ON THE JOINT NUMERICAL STATUS AND TENSOR PRODUCTS

We prove a result on the joint numerical status of the bounded H11bert space operators on the tensor products. The result seems to have nice appllcattons in the multlparameter spectral theory.


INTRODUCTION AND DEFINITIONS.
In a recent paper Buonl and Wadhwa [I] introduced the concept of the joint numerical status, and have shown that the joint numerical status of commuting normal operators equals the closure of their joint numerical range.Based upon the elegant work of Stampfll [2] on the norm of the inner derivation acting on the Banach algebra of all bounded linear operators on a Hilbert space, Fong [3] introduced the concept of essential maximum numerical range to derive the norm of an inner derivation on the Calkin algebra, and proved several interesting results.The results of Fong seem to hold for the case of the joint maximum numerical status as well.
On the other hand, Dash [4] has proved that the joint numerical range of the tensor product of operators is equal to the Cartesian product of their numerical ranges.
Motivated by the work of Dash, we shall prove an analogous result for the Joint numerical status of the tensor product of operators.It seems that results obtained in this paper have nice applications in the multlparameter spectral theory [5].
Let H be a complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H.The algebra L(H) is a Banach space with respect to an operator norm (See Debnath and Mikuslnskl [6]).
Let H H I . ..H denote the Hilbert space n n completion of H H 18 ...SHn with respect to Inner product <, > which is defined by <x,y> <x I, Yl >I <Xn'Yn >n' (1.4) for every x,y E HI ...H DEFINITION 1.4.For the operators Aj e L(Hj), the tensor product operators, (1 J n) is defined Aje Ile ...e Ij_l e AjI Ij+le ... 8 In, (1.5) where lj is the identity operator on Hi.
Clearly, A @ffi (AI .....An is a commuting system of operators on .
2. A MAIN RESULT ON THE JOINT NUMERICAL STATUS.
We now prove the following result on the Joint numerical status: S(A e n e .... S(AI) x...x S(). (2.1) PROOF.In order to prove the inclusion s(A ..... c_S(A I) x...x S(An) e first prove that S(A)__ S(Aj) (J l,..,n).Since this follows by induction from the case of n 2, we first show that S(A 112 S(AI).
We next define a functional f for each A C*(L(HI)) such that f(A) h(AO 12).Then f(I I) h(l O12) I, ]h(AO !2) a.u.p Hence , h(A 1OI 2) f(A l) S(A l).
This implies that S(A! O12)S(AI).This completes the proof of this part.
Similarly, we can prove that S(l 3OAf) S(AI).This result combined with inclusion (1.3) establishes the theorem one way.
To prove converse part of the Theorem, let (h I,... ,n be an element of S(A I) x...xS(An).Then there exists soe state fj for A._ such that j -fj(Aj) for all J, J n.This completes the proof.
REMARK.For AI,...,A commuting normal operators, we find the connection n where the bar denotes the closure, and co(o (.)) denotes the convex hull of the joint aproximate point spectrum Ill.
n the joint numerical range of A is n denoted by W(A) and defined by (1.1) Now, if we set fO rio ...Ofj_lO fj O fj+l O...Ofn, , n.This implies that k : S(A?. A:), and SCA t) x... SCAn) _ S(A ..... An).