Inequalities for Modules and Unitary Invariant Norms

Transcription

1 Int. J. Contemp. Math. Sciences Vol no Inequalities for Modules and Unitary Invariant Norms Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei No Timisoara Romania ciu ls@yahoo.com Abstract. An elementary proof in the case of two operators and forms of the classical inequality of Bohr are presented for linear bounded operators on Hilbert spaces or on pseudo-hilbert spaces in Lemma 2 Theorem and Theorem 2. We will also investigate some variant of the inequalities of O. T. Pop see [9] for linear and bounded operators on Hilbert spaces and then for linear and bounded operators with orthogonal ranges. Then several applications of these results for unitary invariant norms and generalized derivative are given in Proposition 5 and 4 and a generalization of Lemma see [6]. Mathematics Subject Classification: Primary 47A45; Secondary 42B0 Keywords: pseudo-hilbert spaces(loynes spaces) Bohr s inequality superquadratic functions orthogonal ranges. Introduction We recall the classical Bohr s inequality see [3]. p q > with + = p q z + w 2 p z 2 + q w 2 For any z w C and with equality if and only if w =(p )z. Many interesting generalizations of Bohr s inequality have been obtained last years. In this paper we will discuss some forms of the inequality in the case when we have two operators or two positive numbers. We need the following definition which was presented in [7]. Let Z be an admissible space in the Loynes sense. A linear topological space H is called pre-loynes Z space if it satisfies the following properties: H is endowed with a Z valued inner product (gramian) i.e. there exists a mapping H H (h k) [h k] Z having the properties: [h h] 0;

2 772 L. Ciurdariu [h h] = 0 implies h = 0; [h + h 2 h] = [h h]+[h 2 h]; [λh k] = λ[h k]; [h k] =[k h]; for all h k h h 2 Hand λ C. The topology of H is the weakest locally convex topology on H for which the application H h [h h] Z is continuous. Moreover if H is a complete space with this topology then H is called Loynes Z space (pseudo-hilbert space). Let B(H) be the space of linear and bounded operators on Hilbert space H and B (H) the space of linear and bounded operators which admit adjoint on H if H is a pseudo-hilbert space. 2. The results We can prove a similar result to Lemma 2.2 see [0] in pseudo-hilbert spaces for example or Hilbert spaces for the modulus of a linear bounded operator. Lemma. Let A B B (H)(the space of linear and bounded operators which admit gramian adjoint) if H is a Loynes Z- space (or A B B(H) if H is a Hilbert space). If A ± B =( A 2 + B 2 ) 2 then αa ± B =(α 2 A B 2 ) 2 for all α non-negative real numbers. Proof. By A±B 2 =(A ±B )(A±B) = A 2 + B 2 ±B A±A B = A 2 + B 2 it follows that B A + A B = 0 and then αa ± B 2 = α 2 A B 2. We shall give below another proof for a variant of Bohr s inequality that was given in [2] Theorem 6. Lemma 2. Let A B B (H) where H is a Loynes Z- spaceora B B(H) where H is a complex separable Hilbert space. (i) If α γ are three real numbers which satisfies the conditions 0 γ 0 α + γ > 0 and α>0 or α + γ < 0 and α<0 then αa B 2 + A γb 2 ( αγ) A 2 + γ(γ α ) B 2. (ii) If α γ are three real numbers which satisfies the conditions α 0 0γ 0 α + γ > 0 and > 0 or α + γ < 0 and < 0 then γ γ αa B 2 + A γb 2 α(α γ ) A 2 +( αγ ) B 2. (iii) If α γ are three real numbers which satisfies the conditions 0 γ 0 α + γ < 0 and α>0 or α + γ > 0 and α<0 then αa B 2 + A γb 2 ( αγ) A 2 + γ(γ α ) B 2.

