INEQUALITIES FOR THE NUMERICAL RADIUS IN UNITAL NORMED ALGEBRAS

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1 INEQUALITIES FOR THE NUMERICAL RADIUS IN UNITAL NORMED ALGEBRAS S.S. DRAGOMIR Abtract. In thi paper, ome ineualitie between the numerical radiu of an element from a unital normed algebra and certain emi-inner product involving that element and the unity are given.. Introduction Let A be a unital normed algebra over the complex number eld C and let a A Recall that the numerical radiu of a i given by (ee [, p. 5]) (.) v (a) = up fjf (a)j ; f A 0 ; kfk and f () = g ; where A 0 denote the dual pace of A; i.e., the Banach pace of all continuou linear functional on A It i known that v () i a norm on A that i euivalent to the given norm kk More preciely, the following double ineuality hold true (.) kak v (a) kak e for any a A Following [], we notice that thi crucial reult appear lightly hidden in Bohnenblut and Karlin [, Theorem ] together with the ineuality kxk e (x) ; which occur on page 9. A impler proof wa given by Lumer [5], though with the contant 4 in place of e For a imple proof of (.) that borrow idea from Lumer and from Glickfeld [6], ee [, p. 34]. A generaliation of (.) for power ha been obtained by M.J. Crabb [3] which proved that e n (.3) ka n k n! [v (a)] n ; n = ; ; n for any a A In thi paper, ome ineualitie between the numerical radiu of an element and the uperior emi-inner product of that element and the unity in the normed algebra A are given via the celebrated repreentation reult of Lumer from [5].. Some Subet in A Let D () = ff A 0 j kfk and f () = g For C and r > 0; we de ne the ubet of A by (; r) = fa Aj jf (a) j r for each f D ()g Date 6 April, Mathematic Subject Clai cation. 46H05, 46K05, 47A0. Key word and phrae. Unital normed algebra, Numerical radiu, Semi-inner product.

2 S.S. DRAGOMIR The following reult hold. Propoition. Let C and r > 0 Then (; r) i a cloed convex ubet of A and (.) B (; r) (; r) ; where B (; r) = fa Aj ka k rg Now, for ; U (; ) = C, de ne the et n a Aj Re h ( f (a)) f (a) i o 0 for each f D () The following repreentation reult may be tated. Propoition. For any ; C, 6= ; we have (.) U (; ) = ; j j Proof. We oberve that for any z C we have the euivalence z j j if and only if Thi follow by the euality 4 j j Re [( z) (z )] 0 z = Re [( z) (z )] that hold for any z C. The euality (.) i thu a imple concluion of thi fact. Making ue of ome obviou propertie in C and for continuou linear functional, we can tate the following corollary a well. Corollary. For any ; C, we have n h i (.3) U (; ) = a A j Re f ( a) f (a ) = fa A j (Re Re f (a)) (Re f (a) Re ) o 0 for each f D () (Im Im f (a)) (Im f (a) Im ) 0 for each f D ()g Now, if we aume that Re ( ) Re () and Im ( ) Im () ; then we can de ne the following ubet of A (.4) S (; ) = fa A j Re ( ) Re f (a) Re () and Im ( ) Im f (a) Im () for each f D ()g One can eaily oberve that S (; ) i cloed, convex and (.5) S (; ) U (; )

