Internat. J. Math. & Math. Sci.
VOL. 13 NO. 4 (1990) 807-810
807
ON THE JOINT NUMERICAL STATUS AND TENSOR PRODUCTS
RAM VERMA and LOKENATH DEBNATH
Department of Mathemat
University of Central Florida
Orlando, Florida 32816
(Received September I0, 1989 and in revised form December 20, 1989)
ABSTRACT.
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.
KEY WORDS AND PHRASES.
Joint numerical status, Hilbert tensor product, Multlparameter
spectral theory, Joint Numerical range, Maximal numerical range.
1980 AMS SUBJECT CLASSIFICATION CODE. 47A.
1.
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
product
tensor
ranges.
Joint
of
operators
is
equal to the Cartesian product of their numerical
Motivated by the work of Dash, we shall prove an analogous result for the
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
The algebra L(H) is a Banach space with respect to an operator
norm (See Debnath and Mikuslnskl [6]).
In the sequel, we need the following
linear operators on H.
definitions:
DEFINITION I.I.
For A
(A
..A
n
L(H)
n
the joint numerical
range of A is
denoted by W(A) and defined by
(1.1)
RAM VERMAAND LOKENATH DEBNATH
808
For the detaits about the joint numerical ranges, see references [1],
[7] and
[81.
DEFINITION
AI,...,An.
Let
2
denote
(AI,...,An)
for A
Then,
C*(L(H)
the
C*(L(tl)),
C*-algebra
generated
I
by
and
the joint numerical status of A is
denoted by S(A) and defined by
(1.2)
is a state on C*(L(H).
where
We note that
S(A
l,...,An)
Let
DEFINITION 1.3.
<, >i,...,<,>n
H
18 ...SHn
lj
J
H
I
H
denote
>I
<, >
the
space
Hilbert
which is defined by
<Xn’Yn >n’
(1.4)
For the operators
e
Aj
L(Hj),
the tensor product operators,
Ile ...e Ij_l e AjI Ij+ le ...8 In,
is the identity operator on
Clearly, A @ffi
2.
complex Hilbert spaces with inner products
Let H
n) is defined
Aje
where
..
(1.3)
HI ...H
DEFINITION 1.4.
(1
S(An )"
...x
n
with respect to Inner product
<x I, Yl
<x,y>
E
S(A1)
HI"’’’Hn be
respectively.
completion of H
for every x,y
_c
(AI
..... An
Hi.
is a commuting system of operators on
(1.5)
.
A MAIN RESULT ON THE JOINT NUMERICAL STATUS.
We now prove the following result on the Joint numerical status:
e
S(A
PROOF.
.... ne
S(AI)
(2.1)
S().
x...x
In order to prove the inclusion
s(A
e first prove that
the case of n
.....
c_S(A
I)
S(A)__ S(Aj)
(J
x...x
l,..,n).
2, we first show that S(A
(2.2)
S(An)
112
Since this follows by induction from
S(AI).
ON THE JOINT NUMERICAL STATUS AND TENSOR PRODUCTS
If
f(A1)=
,
then there exists a state f for A
S(A1)
Let g be any state on
),.
(AIO 12)
(fO g)
C*(L(H2)).
C*(L(HI))
on
(2.3)
f(Al).
AIO 12 on C*(L(HIIt2))
This implies that fog [s a state for
such that
Thee
Og(I2)
f(A I)
809
S(AIO I2).
and
Thus
S(A1) c-- S(A
.
12)
(2.4)
I2).
S(AIO 12)
Conversely, I
h(A O
O
then there exists a state h for
C*(L(HI))
We next define a functional f for each A
h(AO
f(A)
Then f(I
12).
I)
,
h(A
This implies that
f(A l)
1OI 2)
S(A l).
S(A! O12)S(AI).
Similarly, we can prove that S(l
such that
]h(AO !2)
a.u.p
Hence
such that
I,
O12)
h(l
AIO 12
3OAf)
This completes the proof of this part.
S(AI).
This result combined with inclusion
Theorem,
let
(1.3) establishes the theorem one way.
To
of S(A
prove
converse
I) x...xS(An).
for all J,
J
n.
part
of
the
Then there exists soe state
Now, if we
._
fj
(h I,...
for
A._
,n
such that
be
an
j -fj(Aj)
set
fO rio ...Ofj_lO fj O fj+l O...Ofn,
is a state on C*(L(H
then
1
:
j
,
n.
This
..OH ))
implies that k :
SCA t)
x...
SCAn)
n
S(A?.
S(A
o
),j
f
A:),
and
and
A:
fj (A
for all J,
..... An).
This completes the proof.
REMARK.
For
AI,...,An
element
commuting normal operators, we find the connection
RAM VERMA AND LOKENATH DEBNATH
810
where the bar denotes the closure, and co(o (.)) denotes the convex hull of the joint
aproximate point spectrum
Ill.
REFERENCES
I.
BUONI, J. and WADHWA, B., On joint numerical ranges, Pacific J. Math. 77 (1978),
303-306.
2.
STAMPFLI, J., The norm of a derivation, Pacific J. Math 33 (1970), 737-747.
3.
FONG, C., On the essential maximal numerical range, Acta Sci. Math. 41 (1979),
307-315.
4.
5.
DASH, A.T., Tensor products and joint numerical range, Proc. Amer. Math. Soc. 40
(1973), 521-526.
VERMA,
Alied
R., Multlparameter spectral theory
Math. Letters 2(4) (1989) 391-394.
of
a
separating
operator
system,
6.
DEBNATH, L., and MIKUSINSKI, P., Introduction to Hilbert Spaces with
Applications, Academic Press, Boston 1990.
7.
DASH, A.T., Joint numerical range, Glasnlk Mat. 7 (1972), 75-81.
8.
HALMOS, P., A Hilbert Space Problem Book, second edition, Springer -Verlag, New
York, 1982.