3 Inequalities for modules and unitary invariant norms 773 (iv) If α γ are three real numbers which satisfies the conditions α 0 0γ 0 α + γ < 0 and > 0 or α + γ > 0 and < 0 then γ γ αa B 2 + A γb 2 α(α γ ) A 2 +( αγ ) B 2. Proof. Using the definition of the modulus of an operator we obtain αa B 2 + A γb 2 =(α ) A 2 +(γ 2 +) B 2 (α+γ)(b A+A B)= =[(α + γ)(α + γ ) α(γ + γ )] A 2 +[(α + γ)( γ + α ) γ(α + α )] B 2 (α+γ)(b A+A B)=(α+γ)[(α+ γ ) A 2 +( γ + α ) B 2 (B A+A B)] α(γ+ γ ) A 2 γ( α + α ) B 2 =(α+γ)[α A 2 + γ A 2 + γ B 2 + α B 2 (B A+ +A B)] α(γ + γ ) A 2 γ( α + α ) B 2. For (i) we will consider the following continuation αa B 2 + A γb 2 =(α+γ){[α A 2 (B A+A B)+ α B 2 ]+[ γ A 2 + γ B 2 ]} α(γ+ γ ) A 2 γ( α + α ) B 2 =(α+γ) αa α B 2 +[(α+γ) γ α(γ+ γ )] A 2 + +[(α + γ) γ γ(α + α )] B 2 ( αγ) A 2 + γ(γ α ) B 2. For the second part of (i) we will write αa B 2 + A γb 2 =(α+γ){[ α A 2 (B A+A B) α B 2 ]+[ γ A 2 + γ B 2 ]} α(γ+ γ ) A 2 γ( α + α ) B 2 = (α+γ) α A+ α B 2 +( αγ) A 2 + For (ii) we have +γ(γ γ ) B 2 ( αγ) A 2 + γ(γ α ) B 2. αa B 2 + A γb 2 =(α+γ){[α A 2 + α B 2 ]+[ γ A 2 (B A+A B)+ γ B 2 ]} α(γ + γ ) A 2 γ( α + α ) B 2 =(α + γ) γ A γ B 2 +[(α + γ)α α(γ + γ )] A 2 +[(α+γ) α γ(α + α )] B 2 α(α γ ) A 2 +( αγ ) B 2. (iii) and (iv) will result from (i) and (ii).

4 774 L. Ciurdariu Theorem. Let A B B (H) where H is a Loynes Z- spaceora B B(H) where H is a complex separable Hilbert space. (i) If a b c d R a b c d 0and satisfies the conditions ab+cd > 0 and a > 0 or ab + cd < 0 and a < 0 then b b aa bb 2 + ca db 2 c 2 ( ad cb ) A 2 + d 2 ( cb ad ) B 2 or if a b c d R abcd 0and satisfies the conditions ab + cd > 0 and c > 0 or ab + cd < 0 and c < 0 then d d aa bb 2 + ca db 2 a 2 ( bc ad ) A 2 + b 2 ( ad bc ) B 2. (ii) If a b c d R a b c d 0and satisfies the conditions ab+cd < 0 and a > 0 or ab + cd > 0 and a < 0 then b b aa bb 2 + ca db 2 c 2 ( ad cb ) A 2 + d 2 ( cb ad ) B 2 or if a b c d R abcd 0and satisfies the conditions ab + cd < 0 and c > 0 or ab + cd > 0 and c < 0 then d d aa bb 2 + ca db 2 a 2 ( bc ad ) A 2 + b 2 ( ad bc ) B 2. Proof. We write the expression aa bb 2 + ca db 2 as b 2 ( a b A B 2 + ca b d b B 2 ) and use Theorem. Next theorem is a generalization of Theorem 2.3 []. Theorem 2. (i) For any r 2 ab R + and u v > 0 with the property uv(u + v) r 2 = we have ( + u v )ar +(+ v u )br (ua + vb) r + 2 b r 2 a r. (ii) For any r 2 ab R + and u v > 0 with the property uv(u + v) r 2 = we have ( + u v )ar +(+ v u )br (ua + vb) r + 2 b r 2 a r. Proof. Using the fact that the functions f(x) =x r x 0are superquadratic for r 2 and subquadratic for 0 r 2 see [] we obtain by Lemma.2 see [] for 0 α ab 0 that α f(a)+( α )f(b) f(α a+( α b)) α f(( α ) b a )+( α )f(α b a ). (i) As in the proof of Theorem 2.3 [] we have α a r + b r (α a + b) r + α ( r + α r ) b a r where = α.

