Internat. J. Math. & Math. Sci.
VOL. 13 NO.
(1990) 115-120
115
SOME SPECTRAL INCLUSIONS ON D-COMMUTING SYSTEMS
R.U. VERMA
Duke University
Department of Mathematics
Durham, NC 27706, U.S.A.
and
S.S. JOU
Salem State Unlver[ty
Winston
Department of
Matllemat[cs and
Winston-Salem, NC
Computer Science
U.S.A.
27|10,
(Received January 4, 1989)
Some
ABSTRACT.
spectral
for
inc]uslons
commuting systems are establshed.
the
Taylor
joint
Browder
spectra
of
D-
The obtained results are applied in generalizing
the spectra] mapping theorems.
KEY WORDS AND PIIRASES.
complex, densely
1.
Taylor joint Browder spectrum, D-Commuting systems, Fredholm
defined operators.
INTRODUCTION.
For two complex IIilbert spaces X and Y, let L(X,Y) denote the set of all closed
linear operators,
denote
hence
defined on
For L(X,X)
continuous.
respectively.
X,
linear subspaces of
the subset of those operators from
C(X,X),
and
L(X,Y)
we
with values
Let C(X,Y)
in Y.
which are defined everywhere,
shall
simply
write
and
L(X) and C(X),
For any operator A e L(X,Y), D(A), N(A), and R(A) denote, respectively,
the domain, null space, and range of A.
Let
let a
p.
(p)
e
_
(where Z is the ring of integers) be a family of
L(xP,Y p)
be a family of operators such
that R(a
(p)
Hi[bert
cN(a
(p+)
spaces, and
), for all
Let the following sequence represent the complex of Hilbert spaces:
Let
p,
[XP}peZ
(p-l)
{HP(X’a)}peZ denote
that
>
Xp
a
(p)
__>
xP+l
a
(p+l)
the cohomology oF the complex
s, HP(x,a)=N(a (p) )/R(a(P-1)).
(X,a)=(XP,a (p) ),
for all
.U. VRHA AND S.-S. JOU
If6
(XP,a (p)
A complex (X,a)
is sald to be Fredllolm If inf{r(a
r(a (p))
is the reduced minimum modulus of
HP(X,a)
#
a(P)),
(p))} >
dim(HP(X,a)) <
0 (where
for each pZ, and
0 only for a finite number of indices.
Note that for a Fredholm complex (X,a), the quotient space IIP(X,a) is isomorphic
to the subspace N(a (p)) 0
for all pZ, and hence HP(x,a) will carry this
le also assume without loss of generality that D(a
meaaing i.n the sequel.
is
p
dense in X for each pEZ so that this emphasis is attached to the complex
R(a(P-I)),
(p))
(XP,a(P)).
(X,a)
We
recall
indeterminates,
some
definitions
E(s)
let
and
be
[I].
the
Let
s--(Sl,...,s n
algebra
exterior
over
be
a
the
complex
of
n
field
C,
system
EP(s)
generated by the indeterminates
Sl,...,s n Then, for any Integer p, 0 < p < n,
is the eector subspace of E(s) coutatning all homogeneous oxter[or forms of degree p
(p=O,l,.
,n) in s]
donote
the
tensor
s
product
For any Hilbert space X, E(X,s) (resp. EP(X,s)) shall
(resp. xEP(s)).
An element EP(X,s) wll] be
XE(s)
.
written
E
l<Jl<... jp<n
where x.
e
31’’’Jp
EP(x,s)
space
xj
X, and x.
Jl’’’Jp 5j
A...A
.
jp I
Jp
stands for x.
d
Jl’" "Jp Jl
3p
(n)
is identified with a direct sum of
P
copies of
can be endowed with a natural structure of Hilbert space.
A...A
X, therefore,
6j p
The
EP(x,s)
For two Htlbert spaces X
Y, there is a natural identification of the space E(X,s)
E(y,t) with space
E(X @ Y,(s,t)), where (tl,...,t) is another system of indeterminates.
and
n
Let
H.:E(s)2
>
E(s) be defined by
Hjrt=sj
n, nE(s).
It is obvious that
HjHk+HkHj=O,
DEFINITION l.l.
system
if
there
for j,k
A n-tuple a=(a
exists
a
dense
1,...,n.
a
n
subspace
n
e L (X) is said to be a D-commutlng
n
D of X in
with the following
D(aj)
properties [2]:
(I)
)IE(D,s)
=(a DH +..+a H
is closable.
a
n
n
If
is the canonical closure of
a
a’ th.n R(fl )cN(.-.3. ).
DEFINITION 1.2. A D-commutlng system a=(a
Ln(x) is said to be singular
,a
n
(resp. non-slngular) if R(
N(d
R(6 )=N( )).
