STABILITY OF THE INDEX OF A COMPLEX OF BANACH SPACES
F.-H VASILESCU
1 PRELIMINARIES
Let X and Y be two Banach spaces over the complex field C We denote by
Ạ(X, Y) the set of all linear and closed operators, defined on Jinear submanifolds
of X, assigning values in Y The subset of those operators of @(X, Y) which are everywhere defined, hence continuous, will be.denoted by BX, Y) We write G(X)
and ử(X) for G(X, X) and BX, X), respectively We put also X* = A(X, C), ie the
dual space of X
For every Sằ@(X, Y) we denote by D(S), R(S) and N(S) the domain of defi- nition, the range and the null-space of S, respectively We recall that the index of S
is given by
(1.1) ind S = dim N(S) Ở dim Y/R(S),
provided that R(S) is closed in Y and at least one of the numbers dim N(S),
dim Y/R(S) is finite For every complex vector space M we denote by dim M the algebraic dimension of M If we represent the action of S by the sequence
(1.2) 0>XẾYể0,
not forgetting that S acts only on D(S)<X, then the number (1.1) may be interpret- ed as the Euler characteristic of the complex (1.2) (see [9] or [7]) This remark suggests a more general definition of the index, which will be presented in the sequel
Consider a countable family of Banach spaces {X?}*ồồ , and a family of opera-
tors ỦỢc #(X?, X?+1) such that R(a?) < N(aw?*), for each integer p We represent them by the sequence
pol a? aPt}
(1.3) cee > YP > Yrtl Ở >
Trang 2=(H'(X,ụ))zệ Ấ, where H?(X, Ủ)= N(a?)/R(a?*) Let us assume that dim H?(X, a) < 00 =Ở009
for every integer p and that dim H?(X, ề) = 0 for all but a finite number of indices Then we may define
(1.4) ind (X, a) = x (Ở1)? dim H? (X, a)
p~ỞỦ
The number ind (X, ề), which may be interpreted as the Euler characteristic of the
complex (1.3), will be called shortly the index of the complex (Xj ề)
It is easy to imagine a trick which makes possible the reduction of the case of
unbounded operators {ềỢ} to the case of bounded ones (see the proof of Lemma 2.5
below), and we use occasionally such a procedure However, we do not generalize
that procedure since it involves the transformation of the original topology into a rather artificial one and some estimations become less precise
Let us discuss the significance of the number (1.4) in the finite-dimensional case
If dim X? < Ủ for every p, ề? Ạ BXỢ, Xồ+4) and dim Hồ(X, ề) = 0 if p < 0 and
p > n then one can easily see that
(1.5) ind (X,a)= y (Ở1)? dim X? Ở dim R(a~) + (Ở1)"*! dimxXỢ/N(@Ợ)
p=0
This remark shows that for arbitrary Banach spaces the number (1.4) cannot be, in general, invariant under compact perturbations, as a well-behaved index is expected to be When a! = 0 and a" = 0, the number (1.5) depends just on the geometry of the spaces, therefore only a certain type of complexes of Banach spaces,
namely of finite length, is significant from the point of view of the classical stability
theorems of the index [3], at least for compact perturbations However, the number
(1.4) makes sense and is stable under small perturbations for larger conditions (see
Theorem 2.12)
When dealing with complexes of Banach spaces of the form (XY, ề) = CXỢ, a?)te with X? == 0 for p < 0 and p > n (i.e complexes of finite length), we write them as (X, ề) = (X?, ụ)7.ạ, using freely the assumptions X? = 0 for p < Ở1 orp >n+landae?=Oforp < Ởlorp Sn
