with B, then (3.3) holds true.Another sufficient condition for (A,B) to satisfy (3.4) is incorporated in[3, p. 39].Theorem 3.3. Let A commutes with B and A2 = αI for some α ∈ C.Then (3.4) holds true.Given that A and B are commutative, the question of whetherw (AB) ≤ w (A) ‖B‖was open for about twenty years. The issue was finally resolved by V. Millerin 1988, see for instance [3, p. 40]. Miller’s approach was computational,and a counterexample was found in a 12-dimensional Hilbert space.The related question of the best constant C > 0 for the inequalityw (AB) ≤ Cw (A) ‖B‖(3.5)70 S.S. Dragomirfor commuting A and B has also been considered by K. Okubo and T. Andoin 1976 and by K. Davidson and J. Holbrook in 1988, see for instance [3, p.42].For other results on numerical radius, see the classical problem book [4].The following result may be stated.Theorem 3.4. If A1, . . . , An ∈ B (H), then∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ 4w( n∑k=1Ak) n∑j=1w (Aj) .(3.6)Moreover, if∑nk=1Ak commutes with each A∗j for j ∈ {1, . . . , n}, then∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ 2w( n∑k=1Ak) n∑j=1w (Aj) .(3.7)Proof. For any x ∈ H we have∥∥∥∥ n∑k=1Akx∥∥∥∥2 =∣∣∣∣∣n∑j=1〈( n∑k=1Ak)x,Ajx〉∣∣∣∣∣(3.8)≤n∑j=1∣∣∣∣∣〈( n∑k=1Ak)x,Ajx〉∣∣∣∣∣ =n∑j=1∣∣∣∣∣〈A∗j( n∑k=1Ak)x, x〉∣∣∣∣∣.Taking the supremum over x ∈ H, ‖x‖ = 1 in (3.8 ), we get∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ n∑j=1w[A∗j( n∑k=1Ak)],(3.9)which is an inequality of interest in itself.Now, since, by (3.1) applied for A∗j (j = 1, . . . , n) and∑nk=1Ak we canstate thatw[A∗j( n∑k=1Ak)]≤ 4w (Aj)w( n∑k=1Ak)(3.10)for any j ∈ {1, . . . , n}, then by (3.9) and (3.10) we deduce (3.6).Now, if A∗j (j = 1, . . . , n) commutes with∑nk=1Ak, then on utilising thesecond part of Theorem 3.1 we can state that:w[A∗j( n∑k=1Ak)]≤ 2w (Aj)w( n∑k=1Ak), j ∈ {1, . . . , n}(3.11)Norm and Numerical Radius Inequalities . . . 71which, by (3.9) will imply the desired inequality (3.7).The following particular case may be of interest.Corollary 3.4. If A1, . . . , An ∈ B (H) are normal operators and ∗−com-mute with each other, i.e., AkA∗j = A∗jAk for k, j ∈ {1, . . . , n}, k 6= j, then(3.7) holds true.Remark 3.1. It is useful to observe that the case n = 2 in the inequality (3.9)provides the result‖B + C‖2 ≤ w (B∗B +B∗C) + w (C∗B + C∗C)(3.12)for any B,C ∈ B(H). In particular, for B = A and C = A∗, we get from (3.12) theinequality‖A+A∗‖2 ≤ w (A∗A+ (A∗)2)+ w (A2 +AA∗)(3.13)for any A ∈ B(H).Now, if we assume in (3.12) that B and C are the Cartesian decomposition ofA ∈ B (H), then‖A‖2 ≤ 12[w(A2 +A∗A)+ w(A2 −A∗A)] .The constant 12 here is best possible in the sense that it cannot be replaced by asmaller constant. The equality case is, for instance, realised when A is a self adjointoperator.A more interesting case which provides refinements of the generalizedtriangle inequality for operator norm is incorporated in the following:Theorem 3.5. Let A1, . . . , An ∈ B (H). If A∗j (j = 1, . . . , n) double com-mutes with∑nk=1Ak, then we have the following refinement of the generalizedtriangle inequality:∥∥∥∥ n∑k=1Ak∥∥∥∥ ≤ n∑k=1w (Ak)(≤n∑k=1‖Ak‖).(3.14)Moreover, if∑nk=1Ak 6= 0, then also(1 ≤) ‖∑nk=1Ak‖w (∑nk=1Ak)≤∑nk=1 ‖Ak‖‖∑nk=1Ak‖ .(3.15)72 S.S. DragomirProof. If∑nk=1Ak = 0, then (3.14) is obvious.Now, suppose∑nk=1Ak 6= 0 and A∗j (j = 1, . . . , n) double commutes with∑nk=1Ak. By Theorem 3.2, we then havew[A∗j( n∑k=1Ak)]≤ w (Aj)∥∥∥∥ n∑k=1Ak∥∥∥∥, j ∈ {1, . . . , n}which, by (3.9), implies that∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1Ak∥∥∥∥ n∑j=1w (Aj) ,which is clearly equivalent with (3.14).By the same Theorem 3.2, we can also state thatw[A∗j( n∑k=1Ak)]≤ w( n∑k=1Ak)‖Aj‖ , j ∈ {1, . . . , n} ,which, together with (3.9) imply (3.15).The following corollary may be stated.Corollary 3.5. If A1, . . . , An are normal operators which double commutewith each other, i.e., AkAj = AjAk and A∗kAj = AjA∗k for each k, j ∈{1, . . . , n}; then both (3.14) and (3.15) hold true.Utilising Theorem 3.3 we are able to state the following result as well.Corollary 3.5. Let A1, . . . , An ∈ B (H) such that A∗j commutes with thesum∑nk=1Ak. If either (∑nk=1Ak)2 = βI for some β ∈ C or (A∗j )2 = βjIfor some βj ∈ C, j ∈ {1, . . . , n}; are valid, then both (3.14) and (3.15) holdtrue.From a different perspective, we can state the following result.Theorem 3.6. Let A1, . . . , An ∈ B (H). Then∥∥∥∥ n∑k=1Ak∥∥∥∥2 + 4 n∑k=1w2 (Ak) ≤∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 4( n∑k=1w (Ak))2.