5 Inequalities for modules and unitary invariant norms 775 For α = +γ = γ γ>0it results +γ + γ ar + γ + γ br ( + γ a+ γ + γ b)r + γ ( + γ) [( γ 2 + γ )r +( + γ )r ] b a r Because ( γ +γ )r or = γ +γ which leads to + γ a r + + γ γ +γ +γ 2 r 2 and this implies (a r + γb r ) + γ γ +γ + γ γ (ar + γb r ) br ( [0 ] and r 2 we have ( +γ )r + (a + γb)r + γ b ( + γ) r ( + γ) 2 a r 2r 2 (γ) r ( + γ) r 2 r (a + γb)r γ( + γ) + b r 2 a r 2r 2 a + γ (γ) r ( + γ) r 2 r b) r + 2 r 2 b a r. Now taking γ = v where >0we have γ>0 = u and our inequality u γ v becomes ( + u v )ar +(+ v u )br (ua + vb) r + b a r 2r 2 because u and (γ) r ( + γ) r 2 r = ( 2 v ) r ( + v r 2 ) r u u γ (γ) r ( + γ) r 2 r = = v u r v r (u + v) 2 r by hypothesis. For (ii) we will use the same argument taking into account that α r if r 2. 2 r 2 = u + r Corollary. If we take above u = v(p ) when r 2 and <p 2 then pa r + qb r 2 ((p )a + r 2 b)r + 2 b r 2 a r for any a b R + and + =. p q Proof. From u = v(p ) and <p 2 it results u>0 and pa r + qb r =(+p )a r +(+ p )br v r ((p )a + b) r + 2 b r 2 a r. Replacing u from u = v(p ) in uv(u + v) r 2 = we have v 2 (p )[v(p ) + v] r 2 =orv r (p )p r 2 = that means v r =. Thus (p )p r 2 2 r 2 pa r + qb r =(+p )a r +(+ p )br 2 r 2 ((p )a + b)r + 2 r 2 b a r.

6 776 L. Ciurdariu Now we consider n linear bounded operators having orthogonal ranges as below and we prove the following equality an analogue result of some results from [9] [4] [5]: Theorem 3. If n N n 2 N N 2... N n B (H) with Ni N j = 0 ( ) i j =... n i j i.e the operators have orthogonal ranges and a a 2... a n R \{0} then a i N j a j N i 4 a = N N n 4 a2 n a 2 i<j n i a2 j a a2 n ( 2 a 2 k a ) N k 4 = a2 n = a a 2 n i<j n a i N j + a j N i 4. a 2 i a2 j Proof. Taking into account that the operators have orthogonal ranges and using then the previous theorem with a 2 i instead of a i and N i 2 instead N i i=... n we have a i N j a j N i 4 (a 2 i N j 2 + a 2 j N i 2 ) 2 a = a2 n a a2 n i<j n a 2 i a2 j i<j n a 2 i a2 j = ( N N n 2 ) 2 a a2 n ( 2 a 2 k a ) N k 4 = N N n 4 a2 n a a2 n + ( 2 ) N a 2 k a a 2 k 4. n = As a consequence of the Theorem 3 and Proposition 2.7 see [8] we have: Consequence. If n N n 2 N N 2... N n B (H) where H is a Loynes Z- space with Ni N j =0 ( ) i j =... n i j i.e the operators have orthogonal ranges and a a 2... a n α... α n R + \{0} then i<j n a i N j a j N i 4 a i a j (a i. a k + a i Proof. Taking in Proposition 2.7 r = 4 we have N i 4 α i N i 4. i<j n Using now Theorem 3 we obtain: a i N j a j N i 4 a i a j α i α i α i N i 4 + a k 2) N k 4. ( a i 2) N k 4 = a k