1.1 DEFINITION Let (X, a) = (X?, wỢ)0_.9 be a complex of Banach spaces if R(ề"~4) is closed in X", dim H?(X, a) < oo for 1 <p <n -Ở 1 and at least one of the numbers dim Hồ(X, ề), dim HỢ(X, ề) is finite then (X, a) will be called a semi-
Fredholm complex of Banach spaces
When dim Hồ(X, ề) < o for p = 0,1, .,Ợ then (X, a) is called a Fredholm complex of Banach spaces
Trang 3Note that if (X, ề) Ở (X?, ụf)7Ởg 1s a semi-Fredholm complex of Banach spaces then R(@?) is closed for all p = 0,1, .,2 Ở 1 Indeed, R(ề"}) is closed by defini- tion and R(ềỢ) is closed by the condition dim HỖ(X, ụ) < o, forl<p<nỞJ] (see [3] or [9])
In the next two sections of this work we shall obtain extensions of the usual stability theorems of the index [3], valid for a semi-Fredholm complexes
The fourth section contains some consequences of the stability theorems of the index for finite systems of closed operators, commuting in a sense which will be specified
There is a consensus of the specialists (R.G Douglas, D Voiculescu etc.) that a suitable notion of index for commuting systems of bounded operators on Hilbert spaces must be connected with the Euler characteristic of an associated
complex (this was one of the facts which inspired our Definition 1.1) An approach
to the Fredholm theory in this context has been already developed in [2] With these
conditions, the index of a commuting system turns out to be the index of a certain
operator, therefore the stability theorems can be reduced to the classical ones As
a matter of fact, the index of a Fredholm complex of Hilbert spaces is always equal
to the index of a certain operator, as our Theorem 3.8 shows However, it seems that the case of commuting operators acting in Banach spaces (and, in general, the case of complexes of Banach spaces) cannot be reduced to the case of one opera- tor, while our methods still work
Let us also mention that the Cauchy-Riemann complex of the ô-operator [4] 1s semi-Fredholm in cer(ain condifions (this was another fact which led us to Definition 1.1) and an application related to this result ends the present work 2 THE STABILITY UNDER SMALL PERTURBATIONS
In this section we investigate the stability under small perturbations of the index of a semi-Fredholm complex of Banach spaces
Let X and Y be Banach spaces and Se @(X, Y) We recall that the reduced minimum modulus of S(#0) is given by
JSxI
y(S) = in ỞỞỞỞỞỈ
xents d(x, N(S)
where ỖỖdỢ stands for the distance It is known [3] that R(S) is closed if and only if
yCS) > 0 Jn this case there is a continuous operator
Trang 4which maps &(S) into X/A(S) and with ||S 1|| = y(S) 1
When S c 0 then one defines y(S) = o
2.1 Lemma Let X,Y and Z be Banach spaces, Sằ@(X, Y), Te OY, 2)
with R(S) = N(T) and R(T ') closed Assume that A: D(S) - Y,B: D(T) + Z are bounded operators and R(Ế) c MT), where S=S+A,T=T+H+B If
(2.1) Al] xCS)* + Bl xT) + [All |B xO) p(T) < 1 then R(S) = NỨT)
Proof Take rg > y(S)7 and ry > y(%)Ỏ such that
(2.2) J4llrs + lBlrz + II4lILBlrsrz < 1,
which is possible by (2.1) Consider then y Ạ M(T) arbitrary We shall construct an
element x Ạ X such that Sx = y We shall use a closed graph type procedure inspired from [10, Lemma 2.1] Choose first yỖ Ạ Y such that
TyỖ = Ty
and
I'll < rel Zy|| = rei Byll < |] Bl relly
Since y Ở yỖ Ạ N(T), there is an x, Ạ X such that y Ở yỖ = Sx,; moreover, we may suppose that
all < rsily Ở yl <rs(l + [Bll rally
