INEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES

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1 INEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES S.S. DRAGOMIR ; Abstract. Let (z) := P anzn be a power series with complex coe - cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that kt k ; kv k < R; then we show among others that k (T ) (V )k a (max kt k ; kv kg) kt V k where a (z) := P janj zn : I U; V B (H) are such that the numerical radii w (T ) ; w (V ) < R then w [ (T ) (V )] 4a (max w (T ) ; w (V )g) w (T V ) : Applications in connection with the Hermite-Hadamard inequality or Lipschitzian unctions are provided as well.. Introduction The numerical radius w (T ) o an operator T on H is given by [4, p. 8]: (.) w (T ) = sup jj ; W (T )g = sup jht x; xij ; kxk = g : It is well known that w () is a norm on the Banach algebra B (H) o all bounded linear operators T : H! H: This norm is equivalent with the operator norm. In act, the ollowing more precise result holds [4, p. 9]: Theorem (Equivalent norm). For any T B (H) one has (.) w (T ) kt k w (T ) : Some improvements o (.) are as ollows: Theorem (Kittaneh, 3 [9]). For any operator T B (H) we have the ollowing re nement o the rst inequality in (.) (.3) w (T ) kt k + T = : From a di erent perspective, we have the ollowing result as well: Theorem 3 (Dragomir, 7 [6]). For any operator T B (H) we have (.4) w (T ) h w T + kt k i : The ollowing general result or the product o two operators holds [4, p. 37]: Theorem 4 (Holbrook, 969 [6]). I A; B are two bounded linear operators on the Hilbert space (H; h; i) ; then w (AB) 4w (A) w (B) : In the case that AB = BA; then w (AB) w (A) w (B) : The constant is best possible here. 99 Mathematics Subject Classi cation. 47A63; 47A99. Key words phrases. Bounded linear operators, Functions o operators, Numerical radius, Power series.

2 S.S. DRAGOMIR ; The ollowing results are also well known [4, p. 38]. Theorem 5 (Holbrook, 969 [6]). I A is a unitary operator that commutes with another operator B; then (.5) w (AB) w (B) : I A is an isometry AB = BA; then (.5) also holds true. For other results on numerical radius inequalities see [], [3]-[7], [8]-[], [3] [7]-[3]. Now, by the help o power series (z) = P a nz n we can naturally construct another power series which will have as coe cients the absolute values o the coe- cient o the original series, namely, a (z) := P ja nj z n : It is obvious that this new power series will have the same radius o convergence as the original series. We also notice that i all coe cients a n ; then a = : In this paper we establish some Lipschitz type inequalities, i.e., bounds or the quantities k (T ) (V )k ; k (kt k) T (kv k) V k w ( (T ) (V )) etc. where T; V B (H), in terms o di erent values o a, kt V k w (T V ) ; respectively: Applications in relation with the operator version o the Hermite-Hadamard inequality that compares the operators (U) + (V ) ; Z (( s) U + tv ) ds are also given. Some examples or particular unctions o interest such as the exponential, logarithmic trigonometric unctions are also presented.. Norm Inequalities We start with the ollowing result that provides a quasi-lipschitzian condition or unctions de ned by power series: Theorem 6. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that kt k ; kv k < R; then (.) k (T ) (V )k a (max kt k ; kv kg) kt V k : Proo. We show rst that the ollowing power inequality holds true or any n N (.) kt n V n k n (max kt k ; kv kg) n kt V k : We prove this by induction. We observe that or n = n = the inequality reduces to an equality. Assume now that (.) is true or k N, k let us prove it or k + :