(resp.
a
a
a
a
REMARK 1.3. Note that to each D-commuting system we can associate a complex of
Hilbert spaces (EP(x s)
a
p=O where
The restriction
([i)
d(p))n
(P)
a
a
EP(x
s)flD(6 ),
SOME SPECTRAL INCLUSIONS ON D-COMMUTING SYSTEMS
DEFINITION 1.4. Let a=(a
a
complex
of
n
L (X) be a D-commuting system associated wth
...,a
spaces
H[bert
117
(EP(:K,s), (p))n
p=O"
a
Then
a=(al,
..,a
n
is
said
to
be
Fredholm if the corresponding complex is Fredholm.
DEFINITION 1.5.
(a
a
denoted
...,a
The joint spectrum of a D-commuting system
n
C such that a-hi is slngular,
n
L (X) is the set of those points h
n
T(a, X).
DEFINITION 1.6.
a=(al,.
.,a
The Fredholm spectrum of a D-commuting system
Ln(X)
n
t the set of those points
such that a -hi is not Fredholm,
denoted
DEFINITION 1.7.
a=(a
The joint Browder spectrum of a D-commuting system
n
an)
L (X) is defined
i(a)
U{accumulation points of
denote
the
spectrum.
commutant
bv
the relation
}, for
spectrum,
iB(a)
Dash
=(a)
I,D,P and T,
where
i=I,D,P,T,
spectrum,
respectively,
and
spectrum,
polynomial
Taylor
For detail, see [3], [I] and [4].
The prime aim of this paper Is to establish some inclusions for the Taylor joint
Browder spectra of
The obtained results are also applied In
D-commuting systems.
generalizing the spectral mapping theorems on tensor products.
Unlike the case of the
bounded linear operator systems, the situation is quite different in the case of the
closed densely-defined operator systems [3].
2.
SOME LEMMAS.
We state and prove (in certain case) some results for our main theorems.
LEMMA 2.1. (Grosu and Vasilescu [211.
system, and b=(b
Lm(y)
l,...,bm
T
o (a @ b,X I}
Hilbert
n
system.
a
Then
o-(b,Y).
Let (X,a) and (Y,b) be two complexes of
Then their tensor product (X Y,a
b) is Fredholm and exact
spaces.
_
is Fredholm and exact.
n
L (X) be D -commuting system and let b=(b ,...,b
.,a
n
a
be a Db-COmmuting system. Then
Let a=a
Y)
o (a } b,X
e
PROOF.
(a X)
J(b
Y)I o
(T(a
X)
(b Y)].
(2.2)
In order to prove the [nclusion, let us take a point
(h, IJ)
0
Y)=o’(a,X)
(X,a) or (Y,b)
LEMMA 2.3.
Lm(Y)
Db-COmmuting
n
L (X) be D -commuting
,a
(Grosu and Vasilescu [211.
LEMMA 2.2.
either
be a
Let a=(a
’l: [Tea
e
X)
JOb
Y)] u
((a
x)
](b,Y)].
e
We may assume without loss of generality that h=O, and =0. If 0
(a,X), thon from Lemma 2.2, t follows that a b ts Fredholm.
happens when 0
:(b,Y).
Therefore,
o:(a,X)
and
The same thing
R.U. VRMA AND S.-S.
ll8
LEMIA 2.4. (Fa[qshtein [5]).
Let
Let a=(a
c
..,a
Lq(X)
a I) -commuting system.
be
be a family of functions, which are ho!,),nocphie in a neighbolrhood of
o(a l) x...x o(a ).
Then
n
gc[f(a 1’
LEMMA 2.5.
a
)l
’-[
or(a
o_
)].
l(a I’
" n )l
(2.3)
rl
Lemna 2.4, e have
[Jnder the assumptions of
o[(a
3.
n
a
q
)1.
SPECTRAL INCLUSIONS.
We are about to establish tle
Let a=(a
THEOREM 3.1.
b=(bl,
a Ob
c
,bin
l.(?)
(alr
be a
al,lb
results.
,nain
Ln(x) be a D -com,mttng
..,a
and let
Db-COmmuting system,
n
(XY) be a D
c L
...,lb
a
system, let
Db-COmmuting
system.
Then the Taylor joint Browder spectrum
oB(aOb,XOY) c_[
COROLLARY 3.2.
(a,X) x
(al@
For a@b
r
_
If (,U) c
by
Lemma
isolated
then
), I) e:
(a X)
point
either k or
And,
of
T
[o(a,X)
if
(%,)c
oT(aOb,XY).
o(b
1,Y)I
.b,Y). LI
O [o(a
T
[oB(a,X)
or.a(X)
in
turn,
(3. t)
we find
oB(b 1,Y)].
I,X)
(ab,XY) 2
(3.2)
(alb,XtY).
(b,Y)],
oT(ab,XY)\ oT(ab,X
e
Thus,
(b,Y)].
)
.b
e
oT(b,Y)] U[oT(a,X)
is not an isolated point.
(,,U) c
c L
Let us assume that (),la) c
(a b,X ttY),
2.3.
