一般化された数域について

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(1)Title. 一般化された数域について. Author(s). 大久保, 和義. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 29(2) : 7-10. Issue Date. 1979-02. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6028. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Sectinn IlA)VoI. 29, No. 2 February 1979. -ffcfSiaftfl^^ffi^ (^ 2 SliA) ® 29 ^ ^ 2 ^ Hgfl] 54 ^ 2 H. On a Generalized Numerical Range. Kazuyoshi OKUBO Mathematics Laboratory, Sapporo College, Hokkaido University of Education Sapporo 064. ^kCTtt: -^b?;?z^M(^^-c. w^n±w?^w^m Abstract Let T be a bounded linear operator on a complex Hilbert space. In this paper, we will denote the invstigation of a generalized numerical range Wp( T) that was defined by S.-M. Lin.. In particular we will give some characterization of Wp( T) and the equivalent condition of the convexityofT7/,(T). 1. Let B(,H) denote the algebra of all bounded linear operators on a separable complex Hilbert space H. In this paper, an operator means an element of B(H). Sz. Nagy and Foia?. [6] said that an operator T belongs to the class C/>(0<io<oo) if there exists a unitary operator U on some Hilbert space K contains H as a subspace and such that T"h=pPU"h for h^H and w=l, 2, ••••••,. where P is the orthogonal projection of K onto H. Later, Holbrook [3] defined the operator radius wp( T) of an operator T as Wp(T):=mf[u\u>0, u~lT^Cp} (0<p<oo), and w^(T)=limwp(T). pôo. It is known that wp( D is a non-increasing con vex function of p; in particular, wi(T)=\\T\\, wz( T)=w( T):=sup{|/l| |^£1V( D} (numerical radius) where W(. T) is a numerical range of T i.e.. W(T):=((Th,h)\ \\h\\=l}, and w<o( T)=r( T) (spectral radius) (see Ando and Nishio [I], Holbrook [3], [4]). Yet the generalized numerical range Wp( T) of an operator T is defined by Lin [5] as. (7).

(3) K. OKUBO. W^T)=n{A\\A-u\<wp(T-u), u(E0)(Kp^^) u. where 0 is the complex plane.. Un showed the following facts in [5]; for p>l, (1) Wp( T) is a compact convex subset of 0; (2) Wp( D=)co ff( D where o-( D is the spectrum of T and co At is the convex hull. of M; (3) W,(aT+/3)=aWp(T)+/3, a and ^£0; (4) Wp{ T)^WA T) for l^p<p; (5) Wp( T)=Wz( T) for Kp<2, where M is the closure of M;. (6) limW,(T)=W^(T)=coff(T). p-.oo. Moreover Un defined the function w?( T) as w?(T):=sup{\A\A(=W^T)}forl^p<.oo. w?( T) has the following properties;. (1) r(T)^w?(T)^wAT); (2) Yimw?(T)=r(T); (3) w?(uT)=\u\w?(T) for u(=0 ; (4) w?(T)=W2(T)iorl<p<.2. In this paper, we shall give some characterizations of generalized numerical ranges.. 2. We call the bounded linear functional 4> on B(H) to be p-state if <f> satisfies. i) |<A( T)\<Wp( T) for all T(=B(H) ; and. ii) <f>(I)=l where I is the identity operator on H. In particular, we say 0 is state instead of 1-state <f>. We define MT={aT+0\ a, 0(=0] for T^B(H). Theorem 1. Let p^l, then. Wf,(T)={</>(T)\if> is state, \</>(S)\<Wp(S) for all S^Mr}.. Proof. Let ?r^{(A(T)[^ is state, |^(S)|^w,(S) for all SeMr}. If ^(T) is an element of W, then we have. \</>(T)+a\=\<f>(T+a)\<£.w,(T+a) for all a^0. Hence from the definition of Wp( T), it is clear </>( T)<EWp( T). Suppose conversely u(=Wp( T). If we define </> on M r by </>(S):=au+0 where S=aT+f!, then we have \</>(S)\^Wp(S) since u^Wfi(T), and </> (/)=!. By the Hahn-Banach theorem, ^ can be extended to the functional on B (H), hence we have u=<f>( T)^W. This completes the proof.. (8).