3 INEQUALITIES OF LIPSCHITZ TYPE 3 Utilising the properties o the operator norm, the induction hypothesis, we have successively T k+ V k+ = T k (T V ) + T k V k V T k (T V ) + T k V k V T k kt V k + T k V k kv k kt k k kt V k + k (max kt k ; kv kg) k kt V k kv k n max kt k k ; kv k ko kt V k + k (max kt k ; kv kg) k kt V k max kt k ; kv kg = (max kt k ; kv kg) k kt V k + k (max kt k ; kv kg) k kt V k = (k + ) (max kt k ; kv kg) k kt V k the inequality (.) is proved. Now, or any m, by making use o the inequality (.) we have m X (.3) a n T n a n V n ja n j kt n V n k kt V k n ja n j (max kt k ; kv kg) n : Since the series P a nt n ; P a nv n P n ja nj (max kt k ; kv kg) n are convergent, then by letting m! in (.3) we get the inequality (.). Remark. We observe, rom the proo o the above theorem, that the inequality (.) remains valid in the more general setting o a Banach algebra. However the details are not considered here. We de ne the absolute value o an operator A B (H) de ned as jaj as the square root operator o the positive operator A A. With this notation, we have: Corollary. With the above assumptions or, we have (.4) k (T ) (T )k a (kt k) kt T k i T B (H) with kt k < R (.5) jn j jnj a knk jn j jnj i N B (H) with knk < R: Remark. With the assumption o Theorem 6 we have k (jt j) (jv j)k a (max kt k ; kv kg) kjt j jv jk provided kt k ; kv k < R; in particular, the ollowing power inequality that we will use below (.6) kjt j n jv j n k n (max kt k ; kv kg) n kjt j jv jk that holds or any T; V B (H) :

4 4 S.S. DRAGOMIR ; We notice that i (.7) ( ) n (z) = n zn = ln n= g (z) = h (z) = l (z) = ; z D (; ) ; + z ( ) n (n)! zn = cos z; z C; ( ) n (n + )! zn+ = sin z; z C; ( ) n z n = ; z D (; ) ; + z then the corresponding unctions constructed by the use o the absolute values o the coe cients are (.8) a (z) = g a (z) = h a (z) = l a (z) = n= n! zn = ln ; z D (; ) ; z (n)! zn = cosh z; z C; (n + )! zn+ = sinh z; z C; z n = ; z D (; ) : z (.9) I T; V B (H) are such that kt k ; kv k < ; then by (.) we have (I T ) (I V ) ( max kt k ; kv kg) kt V k (.) ln (I T ) ln (I V ) ( max kt k ; kv kg) kt V k : I T; V B (H) ; then by (.) we also have (.) max ksin T sin V k ; ksinh T sinh V kg cosh (max kt k ; kv kg) kt V k (.) max kcos T cos V k ; kcosh T cosh V kg sinh (max kt k ; kv kg) kt V k :

5 INEQUALITIES OF LIPSCHITZ TYPE 5 Other important examples o unctions as power series representations with nonnegative coe cients are: (.3) exp (z) = n! zn ; z C; + z ln = z n zn ; z D (; ) ; n= sin n + (z) = p z n+ ; z D (; ) ; (n + ) n! tanh (z) = F (; ; ; z) = n= z D (; ) ; n zn ; z D (; ) ; (n + ) (n + ) () n! () () (n + ) zn ; ; ; > ; where is Gamma unction. I T; V B (H) are such that kt k ; kv k < ; then by (.) we have (.4) tanh T tanh V h (max kt k ; kv kg) i kt V k (.5) sin T sin V I T; V B (H) ; then by (.) we also have h (max kt k ; kv kg) i = kt V k : (.6) kexp (T ) exp (V )k exp (max kt k ; kv kg) kt V k : The ollowing result also holds. Theorem 7. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that kt k ; kv k < R; then (.7) k (kt k) T (kv k) V k [ a (max kt k ; kv kg) + max kt k ; kv kg a (max kt k ; kv kg)] kt V k : Proo. We show rst that the ollowing power inequality holds true or any n N (.8) kkt k n T kv k n V k (n + ) (max kt k ; kv kg) n kt V k : For n = ; the inequality becomes an equality. Assume that n ; then we have (.9) kkt k n T kv k n V k = kkt k n T kt k n V + kt k n V kv k n V k kkt k n (T V )k + k(kt k n kv k n ) V k = kt k n kt V k + jkt k n kv k n j kv k (max kt k ; kv kg) n kt V k + jkt k n kv k n j max kt k ; kv kg :