7 Inequalities for modules and unitary invariant norms 777 = [α i + a k 2] N i 4. α k a i Let x =(x... x n ) R n and we define see [5] an n n matrix Λ(x) = x x =(x i x j ) and D(x) =diag(x... x n ). Now we will give an analogue of Theorem 3. from [5] for operators with orthogonal ranges. Proposition. If Λ(a 2 )+Λ(b 2 ) D(c 2 ) for a b c R n then a i A i 4 + b i A i 4 c i A i 4 for arbitrary n-tuple (A i ) of operators with orthogonal ranges (i.e A ja i = 0 ( ) i j i j n)inb (H) or B(H) respectively. In addition if Λ(a 2 )+Λ(b 2 ) D(c 2 ) for a b c R n then a i A i 4 + b i A i 4 c i A i 4 for arbitrary n-tuple (A i ) of operators with orthogonal ranges in B (H) or B(H) respectively. Proof. We take into account that a i A i 4 + b i A i 4 = and then use Theorem 3. [5]. Using Theorem 28 from [2] we have: a 2 i A i b 2 i A i 2 2 Proposition 2. Let A i B (H) with A i A j = 0 i j n and α ik p i R for i = 2... n and k = 2... m. Define X =(x ij ) where x ij = m α4 ik p i if i = j and x ij = m α2 ik α2 jk if i j. If X 0 then m α ik A i 4 p i A i 4. If X 0 then m α ik A i 4 Proof. We use that m α ik A i 4 = if X 0 i.e. use Theorem 28 [2]. p i A i 4. m ( α 2 ik A i 2 ) 2 p i A i 4

8 778 L. Ciurdariu Proposition 3. If n N n 2 N N 2... N n B(H) and α α 2... α n R \{0} with α + α α n 0and Λ(a) +Λ(b) D(c) for a b c R n then i<j n α i a j N j α j a i N i 2 + α i b j N j α j b i N i 2 α i α j ( a i + b i α i α k c i ) N i 2. As in the case of real and complex numbers see [] it can be shown the following two inequalities if we take into consideration n = 2: Remark. For any linear gramian bounded operators N N 2 B (H) we have N + N 2 2 (2) N 2 u + v u + N 2 2 v if u v 0u+ v 0uv(u + v) > 0 and N + N 2 2 (3) N 2 u + v u + N 2 2 v if u v 0u+ v 0uv(u + v) < 0. In addition if N N 2 =0 then N 4 (4) u + N 2 4 N + N 2 4 v u + v if u v 0u+ v 0uv(u + v) > 0 or N + N 2 4 (5) N 4 u + v u + N 2 4 v if u v 0u+ v 0uv(u + v) < 0. We can obtain below another generalization of a theorem presented in [9]. Remark 2. (i) If n N n 2 N i B (H) i = n and a a 2... a n R \{0} with a + a a n 0 and a m (a k + a l ) > 0 ( ) m = n m k l a k a l 0 then N 2 + N N n 2 N + N N n 2 a a 2 a n a + a a n a i N j a j N i 2 N kl + a + a a n a i a j i<j n ij kl where a i N j a j N i 2 N kl = max i<j n a i a j (a i + a j ) = a kn l a l N k 2 k<l n a k a l (a k + a l ) if there exist.

9 Inequalities for modules and unitary invariant norms 779 (ii) If n N n 2 N i B (H) i= n and a a 2... a n R \{0} with a + a a n 0 and then a m (a k + a l ) < 0 ( ) m = n m k l a k a l 0 N 2 + N N n 2 N + N N n 2 a a 2 a n a + a a n N kl + a + a a n i<j n ij kl a i N j a j N i 2 a i a j where a i N j a j N i 2 N kl = min i<j n a i a j (a i + a j ) = a kn l a l N k 2 k<l n a k a l (a k + a l ) if there exist. Proof. The proof will be as in the case of real numbers see [9]. We will present now a generalization of Theorem 4 when the operators N i i=... n have the orthogonal ranges. Theorem 4. If n N n 2 N i B (H) i= nwith N i N j =0 ( ) i j =... n i j i.e the operators have orthogonal ranges and a a 2... a n R \{0} then N N n 4 a a2 n + ( 2 a 2 k a ) N k 4 a2 n where N kl = a kn l a l N k 4 a 2 k a2 l (a2 k + a2 l ) + a a 2 n i<j n a i N j a j N i 4 a 2 i a2 l a i N j a j N i 4 max i<j n a 2 i a2 j (a2 i + a2 j ) = a kn l a l N k 4 a 2 k a2 l (a2 k + k<l n a2 l ) if there exist. Proof. The proof will be as in the case of real numbers see [9] using Remark 2 and Theorem 3.