(4) On a Generalized Numerical Range. Corollary 2. Let l^p<.2, then Wp( T)={</>(. T)\4> is p-state}. Proof. It is well known that Wp(') is the norm on B(H^ if \<.p<.2 (see Holbrook [3]). Hence the proof easily follows from Theorem 1. j Now we will consider the convexity of w°p{. T) for p ^2. As we remarked before, Wp( T) is ! convex function for p >0, but it is unknown about the convexity of w°p{ T). However we will show the closed relation of the convexity of w°p{ T) and Wp(. T). Lemma 3. w°p{ T) is seminorm on MT for p ^1. i Proof. Let p S>1. We denot S.GMr by S,=ff,T+/?, where a,, 0{<=0 for ;=1, 2. Then we have. W^S,+S2)=WâiT+/3,+a2T+02)=. =(ffi+ff2)^( T)+/?i+^Cffi^( D+ff2T7,( D+î+/?2 =Wft(S,)+W,(S2). Hence we have w?(Si+S2)<S.w?(Si)+w?(S2). Theorem 4. In the following conditions a) and b) are equivalent. a) Wp( T) is convex for p >2, that is, Wp( T) satisfies. AW^( T)+(1-A)W^ T)^W^w-^( T) for OÂ^l and for 2<pi<p2. b) w?{ T— z) is convex for p > 2 and for all ^ £ ^. Proof. a)=^b) is clear. We will show b)=?-a). Let :r!(=WAp,w-A)p2(T) for Q<A<\ and 2<pi^p2. Then by Theorem 1, we have T/=^( T). for some bounded linear functional </> on B{H) where ^(/)=1 and \</>(S)\^Wip,+a-^p^S) for all S^ Mr- From the definition of the generalized numerical range, ^(S) is an element of WAP,+(I-HP^S), hence it follows \^>(S)\^w?p,+n-Hp, (S) for all S<=Mr. From this assumption,. we have \</>(S)\<.Aw?,(S)+(l—A)w?i(S). By using Lemma 3 we define the seminorm on ^rCMr by ||(Si, 52)||:=/1wô(Si)+(l-^)w^(52) where (5i, S^MT^MT. Moreover we define the subspace M of MrQMrby M={(S,S)\S^MT], and define the bounded linear functional (P on M by 0(S,S):=<f>(S). Then by the Hahn-Banach theorem, there exists a bounded linear functional d>' on .Mr9 Mr. which satifies the following properties,. i) |<?/(Si, S2)|^||(Si, S2)||(Si, SzC^r) and ii) Q)'\M=<I>. Therefore there exist bounded linear functionals ^,(z=l, 2) on Mr such that | if>i(S)\</[w^(S),. (9).

(5) K. OKUBO. \<F2(S)\<.(l-A)w^(T) foraUS(=MT,and(A(S)=î(S). Since 1=</>(I)= </>i(I}+</>2{I)< Aw?(I)+(l-A)w^(I)=land </',(I)<A, ^(Z)<S(1-/0, we have <p,(I)=A, (W)=l-/l. Let î=-^1-, ^2=-]^-. then it is clear </>,(I) =1 and |^,(S)| <Wp°,(S) for z=l,2 and for Se Mr. Thus it follows ^,( T) (=T7^(D(?=l,2)byTheoreml. Hence we have <f>( D=0i( T)+ ^( T)=A<1>^ T)+(1-A)</>2{ T)c:AW^{ T)+(1-A)W^( T), so the proof is complete.. It is known that Wp( T)^w°p( T) for p ^2. But in general wp( T)^w?( T) (p >2). We will give the next simple example. Let H=S't, and let p >2. We define an operator T on H. by T=(^ ^) (ff<=0). Then we have , _ 2(p -l)|ff|+l+/(2|ff|-l)2+4p|ff|. ~w. and w°p{ T)=\a\+— by elementary computations. Consequently we obtain that Wp{ T)>w?( T) (ff+0). IfWp(T) isconvexforp^2,itisclearthat^(D=^(D for p'>p S>2 implies Wp"(T)= co c? (T) for all p" >p. But T. Furuta showed the next theorem in [2]. We will give the elementary proof for this theorem.. Theorem 5. Let p >2. If Wz( T}=Wp{ D, then Wz{ T}=co<j( T), that is Wp\ T)= coa(T) for 2>p'>oo.. Proof. SmceW2(T}=W,(T),wehawW2(T+a)=W,(T+a) for all a<=0, hence it follows. W2(T+a)=wS(T+a}=w?(T+a) <wp( T+a)<W2( T+ff) for all a^0. Therefore we have W2< T+ff)=w/>( T+ff) for all ff£^. Since wp( T) is a non-increasing convex function for p >0, we have wz( T+a)=r( T+a) for all a<=0. Consequently, from the. definition of Wp( T), we have Wi( T)=coa( T). The proof is complete.. References C 1 ] Ando, T. and Nishio, K. (1973), Convexity properties of operator radii associated with unitary p-dilations. Michigan Math. J., Vol. 20, p. 303-307. [ 2 } Furuta, T. (1978), Conuexoid operators and generalized growth unditions associated with unitary /»-dilations of Sz.Nagy and Foias, Acta Sci. Math. (Szeged), Vol. 40, p. 53-61. ( 3 ] Holbrook, J. A. R. (1968), On the power-bounded operators of Sz.-Nagy and Foias, Acta Sci. Math. (Szeged), Vol. 29, p. 299-310. [ 4 ] Holbrook, J. A. R. (1971), In equalities governing operator radii associated with unitary p-dillations, Michigan Math. J. Vol. 18, p. 149-159. [ 5 ] Lin, C.-S. (1974), On a family of generalized numerical ranges, Can. J. Math., Vol. 26, p. 678-685. [ 6 ) Sz.-Nagy, B. and Foias, C. (1966), On certain classes of power-bounded operators in Hilbert space, Acta Sci. Math. (Szeged) Vol. 27, p. 17-25.. (10).

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