6 6 S.S. DRAGOMIR ; On the other h (.) jkt k n kv k n j = jkt k kv kj kt k n + ::: + kv k n n kt V k (max kt k ; kv kg) n : Using (.9) (.) we have kkt k n T kv k n V k (max kt k ; kv kg) n kt V k + n kt V k (max kt k ; kv kg) n = (n + ) (max kt k ; kv kg) n kt V k the inequality (.8) is proved. Now, or any m, by making use o the inequality (.8) we have (.)!! a n kt k n T a n kv k n V ja n j kkt k n T kv k n V k kt V k (n + ) ja n j (max kt k ; kv kg) n = kt V k + ja n j (max kt k ; kv kg) n! n ja n j (max kt k ; kv kg) n = kt V k + ja n j (max kt k ; kv kg) n! n ja n j (max kt k ; kv kg) n : n= Since the ollowing series are convergent a n kt k n = (kt k) ; a n kv k n = (kv k) ; ja n j (max kt k ; kv kg) n = a (max kt k ; kv kg) n ja n j (max kt k ; kv kg) n = max kt k ; kv kg a (max kt k ; kv kg) n= then by letting m! in (.) we deduce the desired result (.7). Remark 3. We observe that, since no norm inequality or the operator product has been used in the proo o the inequality (.7), it holds true or any norm on B (H) :

7 INEQUALITIES OF LIPSCHITZ TYPE 7 Corollary. Let be as in Theorem 6, then (.) k (kabk) AB (kbak) BAk kab BAk [ a (max kabk ; kbakg) + max kabk ; kbakg a (max kabk ; kbakg)] [ a (kak kbk) + kak kbk a (kak kbk)] kab BAk or any A; B B (H) with kak kbk < R: From a di erent perspective we have: Theorem 8. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that kt k ; kv k < R; then (.3) k (jt j) T (jv j) V k a (max kt k ; kv kg) kt V k + max kt k ; kv kg a (max kt k ; kv kg) kjt j jv jk : Proo. Observe that or n ; we have (.4) kjt j n T jv j n V k = kjt j n T jt j n V + jt j n V jv j n V k kjt j n (T V )k + k(jt j n jv j n ) V k kt k n kt V k + kjt j n jv j n k kv k (max kt k ; kv kg) n kt V k + kjt j n jv j n k max kt k ; kv kg : Since by (.6) we have (.5) kjt j n jv j n k n (max kt k ; kv kg) n kjt j jv jk ; then by (.4) (.5) we have (.6) kjt j n T jv j n V k (max kt k ; kv kg) n kt V k + n (max kt k ; kv kg) n kjt j jv jk : The inequality (.6) also holds or n = ; so by employing an argument similar to the one rom the proo o Theorem 6 we deduce the desired result (.3). Corollary 3. With the assumptions o Theorem 8 or we have (.7) k (jt j) T (jt j) T k a (kt k) kt T k + kt k a (kt k) kjt j jt jk : i T B (H) with kt k < R. Proposition. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T B (H) are such that kt k < R; then (.8) k (T ) T (T ) T k [ a (kt k) + kt k a (kt k)] kt T k :

8 8 S.S. DRAGOMIR ; Proo. For any n N we have kt n T (T ) n T k = T n T T n+ + T n+ (T ) n T = kt n (T T ) + (T n (T ) n ) T k kt n (T T )k + k(t n (T ) n ) T k kt k n kt T k + kt n (T ) n k kt k (max kt k ; kt kg) n kt T k + kt n (T ) n k max kt k ; kt kg = kt k n kt T k + kt k kt n (T ) n k =: J Utilising the inequality (.4) or the power unction we have (.9) kt n (T ) n k n kt k n kt T k : By (.9) we have Thereore J kt k n kt T k + kt k kt n (T ) n k kt k n kt T k + n kt k n kt T k = (n + ) kt k n kt T k : kt n T (T ) n T k (n + ) kt k n kt T k ; or any n N. On making use o similar argument to that in Theorem 7 we obtain the desired result (.8). Remark 4. By employing the examples o power series presented beore the reader may state other operator inequalities. However the details are omitted. The ollowing result holds: 3. Numerical Radius Inequalities Theorem 9. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that w (T ) ; w (V ) < R; then (3.) w ( (T ) (V )) 4 a (max w (T ) ; w (V )g) w (T V ) : Moreover, i T V are commutative, then we have a better inequality (3.) w ( (T ) (V )) a (max w (T ) ; w (V )g) w (T V ) : Proo. We show rst that the ollowing power inequality holds true or any n N (3.3) w (T n V n ) 4n (max w (T ) ; w (V )g) n w (T V ) : We prove this by induction. We observe that or n = n = the inequality reduces to an equality. Assume now that (3.3) is true or k N, k let us prove it or k + :