10 780 L. Ciurdariu 3. Applications Let B(H) denote the algebra of all bounded linear operators on a complex separable Hilbert space H. We shall consider as in [6] a unitarily invariant norm. which is a norm on an ideal C. of B(H) making C. into a Banach space and satisfying UXV = X for all X B(H) and all unitary operators U and V in B(H). Lemma 3 ([6]). Let A B X B(H) where H is a Hilbert space with A and B self-adjoint and X γi in which γ is a positive real number. Then γ A B AX BX. In the following we shall give some generalizations of Theorem 2 of O. Hirzallah see [6]. Theorem 5. Let A B X B(H) where H is a complex separable Hilbert space and X γi for positive real number γ. If a b c d R abcd 0and satisfies the conditions ab + cd < 0 and c > 0 or ab + cd > 0 and c < 0 then d d γ aa bb 2 + ca db 2 a 2 ( bc ad ) A 2 X + b 2 ( ad bc )X B 2. Proof. We replace in Lemma 3 A by a 2 ( bc ad ) A 2 and B by b 2 ( ad bc ) B 2 and we use Theorem (ii). Remark 3. (a) Let A B X B(H) where H is a Hilbert space as in [6] and +γ < 0. IfX γ I γ is a positive real number then γ A B 2 + A γb 2 γ A 2 X γx B 2. (b) If we take in Theorem 3 (i) a = p b = c = d =we obtain γ ( p)a B 2 + A B 2 p A 2 X + qx B 2 when <p<2 and + =. p q (c) If we take in Theorem 3 (ii) a = b = c =and d = q we obtain γ A B 2 + A ( q)b 2 p A 2 X + qx B 2 when p>2 and + =. p q We recall that the generalized derivation δ AB of two operators A B B(H) is defined by δ AB (X) =AX XB for all X B(H). Also δ 2 AB(X) = δ AB (δ AB (X)). We shall give a generalization of Theorem 4 from [6]. The constants p and q from Theorem 4 can be replaced by a b c d as below.

11 Inequalities for modules and unitary invariant norms 78 Proposition 4. Let A B X B(H) where H is a complex separable Hilbert space and A B are normal operators. (i) If a b c d R a b c d 0and satisfies the conditions ab+cd < 0 and a > 0 or ab + cd > 0 and a < 0 then b b δ 2 aabb (X) δ2 cadb (X) 2 2 c2 ( ad bc ) A 2 X + d 2 ( cb ad )X B (ii) If a b c d R a b c d 0and satisfies the conditions ab+cd > 0 and a > 0 or ab + cd < 0 and a < 0 then b b δ 2 aabb (X) δ2 cadb (X) c2 ( ad bc ) A 2 X + d 2 ( cb ad )X B Proof. It will be as in [6] Theorem 4 but we will use Consequence 3 (ii) (i) instead of inequality (7) see [6]. Also we use the elementary inequalities (x + y) 2 2(x 2 + y 2 ) and x 2 + y 2 (x + y) 2 for all x y 0. We know see [] for example that for uv(u + v) > 0 and r> z z 2 C we have z + z 2 r z r u + v u + z 2 r v and for uv(u + v) < 0 and r> z z 2 C we have z + z 2 r u + v z r u + z 2 r v. Considering r N r 0 we can define δ r AB(X)byδ r AB(X) =δ AB (δ r AB (X)) and by induction we can show that δ r AB (X) =Ar X Cr Ar XB ( ) r Cr r AXB r +( ) r XB r. We will give two similar properties for the generalized derivation δ AB of two normal operators A B B(H). Proposition 5. Let A B X B(H) where H is a complex separable Hilbert space and A B are normal operators. (i) If uv > 0 and r r N then (u + v) 2 δr A B (X) 2 2 u A r X + v X B r 2 2. for every X B(H). (ii) If r N r 2 and w w 2 R + then w δ r A B(X) 2 2 A r X + X B r 2 2. w + w 2 w + w 2 (iii) If r 3 r N and D D 2 are two diagonal positive operators in B(H) with u v as in Theorem 2 then δ r ud vd 2 (X) (r 2) δr D D 2 (X) 2 2 ( + u v )Dr X +(+v u )XDr w 2