3 NUMERICAL RADIUS IN UNITAL NORMED ALGEBRAS 3 3. Semi-Inner Product and Lumer Theorem Let (X; kk) be a normed linear pace over the real of complex number eld K. The mapping f X! R, f (x) = kxk i obviouly convex and then there exit the following limit ky txk kyk hx; yi i ; t!0 t ky txk kyk hx; yi t!0 t for every two element x; y X The mapping h; i (h; i i ) will be called the uperior emi-inner product (the interior emi-inner product) aociated to the norm kk We lit ome propertie of thee emi-inner product that can be eaily derived from the de nition (ee for intance [4]) (i) hx; xi p = kxk ; hix; xi p = hx; ixi p = 0; x X; (ii) hx; yi p = hx; yi p ; hx; yi p = hx; yi p for 0; x; y X; (iii) hx; yi p = hx; yi ; hx; yi p = hx; yi for < 0; x; y X; (iv) hix; yi p = hx; iyi p ; hx; yi = hx; yi if 0; x; y X; (v) h x; yi p = hx; yi p = hx; yi ; x; y X; (vi) hx; yi p kxk kyk ; x; y X; (vii) hx x ; yi (i) () hx ; yi (i) hx ; yi (i) for x ; x ; y X; (ix) hx y; xi p = kxk hy; xi p ; R, x; y X; (x) hy z; xi p hz; xi p kyk kxk ; x; y; z X; (xi) The mapping h; xi p i continuou on (X; kk) for each x X; where p; f; ig and p 6= The following reult eentially due to Lumer [5] (ee [, p. 7]) can be tated. Theorem. Let A be a unital normed algebra over K (K = C; R) For each a A; (3.) max fre j V (a) jg = inf [k ak >0 ] [k ak ] ;!0 where V (a) i the numerical range of a (ee for intance [, p. 5]). Remark. In term of emi-inner product, the above identity can be tated a (3.) max fre f (a) jf D ()g = ha; i The following reult that provide more information may be tated. Theorem. For any a A; we have (3.3) ha; i v; = ha; i ; where ha; bi v; t!0 v (b ta) v (b) t i the uperior emi-inner product aociated with the numerical radiu.

4 4 S.S. DRAGOMIR Proof. Since v (a) kak ; we have v ( ta) v () v ( ta) ha; i v; t!0 t t!0 t k tak lim = ha; i t!0 t Now, let f D () Then, for each > 0; giving f (a) = [f ( a) f ()] = [f ( a) ] ; Re f (a) = [Re f ( a) f ()] [jf ( a)j ] [v ( a) ] Taking the in mum over > 0; we deduce (3.4) Re f (a) inf [v ( a) ] >0!0 v ( a) = ha; i!0 v; v ( a) If we now take the upremum over f D () in (3.4), we obtain up fre f (a) jf D ()g ha; i v; which give, by Lumer identity that ha; i ha; i v; Corollary. We have the ineuality (3.5) jha; i j v (a) ( kak) Proof. Schwarz ineuality for the norm v () give that ha; i v; v (a) v () = v (a) ; and by (3.3), the ineuality (3.5) i proved. 4. Revere Ineualitie for the Numerical Radiu Utiliing the ineuality (3.5) we oberve that for any complex number located in the cloed dic centered in 0 and with radiu we have jha; i j a a lower bound for the numerical radiu v (a) Therefore, it i a natural uetion to ak how far thee uantitie are from each other under variou aumption for the element a in the unital normed algebra A and the calar A number of reult anwering thi uetion are incorporated in the following theorem. Theorem 3. Let Cn f0g and r > 0 If a (; r) ; then (4.) v (a) a; r

5 NUMERICAL RADIUS IN UNITAL NORMED ALGEBRAS 5 Proof. Since a (; ) ; then jf (a) (4.) jf (a)j Re f a r j r ; for each f D () ; giving that for each f D () Taking the upremum of f D () in (4.) and utiliing the repreentation (3.), we deduce (4.3) v (a) a; r which i an ineuality of interet in itelf. On the other hand, we have the elementary ineuality (4.4) v (a) v (a) ; which, together with (4.3) implie the deired reult (4.). Remark. Notice that, by the incluion (.) a u cient condition for (4.) to hold i that a B (; r) Corollary 3. Let ; C with 6= If a U (; ) ; then (4.5) v (a) j j a; 4 j j j j Remark 3. If M > m 0 and a U (m; M) ; then (4.6) (0 ) v (a) ha; i (M m) 4 m M Oberve that, due to the incluion (.5), a u cient condition for (4.6) to hold i that M Re f (a) ; Im f (a) m for any f D () The following reult may be tated a well. Theorem 4. Let C and r > 0 with > r If a (; r) ; then * (4.7) v (a) a; r and, euivalently, (4.8) v (a) v (a) r a; r v (a) Proof. Since > r; hence by (4.3) we have, on dividing by (4.9) r On the other hand, we alo have v (a) r v (a) r r a; r > 0; that which, together with (4.9), give (4.0) v (a) a; r