9 INEQUALITIES OF LIPSCHITZ TYPE 9 Utilising the properties o the numerical radius, the induction hypothesis, we have successively w T k+ V k+ = w T k (T V ) + T k V k V w T k (T V ) + w T k V k V 4w T k w (T V ) + 4w T k V k w (V ) 4w (T ) k w (T V ) + 4k (max w (T ) ; w (V )g) k w (T V ) w (V ) n 4 max w (T ) k ; w (V ) ko w (T V ) + 4k (max w (T ) ; w (V )g) k w (T V ) max w (T ) ; w (V )g = 4 (max w (T ) ; w (V )g) k w (T V ) + 4k (max w (T ) ; w (V )g) k w (T V ) = 4 (k + ) (max w (T ) ; w (V )g) k w (T V ) the inequality (3.3) is proved. Now, or any m, by making use o the inequality (3.3) we have m! X (3.4) w a n T n a n V n ja n j w (T n V n ) 4w (T V ) n ja n j (max w (T ) ; w (V )g) n : Since the series P a nt n ; P a nv n P n ja nj (max kt k ; kv kg) n convergent, then by letting m! in (3.4) we get the inequality (3.). are Now, i T V are commutative, then T k T V as well as T k V k V are commutative we have w T k (T V ) + w T k V k V w T k w (T V ) + w T k V k w (V ) perorming an argument similar with the one above, we deduce the inequality (3.5) w (T n V n ) n (max w (T ) ; w (V )g) n w (T V ) ; or any n N. This implies the desired inequality (3.). The details are omitted. Taking into account Remark 3 we can state the ollowing result or the numerical radius: Proposition. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) are such that

10 S.S. DRAGOMIR ; w (T ) ; w (V ) < R; then (3.6) w ( (w (T )) T (w (V )) V ) [ a (max w (T ) ; w (V )g) + max w (T ) ; w (V )g a (max w (T ) ; w (V )g)] w (T V ) : For unitary commuting operators we have the ollowing result o interest as well: Proposition 3. Let (z) := P a nz n be a power series with complex coe - cients convergent on the open disk D (; R) ; R > : I T; V B (H) are two commuting unitary operators z D (; R) ; then (3.7) w ( (zt ) (zv )) jzj a (jzj) w (T V ) : Proo. We show rst that the ollowing power inequality holds true or any n N (3.8) w (T n V n ) nw (T V ) : We prove this by induction. We observe that or n = n = the inequality reduces to an equality. Assume now that (3.3) is true or k N, k let us prove it or k + : Utilising the properties o the numerical radius we have w T k+ V k+ = w T k (T V ) + T k V k V w T k (T V ) + w T k V k V := : Now, since T is a unitary operator, then T k is also unitary or each k N. Since T V are commuting operators then T k T V are also commuting by Theorem 5 we have w T k (T V ) w (T V ) ; or each k N. Since V is a unitary operator T k V k V are also commuting, then by the same Theorem 5 we have w T k V k V w T k V k or each k N. Utilising these two inequalities the induction hypothesis we have w (T V ) + w T k V k w (T V ) + kw (T V ) = (k + ) w (T V ) the inequality (3.8) is thus proved or each n N. Now, or any m, by making use o the inequality (3.8) we have m! X (3.9) w a n z n T n a n z n V n ja n j jzj n w (T n V n ) w (T V ) n ja n j jzj n = w (T V ) jzj n ja n j jzj n : Since the series P a nz n T n ; P a nz n V n P n ja nj jzj n are convergent, then by letting m! in (3.9) we get the inequality (3.7).

