V. G. Kurbatov, Linear Finite--Difference Equations [in Russian], lzdat Voronezh. Univ., Voronezh (1990). 6. V. B. Kolmanovskii and V. R. Nosov, Stability and ...
References 1. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York (1980). 2. J.-L. Lions and E. Magenes, Problems aux limites non homog~nes of applications, Dunod, Paris (1968). 3. N. V. Azbelev, V. P. Maksimov, and 1. F. Rakhmatullina, Introduction to the Theory of Functional-Differential Equations [in Russia.], Nauka, Moscow (1991). 4. J. Hale, Theory of FunctionM Differential Equations, Springer, New York (1977) 5. V. G. Kurbatov, Linear Finite--Difference Equations [in Russian], lzdat Voronezh. Univ., Voronezh (1990). 6. V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Modes in Control Systems with Delay [in Russian], Nauka, Moscow (1981). 7. S. Nakagiri, Osaka J. Math., 85, 353-398 (1988). 8. Di Blasio, K. Kunish and E. Svnestrari, J. Math. Anal. Appl., 102, 38-57 (1984). 9. O. Staffans, J. Differential Equations, 58, No 2, 157-191 (1985). 10. R. Datko, 2. Differential Equations, 29, No 1, 105-166 (1978). 11. J. Wu, Differential Integral Equations, 4, No 6, 1325-1351 (1991). 12. V. V. Vlasov, Mat. Sb. [Russian Acad. Sci. Sb. Math.], 186, No 8, 67-92 (1995). 13. V. V. Vlasov, Dokl. Ross. Akad. Nauk [Russian Math. Dokl.], 34.5, No 6, 733-736 (1995). 14. V. V. Vlasov, lzv. Vyssh. Uchebn. Zaved. Mat. [Russian Math. (lz. VUZ)], No 1, 22-35 (1996). M o s c o w INSTITUTE OF PHYSICS AND ENGINEERING ( M F T I ) Translated by A. M. Chebotarev
Mathematical Notes, Vol. 62, No. 5, 1997
On the Numerical Ranges of a Family of Commuting Operators A. M. Gomilko and G. V. Radzievskii
KEY WORDS: Banach space, semi-inner product, convex hull, numerical range, spectrum, commuting operators.
Let B be a complex Banach space. All operators are assumed to be linear, b o u n d e d , and mapping B into B . Let as introduce the notion of the numerical range of an operator [1]. According to the H a h n Banach theorem, for any element y E B there exists a continuous linear functional (at least one) fu E B* such that fy(y) = Ilvll2 a n d [Ifyll = Ilyll- The choice of a functional f , for each y E B supplies B with a semi-inner p r o d u c t [ . , .] by the rule [x, y] = fu(z), z, y E B. Then by the numerical range of the A we mean the set W(A) : = { [ a x , x] : ~ c B , [~,x] = Ilxll 2 = 1}. If B is a Hilbert space, then this definition coincides with the ordinary definition of the numerical range. However (in contrast with the case of a Hilbert space), the numerical range of an o p e r a t o r in a Banach space need not be convex [1] and generally depends on the choice of the functionals fu" Nevertheless, as was shown in [2], the convex hull cony W(A) of the closure W(A) is independent of the fu's that define the semi-inner p r o d u c t on B . Let or(A) be the s p e c t r u m of the operator A, and let r(A) be the spectral radius. T h e n
a(A) C_cony W(A),
i.e.,
c o n v a ( A ) C_cony W(A).
Translated from Maternaticheskie Zametki, Vol. 62, No 5, pp. 787-791, November, 1997. Original article submitted June 25, 1997. 0001-.t3.16/97/6256-0659318.00
(~1998 Plenum Publishing Corporation
659
Obviously, the s p e c t r u m of the operator does not depend on the replacement of the original norm of the Banach space B b y an equivalent norm. However, some characteristics of an operator, like the norm and the numerical range, can substantially change when the original norm is replaced by an equivalent one. T h e present note deals with the problem of modifying the numerical ranges of families of commuting o p e r a t o r s u n d e r a modification of the original norm. In what follows, the symbols [[A[[. and W ( A , [[-[[,) s t a n d for the n o r m a n d t h e numerical range of an operator A corresponding to the n o r m [[ - I1.- Moreover, as was n o t e d above, the inclusion a ( A ) C_ c o n v W ( A , II 9 II,) establishes the smallest possible b o u n d for the modification of the convex linear hull of the closure of the numerical range of an o p e r a t o r A under an equivalent renormalization [I " [1. of the Banach space. Simple examples show t h a t t h e relation conv W ( A , II " ll*) -- cony a ( A ) fails in general for an equivalent renormalization (in particular, it can fail for a quasi-nilpotent operator). Therefore, the following t h e o r e m is sharp in a certain sense. T h e o r e m 1. Let A1, . . . , A,, be pairwise c o m m u t i n g operators on a Banach space B . T h e n for any e > O, there exists a n o r m [[ 9 [[~ equivalent to the original norm such that the numerical range o f each oie the operators A t , . . . , A,, belongs to the e-neighborhood o[ the convex hull o f the s p e c t r u m o f this operator, i.e., W ( A , , II- I I , ) C {X : [ A - p l < e, # 9 cony a ( A , ) } ,
7 = 1,...,n.