(3.16)If A∗j commutes with Ak for any k, j ∈ {1, . . . , n}, k 6= j, then∥∥∥∥ n∑k=1Ak∥∥∥∥2 + 2 n∑k=1w2 (Ak) ≤∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ 2( n∑k=1w (Ak))2.(3.17)Norm and Numerical Radius Inequalities . . . 73Proof. Let x ∈ H. Then∥∥∥∥ n∑k=1Akx∥∥∥∥2 = ∣∣∣∣ n∑j=1n∑k=1〈Akx,Ajx〉∣∣∣∣(3.18)=∣∣∣∣ n∑k=1〈Akx,Akx〉+∑1≤k 6=j≤n〈Akx,Ajx〉∣∣∣∣≤∣∣∣∣ n∑k=1〈A∗kAkx, x〉∣∣∣∣+ ∑1≤k 6=j≤n∣∣〈A∗jAkx, x〉∣∣=∣∣∣∣∣〈( n∑k=1A∗kAk)x, x〉∣∣∣∣∣+ ∑1≤k 6=j≤n∣∣〈A∗jAkx, x〉∣∣ .Taking the supremum in (3.18) over x ∈ H, ‖x‖ = 1, we deduce the followinginequality ∥∥∥∥ n∑k=1Ak∥∥∥∥2 ≤ ∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ ∑1≤k 6=j≤nw(A∗jAk),(3.19)that is of interest in itself.Since, by Theorem 3.1, in general we havew(A∗jAk) ≤ 4w (Aj)w (Ak) , k, j ∈ {1, . . . , n} , k 6= jthen ∑1≤k 6=j≤nw(A∗jAk) ≤ 4∑1≤k 6=j≤nw(Aj)w(Ak)= 4[∑i,j=1w(Aj)w(Ak)−n∑k=1w2(Ak)]= 4[( n∑k=1w(Ak))2−n∑k=1w2(Ak)],which, together with (3.19) provides the desired result (3.16).The second inequality follows by making use of the second part of Theo-rem 3.1. The details are omitted.Remark 3.2. The case n = 2 in (3.19) will produce the following sharp inequality∥∥∥∥B + C2∥∥∥∥2 ≤ 12[∥∥∥∥B∗B + C∗C2∥∥∥∥+ w (B∗C)] ,(3.20)74 S.S. Dragomirfor any B,C ∈ B(H). Here, the multiplicative constant 1/2 is best possible. Weget equality in (3.20) if B = C.If we choose B = A∗ and C = A in (3.20), then we get∥∥∥∥A+A∗2∥∥∥∥2 ≤ 12[∥∥∥∥A∗A+AA∗2∥∥∥∥+ w (A2)](3.21)for any A ∈ B(H). Here the constant 1/2 is also best possible. The equality caseholds if A is self adjoint.Finally, if B and C are chosen in (3.20) to be the Cartesian decomposition ofan operator A, then we have‖A‖2 ≤ 12[‖A∗A+AA∗‖+ w [(A+A∗) (A−A∗)]] .The constant 1/2 is sharp.If one were to place more conditions on the operators involved, the fol-lowing result could be stated as well:Theorem 3.7. If A1, . . . , An ∈ B (H) are such that A∗j double commuteswith Ak for k, j ∈ {1, . . . , n}, k 6= j, then∥∥∥∥ n∑k=1Ak∥∥∥∥2 + n∑k=1w (Ak) ‖Ak‖ ≤∥∥∥∥ n∑k=1A∗kAk∥∥∥∥+ n∑k=1w (Ak) ·n∑k=1‖Ak‖ .(3.22)Proof. Since A∗j double commutes with Ak for k, j ∈ {1, . . . , n}, k 6= j,then, by Theorem 3.2 we havew(A∗jAk) ≤ w(Aj)‖Ak‖,for k, j ∈ {1, . . . , n}, k 6= j. This implies that∑1≤k 6=j≤nw(A∗jAk) ≤ ∑1≤k 6=j≤nw (Aj) ‖Ak‖(3.23)=∑i,j=1w (Aj)w (Ak)−n∑k=1w (Aj) ‖Ak‖=n∑k=1w (Ak) ·n∑k=1‖Ak‖ −n∑k=1w (Ak) ‖Ak‖ .Now, utilising (3.19) and (3.23), we deduce the desired result (3.22).Norm and Numerical Radius Inequalities . . . 75REFERENCES1. S.S. Dragomir: Some Schwarz type inequalities for sequences of operatorsin Hilbert spaces. Bull. Austral. Math. Soc. 73 (2006), 17–26.2. S.S. Dragomir: Norm inequalities for sequences of operators related tothe Schwarz inequality. RGMIA Res. Rep. Coll. 9 (2006), Supplement,Article 2 [Online http://rgmia.vu.edu.au/v9(E).html].3. K.E. Gustafson and D.K.M. Rao: Numerical Range. Springer-Verlag,New York, Inc., 1997.4. P.R. Halmos: A Hilbert Space Problem Book. Springer-Verlag, New York,Heidelberg, Berlin, Second Edition, 1982.5. F. Kittaneh: Norm inequalities for sums and differences of positive op-erators. Lin. Alg. & Appl. 383 (2004), 85–91.6. F. Kittaneh: Numerical radius inequalities for Hilbert space operators.Studia Math. 168 (2005), No. 1, 73–80.School of Computer Science and MathematicsVictoria University, PO Box 14428Melbourne City, 8001VIC, Australiaserver.dragomir@vu.edu.auhttp://rgmia.vu.edu.au/dragomir