11 INEQUALITIES OF LIPSCHITZ TYPE Remark 5. I in the above Proposition 3 we assume that R > ; then we have the inequality w ( (T ) (V )) a () w (T V ) or any two commuting unitary operators T; V B (H) : Remark 6. We observe that, as particular cases o interest, we have or any two commuting unitary operators T; V B (H) that w (I ut ) (I uv ) juj ( juj) w (T V ) or any u D (; ) or any v C. w (exp (vt ) exp (vv )) jvj exp (jvj) w (T V ) We have also the ollowing result that provides an upper bound or w ( (zt ) (zv )) in terms o kt V k : Theorem. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I T; V B (H) n g z C are such that kt k ; kv k ; jzj < R; then (3.) w ( (zt ) (zv )) kt V k jzj a (jzj max kt k ; kv kg) + h n a max kt k ; kv k o n + max kt k ; kv k o a Proo. Observe that, or any m, we have m! X (3.) w a n z n T n a n z n V n h a jzj i = n max kt k ; kv k oi = : ja n j jzj n w (T n V n ) : By Kittaneh s inequality (.3) by (.) we have w (T n V n ) kt n V n k + (T n V n ) = (3.) h n (max kt k ; kv kg) n kt V k + (T n V n ) = or any n : I we multiply the inequality (3.) by ja n j jzj n sum over n rom to m; we have (3.3) ja n j jzj n w (T n V n ) kt + m V k jzj X n ja n j jzj n (max kt k ; kv kg) n ja n j jzj n (T n V n ) =

12 S.S. DRAGOMIR ; or any m : I we use the Cauchy-Bunyakovsky-Schwarz weighted inequality we have ja n j jzj n (T n V n ) = (3.4)! = ja n j jzj n! = ja n j jzj n! = ja n j jzj n! = ja n j (T n V n )! = ja n j kt n V n k! m = X kt V k n ja n j (max kt k ; kv kg) (n ) where or the last inequality, we also used (.). Consequently, by (3.)-(3.4) we have m! X (3.5) w a n z n T n a n z n V n kt V k "jzj + n ja n j jzj n (max kt k ; kv kg) n! = ja n j jzj n! 3 = n ja n j (max kt k ; kv kg) (n ) 5 or any m : Since kt k ; kv k ; jzj < R then the series n ja n j jzj n (max kt k ; kv kg) n n (n ) ja n j (max kt k ; kv kg) are convergent, as above n ja n j jzj n (max kt k ; kv kg) n = (jzj max kt k ; kv kg) : We need to nd the representation or the second series. Consider, or u 6= ; the series n n u n = n n u n : u I we denote g (u) := P nu n ; then ug (u) = n n u n

13 INEQUALITIES OF LIPSCHITZ TYPE 3 However then u (ug (u)) = n n u n : u (ug (u)) = ug (u) + u g (u) n n u n = g (u) + ug (u) or u 6= : Utilising these calculations we can state that n ja n j (max kt k ; kv kg) (n ) = a max nkt k ; kv k o n + max kt k ; kv k o a max nkt k ; kv k o or T; V 6= : Finally, by taking m! in (3.5) we deduce the desired result (3.). 4. Applications or Hermite-Hadamard Type Inequalities The ollowing result is well known in the Theory o Inequalities as the Hermite- Hadamard inequality a + b Z b (a) + (b) (t) dt b a a or any convex unction : [a; b]! R: The distance between the middle the let term or Lipschitzian unctions with the constant L > has been estimated in [] to be Z b a + b (4.) (t) dt b a L (b a) 4 a while the distance between the right term the middle term satis es the inequality [] Z (a) + (b) b (4.) (t) dt b a L (b a) : 4 a In order to extend these results to unctions o operators we need the ollowing lemma that is o interest in itsel as well: Lemma. Let : C B (H)! B (H) be an L-Lipschitzian unction on the convex set C, i.e. it satis es k (U) (V )k L ku V k or any U; V C. For U; V C, de ne the unction ' U;V : [; ]! B (H) by ' U;V (t) := ( t) U + t + t = t U + t t V + U + + ( t) V t V :

14 4 S.S. DRAGOMIR ; Then or any t ; t [; ] we have the inequality (4.3) 'U;V (t ) ' U;V (t ) L ku V k jt t j ; i.e., the unction ' U;V is Lipschitzian with the constant L ku V k : In particular, we have the inequalities (4.4) ' U;V (t) L ku V k ( t) ; (4.5) (4.6) or any t [; ] : (U) + (V ) 3 U + 3V + Proo. We have ' U;V (t ) ' U;V (t ) = ( t ) U + t ( t ) U + t ( t ) U + t + t + ( t ) V L ( t ) U + t + L t ' U;V (t) L ku + V k t ' U;V (t) L ku t t + ( t ) V + ( t ) V ( t ) U + t ( t ) U + t ( t ) U t + ( t ) V ( t ) U t V k t = 4 L ku V k jt t j + 4 L ku V k jt t j = L ku V k jt t j or any t ; t [; ] ; which proves (4.3). We can prove now the ollowing Hermite-Hadamard type inequalities or Lipschitzian unctions o operators. Theorem. Let : C B (H)! B (H) be an L Lipschitzian unction on the convex set C. Then we have the inequalities Z (4.7) (( s) U + sv ) dt 4 L ku V k ; (4.8) (U) + (V ) Z (( s) U + tv ) ds 4 L ku V k