I f the Banach space B is a Hilbert space, then the corresponding n o r m [[ . [1~ can be chosen to be a Hilbere norm. In w h a t follows, I s t a n d s for the identity operator in B , and we set A ~ := I . Before proving T h e o r e m 1, we s t a t e a l e m m a that extends, to the case of n-operators, a well-known fact related to the modification of the n o r m of an o p e r a t o r under an equivalent renormalization of the space. L e m m a . Let A1, . . . , A,, be pa/rw/se c o m m u t i n g operators acting in a Banach space B . each c > 0, there exists a norm [[ 9 [[~ equivalent to the or/ginaJ n o r m such that IIA,II, < r ( A , ) + e,
T h e n for
7 = 1 , . . . , n.
I f the Banach space B is a HiIbert space, then the corresponding n o r m [[ 9 lie can be chosen to be a Hilbert norm.
=oRykIIA II
P r o o f . T h e series norm [[. [[~ given b y the relation
is convergent for R-r > r ( A , ) .
I1 11= . . . .
Hence for R =
(R1,...,R,),
the
-h k; _
kn=0
"~1
-~n
is defined for all elements z 9 B provided that R-r > r ( A , ) , and moreover the norm [I " 1[~ is equivalent to the original n o r m II " I[- tt is also obvious that if the original norm ][ " [I is a Hilbert norm, then [1 " [1~ is a Hilbert n o r m as well. T h e definition of II implies the inequalities
II&=ll
R II II
for
= 1,...
On setting R.y = r(A.r) + e / 2 , we obtain the assertion of the lemma.
[]
Let us present a consequence of the lemma related to the condition number of the o p e r a t o r A, i.e., the n u m b e r IIAII" II A- II, defined under the assumption that the o p e r a t o r .4 has b o u n d e d inverse (see, e.g., [3, p. 237]). 660
C o r o l l a r y 1. Let A be an operator with bounded inverse on a Banach space B . Then for any e > 0 there is a n o r m equivalent to the originM norm such that the condition n u m b e r of the operator A related to this n o r m is less than r( A )r( A - I ) + e . If the Banach space B is a Hilbert space, then the corresponding norm can be chosen to be a Hilbert norm. P r o o f o f T h e o r e m 1. Let ~Pr(a) = {A: IA - a I < r} be the circle centered at a point a 9 C with radius r > 0. It follows from the approximation theorem for compact convex sets [4, p. 18] that for an arbitrary r > 0 there axe disks ~I%,.,,(a-t,,,,) for which the convex hull of the s p e c t r u m of each operator A-t lies in the intersection of the disks ~P,-,.,,(a-t.,,,), m = 1 , . . . , s-t, and this intersection itself belongs to the e-neighborhood of the convex hull of the spectrum of the operator A-t, i.e.,
conva(A-t) C N
~ r . , . . ( a - t , , . ) C {~: I,~-/~1 < e, ~ 9 conva(A-t)},
7= 1,...,m.
(1)
m=l
According to the first inclusion, the s p e c t r u m of each of the operators A-t + ot-t,mI lies strictly inside the disk ~ r . , . , ( 0 ) (with c e n t e r at the origin). Applying the lemma to the c o m m u t i n g operators A 7 + a - t , , . I , 7 = 1 , . . . , n , rn = 1 , . . . , s-t, we deduce the existence of a norm I1" II, for which ]lA-t + a-t,mIII. < r-t,m, and hence A 9 W ( A - t -t- a.r,mI, II 9 II,) '.- IAI < Therefore, the numerical range W ( A - t , II 9 II,) of the operator A-t belongs to the intersection of the disks ~r,.,~(a-t,m), m = 1 , . . . , s-t. This, together with the second inclusion in relations (1), yields the assertion of the theorem. [] T h e following two assertions are consequences of Theorem 1 and extend the results obtained earlier in the case of Hilbert spaces to Banach spaces. C o r o l l a r y 2 [5, T h e o r e m 4]. Let A be a bounded operator acting in a Banach space B .