Let us define py, = y Ở Sài Then we have
yall S ly Ở S3il| + I4xHl < II rrlyl + II4Ilxill < < (l4lIrs + I|Bllrz + II4IIIIBil rez) Lyi)
Note that y, Ạ NT), therefore we may apply the same construction for y, and find yạẠ NỂT) and x;c X such that Ys Ở Vị Ở Sx, = yỞ Six + x) We obtain in
general the sequences {y,}, < N(T) and {x,}, < D(S) such that y, = y Ở S(x, +
+ + x,) Moreover,
Well < (All rs + Bll er + I4 ILBltrsrz)Ý lly
(2.3)
llxx|Í < rạ(1 + II rzJ|⁄4||zs + IBlrr + I|4ILILB|I rsrz)Ế~ !y|,
for any natural k By the relation (2.2) the series })x, is convergent in X and let x
k
be its sum As y, > 0 when k > ẹ, we obtain that }) Sx, is also convergent, hence
k
Trang 52.2, COROLLARY Consider S,T, A, B, S, T as in Lemma 2.1 If ree WS), > Ữ(T) "1, &,> |All, en = || Bll and eyrs + tgrr ExEgFsrr < 1 then
re(L + egrr)
2.4 Syn < ỞỞ ,_ :.: , `
C4) x8) S70 + Ạ4rs) (1 mm
Proụf Assuming momentarily r > y(S) 1 and r+ > y(T) 1, we obtain that the solution x of the equation y = Sx constructed in the previous lemma satisfies
the estimations
rs(l + &gr7)
2Ở (1 + E4rs) + Egry) yi
Ixll< Ế lx:ll <
k=1
obtained from (2.3) As rs, rp are arbitrarily close to y(S)7}, Ừ(T)7 respectively,
we infer easily the relation (2.4)
2.3 CoROLLARY Consider S,T, A, B, S, T as in Lemma 2.1 Then there is a constant t e9(S, 7) > 0 sách that Ặ || All < e9(S, T) and II! - < @(S, T) then the inclu- sion R(S) c Nữ) is equivalent to the equality R(S) = NỨ )
Proof If at least one of the operators S, T is non-null then we can choose
Ạ((S, T) = (2 Ở 1) min {?(S), ()}
Indeed, if ử0 = max {?(S)"1, y(7)"1} and e> ||4||, e>||B|| then the condition 20 +-
+ s*ậ2 < 1 implies the condition (2.1), therefore we may take
éo(S, T) = sup {e > 0; 208 + 226? <1} = (/2 Ở 18
If both S and T are null then e(S, T) may be any positive number
The bounded perturbations from Lemma 2.1 may be replaced with relatively
bounded perturbations in the sense of the following
2.4, DEFINITION Consider Se @(X, Y), TẠ@(Y, Z) and A a linear operator
with D(A) > D(S) and R(A) < D(T) We say that A is (S, T)-bounded if
(2.5) l4xll + I74xI < zl|x| + Sl] Sx|], x Ạ DCS),
where a, b are nonnegative constants
The operator A is (S, 0)-bounded if and only if 4 is S-bounded in the sense of
13, Ch IV]
Let us also note that the operator A from Lemma 2.1 satisfies the evaluation
|Axf] + |[TAx{] < (All + (BAI) ÍIxll + [BI Sxl], xe ử9),
Trang 6We shall obtain a variant of Lemma 2.1 for relatively bounded perturbations
2.5 LEMMA Let (X?, ề?)3_9 be a complex of Banach spaces with R(ềệ) = N(ề)
and R(a') closed Assume that B? is an (a?, ề?++)-bounded operator (p = 0, 1) satisfying RO? + Bồ) < N@! -+ B) and
(2.6) I[Bồx|] + fart *Bex|| < apllxll + byllaỖxl], x Ạ D(aỖ)
If cy = max {a,, b,} and
(2.7) (0 + y@9) + a0 + y@)Ẽ9 + ca + y@979(1 + 2) < 1 then R(aệ + Bồ) = N(t + ỷ9
Proof The present statement can be reduced to the case of Lemma 2.1 by a well-known procedure Namely, consider xe = Dw?) and define on x? the norm
(2.8) lxlù = l\xll + lwzl xeXỘ
Then X?, endowed with the norm (2.8), becomes a Banach space (p = 0, 1,2) More-
over, if ề? : 0Ừ XP+1 is the operator induced by ề? then lỢll,< 1 Analogously,
if pe : X? Ở X?*1 js the operator induced by ổ? then, by (2.6), we obtain that |j pp ll < < c,(p = 9, 1)
Note also the equalities
A Px P P
y(z)= in lle _ = inf | F lẽ An: xi re?)
xeSe d(x, N(2")) ERG ant Ix Ở yl] x 1 + y(a?)
xằN(a?)