15 INEQUALITIES OF LIPSCHITZ TYPE 5 (4.9) Z 3 U + 3V + (( 8 L ku V k : s) U + sv ) ds Proo. First, observe that : C B (H)! B (H) is continuous in the norm topology o B (H) ; thereore the integral R (( t) U + tv ) dt exists or any U; V C. Utilising the inequality (4.4) the norm inequality or norm, we have Z Z ' U;V (t) dt (4.) ' U;V (t) dt or any U; V C. By the de nition o ' U;V we have Z ' U;V (t) dt = Z ( t) U + t dt + Z L ku V k ( t) dt = 4 L ku V k Z Now, using the change o variable t = s we have Z ( t) U + t dt = Z = by the change o variable t = v we have Z t + ( t) V dt = Z t (( ( v) + ( t) V dt : s) U + sv ) ds + vv dv: Moreover, i we make the change o variable v = s we also have Z ( v) Z + vv dv = (( s) U + sv ) ds: Thereore Z ' U;V (t) dt = = Z = Z = Z (( s) U + sv ) dt + (( = (( s) U + sv ) dt s) U + sv ) ds by (4.) we deduce (4.7). The other inequalities (4.8) (4.9) ollow in a similar way the details are omitted. Remark 7. The inequalities (4.7) (4.8) provide operator generalizations or the known inequalities (4.) (4.). Moreover the technique employed above by utilizing the auxiliary unction ' U;V is di erent rom the original proos in []

16 6 S.S. DRAGOMIR ; []. It also o ers an uniying tool which allows to obtain a better approximation o the integral R (( s) U + sv ) ds as provided by the inequality (4.9). Corollary 4. Let (z) := P a nz n be a power series with complex coe cients convergent on the open disk D (; R) ; R > : I U; V B (H) are such that kuk ; kv k M < R; then Z (4.) (( s) U + sv ) ds 4 a (M) ku V k ; (4.) (4.3) (U) + (V ) Z (( s) U + tv ) ds 4 a (M) ku Z 3 U + 3V + (( 8 a (M) ku V k : V k s) U + sv ) ds Corollary 5. With the assumptions o Corollary 4 we have (4.4) Z (k( s) U + sv k) (( s) U + sv ) ds 4 [ a (M) + M a (M)] ku V k ; (4.5) (4.6) [ (kuk) U + (kv k) V ] Z (k( s) U + sv k) (( s) U + sv ) ds 3 Z 4 [ a (M) + M a (M)] ku 3 U + 3V + U + 3V (k( s) U + sv k) (( s) U + sv ) ds 8 [ a (M) + M a (M)] ku V k V k or U; V B (H) such that kuk ; kv k M < R: Remark 8. Similar results may be stated i one uses the numerical radius inequalities obtained above. However the details are omitted.