6 6 S.S. DRAGOMIR Taking the uare in (4.0), we have v (a) r a; ; which i clearly euivalent to (4.7). Corollary 4. Let ; C with Re ( ) > 0 If a U (; ) ; then, * (4.) v (a) p Re ( ) a; Remark 4. If M m > 0 and a U (m; M) ; then (4.) v (a) M m p mm ha; i ; or, euivalently, (0 ) v (a) ha; i pm p m p mm ha; i The following reult may be tated a well. 0 pm p m p mm Theorem 5. Let Cn f0g and r > 0 with > r If a (; r) ; then (4.3) v (a) a; r a; Proof. Since (by (4.)) Re f a > 0; then dividing by it in (4.) give jf (a)j Re f a Re f a r Re f a ; which i clearly euivalent to jf (a)j (4.4) Re f a Re f a r Re f a Re f a Since (4.5) Re f a I = r = r Re f a Re f a 4 r A ; 3 r Re f a 5 C kaka Re f a = I

7 NUMERICAL RADIUS IN UNITAL NORMED ALGEBRAS 7 hence by (4.4) and (4.5) we have (4.6) jf (a)j Re f a r A Re f a Taking the upremum in f D () and utiliing Lumer reult, we deduce the deired ineuality (4.3). Corollary 5. Let ; C with Re ( ) > 0 If a U (; ) ; then, v (a) j j a; p Re ( ) j j a; Remark 5. If M > m 0 and a U (m; M) ; then p (0 ) v (a) ha; i p p M m ha; i M Finally, the following reult can be tated a well. Theorem 6. Let C and r > 0 with > r If a (; r) ; then (4.7) v (a) r r a; Proof. From the proof of Theorem 3 above, we have r (4.8) jf (a)j Re f a r which i euivalent with (4.9) jf (a)j r Re f a r r = Re f a r p m kak r r Taking the upremum in (4.9) over f D () and utiliing Lumer repreentation theorem, we get (4.0) v (a) r a; r Since r 6= 0; then (4.) r > 0; giving r v (a) v (a) r

8 8 S.S. DRAGOMIR Now, utiliing (4.0) and (4.), we deduce r v (a) a; r ; r which i clearly euivalent with the deired reult (4.7). Remark 6. If M > m 0 and a U (m; M) ; then M m v (a) pm p ha; i m M m p mm In particular, if a U (0; ) with > 0; then we have the following revere ineuality a well (0 ) v (a) ha; i 4 Reference [] H.F. BOHNENBLUST and S. KARLIN, Geometrical propertie of the unit phere of Banach algebra, Ann. of Math., 6 (955), 7-9. [] F.F. BONSALL and J. DUNCAN, Numerical Range of Operator on Normed Space and of Element of Normed Algebra, Cambridge Univerity Pre, 97. [3] M.J. CRABB, Numerical range etimate for the norm of iterated operator, Glagow Math. J., (970), [4] S.S. DRAGOMIR, Semi-inner Product and Application. Nova Science Publiher, Inc., Hauppauge, NY, 004. x pp. [5] G. LUMER, Semi-inner product pace, Tran. Amer. Math. Soc., 00 (96), [6] B.W. GLICKFELD, On an ineuality of Banach algebra geometry and emi-inner product pace theory, Illinoi J. Math., 4 (970), School of Computer Science and Mathematic, Victoria Univerity, PO Box 448, Melbourne, Victoria, 800, Autralia. addre ever.dragomir@vu.edu.au URL http//

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