Then the condition (2.7) implies the inequality
BP y( Gy + YB) yA |B BE v8) GN <1,
which in turn implies, by Lemma 2.1, the equality
Rw? + fồ) = Na + B)
The proof of Lemma 2.5 shows that we can reduce the case of relatively bounded
perturbations to the case of bounded perturbations Moreover, actually the perturbed
Trang 7For any pair of closed subspaces M and N in the Banach space X we set
6(M, N) = sup d(x, N)
xeM
lIxii<1
and 5(M, N) = max{ỏ(M, N), đ(N, M)} When ô(M, N) < 1 then đim # = dim N (see [5] or [3])
2.6 LEMMA If Sằ@(X, ầ), A : D(S) > Y is a bounded operator, S = S+ A
and R(S) is closed in Y then
5(N(S), N(S)) < |}Al] 1S)
Proof Taking r > y(S) 1 and xe N(S) arbitrary then we can find ve N(S) such that
Ix + || < rJSxIl= rlI4xi| < rII4l IIxll therefore
d(x, N(S)) < |||] 1S) x1
If X and Y are two Banach spaces then we denote by X @ Y their direct sum,
endowed with the norm ||x @ y|? = ||x|? + lly|lÊ (xe X, y< Y) We identify some- times XY with ầ @ 0 and YwithO @ Y
2.7 LEMMA Consider S Ạ@(X, Y) and take a finite dimensional Banach space M and Aé BM, Y) Define then S,ằ@(X ệ M, Y) by the relation S,(x ệ v) =
= Sx -+ Av, for every x Ạ D(S) and v Ạ M Then we have
dim M(S,)/N(S) + dim R(S,)/R(S) = dim M
Proof Let us write R(A) = N, + N., where N, = R(S) n R(A) and N, n No= = 0 Clearly, R(S,) = R(S) + No, hence dim RCS ,)/RCS) = dim Np
Consider then M,= AỘ(N,), M, < AT(N,) such that M,+ M,=M, M, 1 M,=0 and with 4:M,Ở- N, an isomorphism Take xẠ D(S), v,Ạ M, and v,ằ M, such that S,(x ệ (v, + v)) =0 = Sx + Av, + Av, Then Av, = 0, thus v, = 0 We can write
N(S,) = {x @ bị; xe D(S), ve My, Sx + Av, = 0}
If we consider the space X/N(S) and the linear operator
A
Trang 8we Infer the equality
N(S,IN(S) = (S40, + N(S), m); me MỊ),
showing that N(S,)/NCS) is isomorphic to M, We conclude that
dim N(S4)/N(S) + dim R(S4)/R(S) = dim M, + dim M, = dim M
Let us mention that a variant of this lemma can be found in [5], for S$ injective 2.8 DEFINITION Let (X, ề) = (X?, ề)}_Ừ9 be a complex of Banach spaces
and {Y, y} = {Y?, y?}ầ_Ừ a system with the following properties: Each Y? is a finite
dimensional Banach space and each y? Ạ @(Y?, X?*1) Let us define B(x @ y) =
= a?x + y?y, where x Ạ D(a?) and ye Y?, and assume that (Y@Y, 6)=(X? @ Y', P?)7 a1s a complex of Banach spaces In this case we say that (Xầ ệ Y, 8) is an extension
of (X, a) by the system {Y, y}
2.9 PROPOSITION Let (X, ề)=(X?, wỢ)3_o be asemi-Fredholm complex of Banach
spaces If (X ệ Y, B) is an extension of (X, ề) by the system {Y, y} = {Yầ?, yP}g-o
then (X @ệ Y, B) is also semi-Fredholm and
ind (X @ Y, B) = ind (X, a) + J) (Ở1)? dim ye p=0
Proof By Lemma 2.7, it will be enough to prove the assertion when (X, Ủ) is actually Fredholm
Note that for an arbitrary p we have the equalities
dim N(B?)/R(B?~) = dim N(B?)/R(a?*) Ở dim R(B?Ỏ)/R@?Ỏ) =
= dim N(?)/R(Ủ?~Đ + dim N(Ữ?)/N(Ủ#) Ở dim R(0?~)/R(ỦP^3),
By Lemma 2.7 we have also
dim N(#?)/N(Ủ?) + dim R(B?)/R (a?) = dim Y?
