813
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 813-814
RESEARCH NOTES
ON MULTIPARAMETER SPECTRAL THEORY
LOKENATH DEBNATH and RAM VERMA
Department of Mathematics
University of Central Florida
Orlando, FL 32816, U.S.A.
(Received March 10, 1991)
ABSTRACT. A connection between the numerical range of the multipararneter linear system
and the joint numerical range of the (commuting) separating operator system is given.
Multiparameter operator system, uniform reasonable crossnorm, numerical range, and spectral problems.
,1991 AMS SUBJECT CLASSIFICATION CODES. Primary 47A.
KEY WORDS AND PHRASES.
INTRODUCTION. We consider the following multiparameter linear operator system:
P(A) (PI(A),...,Pn(A)),
where
PI(A) B A All
P2(A) B2 A A21
A n Aln,
A n A2n,
A n Ann,
Pn(A) B n A Anl
and the operators Bj and Aflr acting on different (smooth) complex Banach spaces Xj are assumed
to be bounded and linear (j,k 1,...,n).
DEFINITION 1.1. Let X be a (smooth) complex Banach space. Then in this smooth space
there is a unique map X --, X such that (see [1]).
fo
(1)II
1 II, < 1, (1) >
"
Now using the function
,
..
we can define a
,
semi-inner product, on X by
< Xl, (Yl) > 1,
[xa, Yx]l
x.
1, Yl E S 1.
Xn
denote the completion of the tensor product
X (R) (R) Xn with respect to a uniform reasonable cross-norm a. For details, see [2]. Let [., .]j be a
semi-inner product on a Banach space X j with respect to a norm aj. Then there is a semi-inner
product [.,.] on X, defined by
DEFINITION 1.2.
Let
X
[z,y]
where z
Let
z (R)
(R)
-
Yn X.
be bounded linear operators on Xj, 1 < j < n.
Zn
Pj(A1,...,An)
[xl, Yl]l...[Zn, Yn]n,
X, and y
Yl
(R)
(R)
L. DEBNATH AND R. VERMA
814
DEFINITION 1.3. We define the (spatial) nurerical range of the system P(A)
where
[xj, xj] __V.PI(A),
en(A)]
J
U
f
0
# zj e Xj
{(AI, An
E C": [Pj(A)xj, xj]j
-
as the set
0},
We next introduce the operators which arise in the separation of the spectral parameters.
To each operator Tj" X j X j, we associate an operator acting on X given by
T? =I I(R)...(R)Ij_I(R)Tj(R)Ij+I(R)...(R)I
A
In order to study the linear operator system P(A), we construct the operators A A
and the matrices obtained
which are well-defined determinants of the operator matrix (Ai)in,
n.
0,
1,’",
n,
j
from this matrix by replacing the j-th column by the column of operators B1@ ,...,Bn@ ,j 1,...,n.
For details, see [3,4].
As in the multiparameter spectral theory, the separation of the spectrum is understood to
mean the reduction of the spectral problems for the system P(A) to the spectral problems for the
commuting operator system (A 1A 1-AII,..., -1 An-An/), the system (A-1A j-AjI),
1 _< j _< n, is called a separating system.
In the present problem, we consider the case when the operator A 0 is invertible, and the
separating system (A 1Aj-AjI) is commutative (1 _<j <_n). The aim of this note is to
announce the following result on the multiparameter spectral theory.
2., MAIN RESULTS.
Let (A A1-AI,... A-IA,-AnI) be a commutative family of
operators acting on X.
DEFINITION 2.1. We introduce the concept of the
{spatial) numerical range of the
operator system A A j- Aj I), 1 _< j _< n, as the set
A
-
1
1A n AnI
X, and [z,z] 1.
v(R)[ A-1/k 1- A I,
for z
z (R)
(R) z n
/X
{A (A1,...,An)( Cn’[ A’ 1/X ix, x] Aj[,x]},
connecting the numerical range of the system P(A) and the joint
A j Aj I), 1 < j < n.
numerical range of the (commuting) separating operator system A
We
now write the result
THEOREM 2.1. V[PI(A),...,Ptt(A)]= v(R)[
A’IA 1-AII,..., A-1An-AnI1.
REFERENCES
WILLIAMS, J.P., Spectra of Products and Numerical Ranges, J. Math. Anal. Appl. 17 (1967),
214-220.
2.
3.
SCHATTEN, R., A Theory of Cross-Spaces, Princeton Univ. Press, Princeton, 1950.
VERMA, R.U., Multiparameter Spectral Theory of a Separating Operator System, Applied
Math. Letters 2 (1989), 391-394.
VERMA, R.U., Joint Spatial Numerical Range of Tensor Products, Proc. Royal Irish Acad.
90A (1990), 201-204.