2(x @Y),
l)
[, I@b
oB(allb,XlltY) c__[oB(a I,X)
PROOF OF THEOREM 3.1.
oT(b,Y)] U[gr(a,X)
it
Y), then (%,I)is
implies,
not
by Lemma 2.1,
an
that
This further implies that
oT(b,Y)! O[or(a,X)
T
OB(b,Y)l,
and this completes the proof.
REMARK 3.3.
The question whether the inclusion (3.1) is true for the cases of the
Commutant, Dash, and Polynomial spectra
is still open.
119
SOME SPECTRAL INCLUSIONS ON D-COMMUTING SYSTEMS
Let
THEOREM 3.4.
XI’’’’’Xn
XIO...O
L(X.) be the
S.
(S l,...,Sn
a(’,’)
n
[2
c
U
0
[j-I
0
’i O’j+l
0...0
L(X]),
for j--l,...,n,
In.
[OB(SI,XI)
o($2,X2)..,
[o(SI,XI) OB(S2,X)
l,Xl)x...x
o
S3
o(S
n
I)
O...O
(Sn_l,Xn_l)
D(Sn )’
and
,Xn)]
X3)...x o( S n ,X
We prove the Theorem by [nductlon.
PROOF.
E
closure of tho operator
Ls a D-commuting system, for D=D(S
U [o (S
Sj
be tle completion of the algebraic
n
with respect to the canonical ttilbert norm, and let
canonical,
II
Then S
X
spaces and let
lli|bert
Let X=X O ..,O X
be densely-defined operators.
tensor product
be
)] U...U
B(Sn,Xn )]"
x o
For n=2, it reduces to a special case
of Corollary 3.2.
Let us assume that the theorem holds for any n-I operators, for n>3.
Then, if B
denotes the canonical closure of the operator
I
(R)...O
S
(BIO
Sj
O...O
(j=l,...,n-l),
In_l,
we obtain
(B
If S
B
I
,Bn_
n
D(B)
n-I
In, [iO... ln_lO Sn)
O
D(S
I)
O.
0 D(S
and
n-I
n-I
=X
Theorem 3.1, Lemma 2.1 and the Induction hypothesis imply that
T
oB(B(R)$ n,’)
T
_c [0 B(B,Xn_I)
O
c_
[oT(B,n_l)
{OB(S[,X I)
o
(Sn,Xn)]
OB(Sn,Xn)
o(S2,X2)x...x O(Sn_l, Xn_ l)}
U
{(Sl’Xl)X OB(S2’X2 )x (S3’X3)x"’x(Sn-l’Xn-I )}
u{o(sL,x)...
O[
o(s
n-2’Xn-2 ) B Sn-l ’Xn-1 )}ix
{o(SI,XI)X... O(Sn_ ,Xn_l)
Thls completes the proof.
x
os(Sn,Xn)].
o(s
o...o
n’ xn )1
0
.
then
X
n-l’
R.U. VERMA AND S.-S. JOU
120
THEOREM 3.5.
Let
ff(fl’’’"
f
be a family of functions which are holomorphtc in
a netghbourhood of the Taylor spectrum of the operator family
a @ b
(al@ I,...,an [,
I @
T
o if(a@ b X@Y)] c
e
bl,...,I
{f[oT(a
e
X)
O b ).
Then
m
oT(b,Y)] U{f[oT(a
X)
r
e
(b Y)]}.
(3.4)
Apply Lemma 2.4 to Lemma 2.3.
PROOF.
THEOREM 3.6.
Under the assumptions of the Theorem 3,5., we find the following
spectral result:
T
oB[f(ab,
PROOF.
XSY)] c-
T
{f[oB(a,X), oT(b,Y)]}U {f[oT(a,X),
oBT(b,Y)]}.
The proof follows from an application of the Lemma 2.5 to the Theorem 3.1.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
8.
9.
TAYLOR, J.L., A Joint spectrum for several commuting operators, J. Funct. Anal.
6 (1970), 172-191.
GROSU, C. and VASILESCU, F.-II., The Kunneth formula for Hilbert complexes,
Integral Equations and Operator Theory 5 (1982), 1-17.
DASH, A.T., Joint Browder spectra and tensor products, preprtnt.
VERMA, R.U., Identity of Harte, Taylor and Commutant spectra, Anal. St. Univ.
Iasl 32 (1986), 23-28.
FAINSHTKIN, A., Joint essential spectrum of a family of linear operators,
Functional Anal. Appl. 14 (1980), 152-153.
HARTE, R., The spectral mapping theorem in several variables, Bull. Amer. Math.
Soc. 73 (1972), 871-875.
MACLANE, S., Homology, Springer-Verlag, New York, 1970.
VASILESCU, F.-H., The stability of Euler characteristic for Hilbert complexes,
Math. Ann. 248 (1980), 109-116.
VERMA, R.U., Joint Browder spectra of D-commuting systems, Oemonst. Math. 21
(1988), to appear.