17 INEQUALITIES OF LIPSCHITZ TYPE 7 It is known that i U V are commuting operators, then the operator exponential unction exp : B (H)! B (H) given by exp (T ) := n! T n satis es the property exp (U) exp (V ) = exp (V ) exp (U) = exp ( ) : Also, i A is invertible a; b R with a < b then Z b a exp (ta) dt = A [exp (ba) exp (aa)] : Proposition 4. Let U V be commuting operators with kuk ; kv k M such that V U is invertible. Then we have the inequalities exp (V U) (4.7) [exp (V ) exp (U)] (4.8) ku V k exp (M) ; 4 exp (U) + exp (V ) (V U) [exp (V ) exp (U)] (4.9) 4 ku V k exp (M) 3 U + 3V exp + exp ku V k exp (M) : 8 (V U) [exp (V ) exp (U)] Proo. Follows by Corollary 4 on observing that Z exp (( s) U + sv ) ds = Z exp (s (V Z = exp (s (V U)) exp (U) ds U)) ds exp (U) = (V U) [exp (V U) I] exp (U) = (V U) [exp (V ) exp (U)] : Reerences [] R. BOULDIN, The numerical range o a product. II. J. Math. Anal. Appl. 33 (97), -9. [] K. R. DAVIDSON J. A. R. HOLBROOK, Numerical radii o zero-one matricies, Michigan Math. J. 35 (988), [3] S. S. DRAGOMIR, Reverse inequalities or the numerical radius o linear operators in Hilbert spaces. Bull. Austral. Math. Soc. 73 (6), no., [4] S. S. DRAGOMIR, Some inequalities or the Euclidean operator radius o two operators in Hilbert spaces. Linear Algebra Appl. 49 (6), no., [5] S. S. DRAGOMIR, A survey o some recent inequalities or the norm numerical radius o operators in Hilbert spaces. Banach J. Math. Anal. (7), no.,

18 8 S.S. DRAGOMIR ; [6] S. S. DRAGOMIR, Inequalities or the norm the numerical radius o linear operators in Hilbert spaces, Demonstratio Mathematica, XL (7), No., [7] S. S. DRAGOMIR, Norm numerical radius inequalities or sums o bounded linear operators in Hilbert spaces. Facta Univ. Ser. Math. Inorm. (7), no., [8] S. S. DRAGOMIR, Inequalities or the numerical radius, the norm the maximum o the real part o bounded linear operators in Hilbert spaces. Linear Algebra Appl. 48 (8), no. -, [9] S. S. DRAGOMIR, New inequalities o the Kantorovich type or bounded linear operators in Hilbert spaces. Linear Algebra Appl. 48 (8), no. -, [] S. S. DRAGOMIR, Some inequalities o the Grüss type or the numerical radius o bounded linear operators in Hilbert spaces. J. Inequal. Appl. 8, Art. ID 763, 9 pp. [] S. S. DRAGOMIR, Inequalities or the norm numerical radius o composite operators in Hilbert spaces. Inequalities applications, 35 46, Internat. Ser. Numer. Math., 57, Birkhäuser, Basel, 9. [] S. S. DRAGOMIR, Y. J. CHO S. S. KIM, Inequalities o Hadamard s type or Lipschitzian mappings their applications. J. Math. Anal. Appl. 45 (), no., [3] C. K. FONG J. A. R. HOLBROOK, Unitarily invariant operators norms, Canad. J. Math. 35 (983), [4] K. E. GUSTAFSON D. K. M. RAO, Numerical Range, Springer-Verlag, New York, Inc., 997. [5] P. R. HALMOS, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 98. [6] J. A. R. HOLBROOK, Multiplicative properties o the numerical radius in operator theory, J. Reine Angew. Math. 37 (969), [7] M. EL-HADDAD F. KITTANEH, Numerical radius inequalities or Hilbert space operators. II, Studia Math. 8 (7), No., [8] O. HIRZALAH, F. KITTANEH K. SHEBRAWI, Numerical radius inequalities or operator matrices, Studia Math. (), [9] F. KITTANEH, A numerical radius inequality an estimate or the numerical radius o the Frobenius companion matrix, Studia Math. 58 (3), No., -7. [] F. KITTANEH, Numerical radius inequalities or Hilbert space operators, Studia Math. 68 (5), No., [] M. MATIĆ J. PE µcarić, Note on inequalities o Hadamard s type or Lipschitzian mappings. Tamkang J. Math. 3 (), no., 7 3. [] V. MÜLLER, The numerical radius o a commuting product, Michigan Math. J. 39 (988), [3] K. OKUBO T. ANDO, Operator radii o commuting products, Proc. Amer. Math. Soc. 56 (976), 3-. [4] G. POPESCU, Unitary invariants in multivariable operator theory. Mem. Amer. Math. Soc. (9), no. 94, vi+9 pp. ISBN: , Preprint, Ariv.math.4/449. Mathematics, School o Engineering & Science, Victoria University, PO Box 448, Melbourne City, MC 8, Australia. address: sever.dragomir@vu.edu.au URL: School o Computational & Applied Mathematics, University o the Witwatersr, Private Bag 3, Johannesburg 5, South Arica