Then we
/lave
cony
=
conv
W(A, II
" I1.),
where the intersection is taken over all equivalent norms II 9 II, on the Banach space B . If the Banach space B is a Hilbert space, then the corresponding intersection is taken over MI Hilbert n o r m s II 9 II.-
C o r o l l a r y 3 [6, p. 95]. For the spectra of pairwise c o m m u t i n g operators A1, . . . , An to belong to the left half-plane, it is necessary and sumcient that there exist a norm II - II. equivalent to the original norm for which the sets W ( A I , II 9 I I . ) , - - . , W ( A , , II - II.) lie in the left half-plane. I f a Banach space B is a Hilbert space, then the corresponding n o r m II - II. can be chosen to be a Hilbert norm. R e m a r k 1. Let A.r be a set of pairwise commuting operators that is relatively compact in the operator topology. T h e n , for a r b i t r a r y e > 0, this set contains a finite e-net A - t t , . . . ,-t.. Let us apply the assertion of the l e m m a to the operators A.n , ... , A-t. and note that the n o r m indicated in this lemma has the property IIDII, _ IIDll for any operator D that commutes with A - i t , . . . , A . t . Therefore, the assertion of the l e m m a holds for infinitely m a n y pairwise commuting operators provided that they form a relatively compact set. It is also clear how T h e o r e m 1 and Corollary 3 can be extended to such sets of operators. Let us show t h a t the c o m m u t a t i v i t y assumption for the operators A-t is essential for Theorem 1 be valid. To this end, we consider the operators A1 = (00
01 )
and
.42=(
01
00)
in the space C 2 . We have a ( A , ) = cr(A2) = {0}. If the assertion of T h e o r e m 1 were valid for A1 and A2, then there would exist a n o r m II " H* on the space C 2 such that the sets W ( A x , [I " II*) and W ( A 2 , II " 11.) would lie inside the disk {,k: I; 1 < 1/4}. However, in this case the set W(AI + A2, II 9 II.) would belong to the disk {,k : I:~l < 1/2}, which is impossible because a(A, + -42) = { - 1 , 1}. This example also 661
shows t h a t the c o m m u t a t i v i t y a s s u m p t i o n for the operators is essential for the validity of the lemma. For Corollary 3, such examples can be constructed in just the same way. Let us establish a criterion for the inclusions W(A-t, [[ - I]*) C_ Q-t, where ~-t are given convex sets, and, in particular, a criterion for the validity of the relations
c o n v W ( A . , I1- 11.) = conva(A~),
7= 1,...,n.
In w h a t follows, we write R(A,Q-t)=
inf [ A - # I t~E~-T
and
R ( A , A ~ ) : = ( A , - A I ) -~,
A ~ ~,(AT).
Theorem 2. Let At, . . . , A~, be pairwise c o m m u t i n g operators acting in a Banach space B , and let ill, . . . , Q,, be closed convex sets such that a(A.f) _C Q T , ")' = l , . . . , n . Then for the existence o f a n o r m [[ - [[. equivalent to the originM norm such that
W(A-t, II- II.) g ~-,,
(~.)
-r= t , . . . , n ,
it is necessary and sufficient that the following estimates hold:
m=1,2,..,
"t = 1 , . . . , n.
(3)
P r o o f . Necessity. Let there exist a n o r m II - II. such t h a t the inclusions (2) hold. T h e n it follows from [7, T h e o r e m 6.1] t h a t
IIR(~, A-t)ll. < (pU,, Q-t))-t, which implies inequalities (3) in the original n o r m [l " [ISufficiency. Let estimates (3) hold. We introduce sets of pairs of complex n u m b e r s (r relations O-t := { ( a , / 3 ) : Re(~w - 13) < 0, w e QT}, 7 = 1 , . . . , n.
by the (4)
Each of the sets O- t is a p o i n t e d cone in the space C 2 , i.e., the relations ( a , / 3 ) , (or',/3') E O-t imply the relation (~a + ~cd, _ 0,
for all pairs (~,/3) E O-r. Therefore, by [2, Theorem 2.1], the nume~cal range
W(~A:, - / 3 I ,
II " I1.)
belongs to the half-plane {w : P~ew _< 0} provided that (a,/5) E O-r, and hence the numerical range W(A-t, II 9 II.) belongs to the intersection of the half-planes {w: Re(aw - / 3 )