Norm and Numerical Radius Inequalities for Sums of Bounded Linear Operators in Hilbert Spaces

Some inequalities for the norm and numerical radius of two bounded

linear operators in Hilbert spaces are established

Victoria University Institutional Repository (VUIR)

Norm and Numerical Radius Inequalities for Two Linear Operators in Hilbert Spaces

The main aim of the present paper is to establish some norm and numerical radius inequalities

for the composite operator BA under suitable assumptions for the transform Cα ,β (T) :=

(T∗ −α I)(β I−T) , where α ,β ∈ C and T ∈ B(H), of the operators involved

Norm and Numerical Radius Inequalities for a Product of Two Linear Operators in Hilbert Spaces

S. S. Dragomir

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Sever S. Dragomir

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A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces

Some inequalities for the operator norm and numerical radius of\ud
sums of bounded linear operators in Hilbert spaces are given. Applications for\ud
the Cartesian decomposition of an operator are also provided

Name not available

Some inequalities for the norm and numerical radius of two bounded\ud
linear operators in Hilbert spaces are established

A Hilbert Space Problem Book,

Norm inequalities for sequences of operators related to the Schwarz inequality,

Norm inequalities for sums and diﬀerences of positive operators,

Numerical radius inequalities for Hilbert space operators,

Some Schwarz type inequalities for sequences of operators in Hilbert spaces,

The main aim of the present paper is to establish some norm and numerical radius inequalities\ud
for the composite operator BA under suitable assumptions for the transform Cα ,β (T) :=\ud
(T∗ −α I)(β I−T) , where α ,β ∈ C and T ∈ B(H), of the operators involved

A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix,

A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces,

New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces, Linear Alg. App.

Numerical Range,

On a generalization of an inequality of L.

Reverse inequalities for the numerical radius of linear operators in Hilbert spaces,

Some inequalities of the Gr¨ uss type for the numerical radius of bounded linear operators in Hilbert spaces,

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Norm and Numerical Radius Inequalities for Sums of Bounded Linear Operators in Hilbert Spaces

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Victoria University Institutional Repository (VUIR)

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Victoria University Eprints Repository

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Victoria University Eprints Repository

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