12 782 L. Ciurdariu Proof. We use the same method as in [6]. We use Voiculescu s Theorem. Given ε>0there are diagonal operators D D 2 and Hilbert-Schmidt operators K K 2 such that A = D + K B = D 2 + K 2 K < ε K 2 < ε D e i = λ i e i and D 2 f i = μ i f i i Nfor some orthonormal bases {e i } and {f i } for H and some sequences {λ i } and {μ i } of complex numbers. If we show that the inequality is true for D D 2 then by a limit argument the inequality from (i) (ii) will be true. For (i) we calculate = (u + v) 2 δr D D 2 (X) 2 2 = (u + v) 2 ( λ i + μ j r ) 2 <Xf j e i > 2 u + v For (ii) we have δ r D D 2 (X) 2 2 = <δ r D D 2 (X)f j e i > 2 = ( λ i r u = u D r X + v X D 2 r 2. <δ r D D 2 (X)f j e i > 2 = + μ j r v )2 <Xf j e i > 2 = ( λ i +μ j r ) 2 <Xf j e i > 2 w ( λ w + w i r w 2 + μ 2 w + w j r ) 2 <Xf j e i > 2 = 2 w = D w + w r w 2 X + X D 2 w + w 2 r taking into account that we used inequality (.) from Theorem see []. For (iii) we use Theorem 2 (i) and we obtain: δ r ud vd 2 (X) (r 2) δr D D 2 (X) 2 2 = <δ r ud vd 2 (X)f j e i > (r 2) + 2 2(r 2) <δ r D D 2 (X)f j e i > 2 = < λ i μ j r Xf j e i > 2 = <Xf j e i > 2 < uλ i + vμ j r Xf j e i > 2 + ((uλ i + vμ j ) 2r + 2 λ 2(r 2) i μ j 2r ) ((uλ i + vμ j ) r + 2 λ (r 2) i μ j r ) 2 <Xf j e i > 2 ((+ u v )λr i +(+ v u )μr j) 2 <Xf j e i > 2 = (+ u v )Dr X +(+ v u )XDr

13 Inequalities for modules and unitary invariant norms 783 We also can replace in Theorem 3 from [6] the constants p and q with c and d as below. Proposition 6. Let A B X B(H) where H is a complex separable Hilbert space and c d R {0} with cd <. If X B(H) with X (c d)(c A 2 d B 2 ) then A B 4 c d c A 2 X dx B 2. References [] S. Abramovich J. Baric and J. Pecaric Superquadracity Bohr s inequality and deviation from a mean value The Australian Journal of Mathematical Analysis and Applications Volume 7 Issue Article pp [2] P. Chansangiam P. Hemchote P. Pantaragphong Generalizations of Bohr inequality for Hilbert space operators J. Math. Anal. Appl. 356 (2009) [3] W.-S. Cheung J. Pecaric Bohr s inequalities for Hilbert space operators J. Math. Anal. Appl. 323 (2006) [4] Ciurdariu L. On Bergstrom inequality for commuting gramian normal operators Journal of Mathematical Inequalities 4 No. 4 (200) [5] M. Fujii H. Zuo Matrix order in Bohr inequality for operators Banach J. Math. Anal. 4 (200) no [6] O. Hirzallah Non-commutative operator Bohr inequality J. Math. Anal. Appl. 282 (2003) [7] R. M. Loynes On generalized positive definite functions Proc. London Math. Soc 3 (965) [8] Moslehian M.S. Pecaric J.E. Peric I. An operator extension of Bohr s inequality Bulletin of the Iranian Mathematical Society 35 2 (2009) pp [9] Pop O. T. About Bergstrom s inequality Journal of Mathematical Inequalities 3 No. 2 (2009) [0] Rajna Rajic Characterization of the norm triangle equality in Pre-Hilbert C -modules and applications Journal of Mathematical Inequalities 3 3(2009) [] M. P. Vasic D. J. Keckic Some inequalities for complex numbers Mathematica Balkanica (97) [2] F. Zhang On the Bohr inequality of operators J. Math. Anal. Appl. 333 (2007) Received: April 202