By summing up these equalities multiplied with suitable powers of Ở1 we obtain
ind (X @ Y, B) = ind (X, a) + 3;(Ở1)*(dim W(#?)/N(@#) Ở Ởdim RỂ?-Đ/R(ồ-Đ) = ind (X, a) + Ế (Ở1)? dim Yồ,
p=0
Trang 9Consider a complex of Banach spaces (X, ề) = (XỢ, ụỢ);_o If D(wỢ) is dense
in X? then the adjoint ề?* is defined and belongs to @(X?+1*, X?*) Moreover, R(Ủ?Ẩ1*) c N(Ủ?*), therefore
ẤnỞỪ*+ g0#
qn"Ở1#
0 X#* > XỢ-l# >ẻ +, Ở> X0* Ở
is again a complex of Banach spaces; it will be denoted by (X*, ề*) and called the
dual of (X, ề)
2.10 Lema Let (X, a) = (X?, #P);.o be a complex of Banach spaces with D(xỖ)
dense in Xồ for every p Then (X, a) is semi-Fredholm if and only if the dual complex (X*, ề*) is semi-~Fredholm In this case ind (X*, ề*) = (Ở1)" ind (X, a)
Proof Assume first that (X, ề) is semi-Fredholm Then R(ề?) is closed for every p, therefore R(a?*) = N(@?)+L and N(w?*) = R(ềỖ) (where Ộ*_L.ỢỖ denotes, as usually, the annihilator of the corresponding subspace in the dual) From simple arguments of duality we have that the space
N(@P*)/ Ra") = RwPỎ)+/N (a)
is isomorphic to the space (N(aỢ)/R(ềỖ1))*, therefore we can write
ind (X*, a*) = Ế (Ở1)dimN(w"~?~*)IR(Ợ~P*) = (Ở1)" ind (X, a)
p=0
The converse implication is similar
2.11 THEOREM Assume that (X, a) Ở (X?, %?)7.o 1s a semi-Fredholm complex of Banach spaces Then there exists a positive number &(X, a) such that if yỖ: D(aỖ) >
Ở X?+1 is bounded, |\y?|| < s(X, ụ), ? = Ủ? + y?(p =0,1, ,m) and (X, Bp) =
= (ỂM',?)j-o is a complex oẶ Banach spaces then dim Hồ(X, B) < dim Hồ(X, ề) for every p and ind (X, B) = ind (X, ề)
Proof Notice first that we may suppose dim H"(X, ề) < oo Indeed, there is no loss of generality in assuming that D(a?) is dense in X? for every p; if dim H"(X, ề) = 00, by passing to the dual complex we obtain, by Lemma 2.10, the desired situation We shall obtain our theorem from a more general statement
2.12, THEOREM Assume that (X, a) = (X?, a) is a complex of Banach spaces with dim H?(X, a) < oo for every p> 1 Assume also that H?(X, ề) =0 for all but a finite number of indices Then there exists a sequence of positive numbers
{ép}p>0 Such that if y? : D(a?) + Xồ+ is bounded, ||y"|| < &,, B? =a? + y? and
CX, B) = (X?, Bồ)P9 is a complex of Banach spaces then dim H"(X, B) < dim HỖ(X, 3)
Trang 10Proof Let us define the number
m(X, ề) = min {m; Hồ(X, a) = 0, p > m}
We shall obtain the assertion by an inductive argument with respect to m(X, a) Assume first that m(X, ề) = 0 Then we take
(2.9) Ep < min {E9(@? 4, a7), eg(e?, a?+4)}, p=0,1,2, ,
where é,(aỢ, ề?++) is given by Corollary 2.3 If we have ||y?|| < ằ, for every p then by
Corollary 2.3 we infer that H?(X, B) = 0, hence ind (X, B) = ind (X, a) = 0 The case m(X, ề) = 1 needs a special treatment Take first ề5 > 0 and ề, > 0 small enough in order to have
eva) ỘWL + & 9(@*)Ỏ) 2Ở (1 + & ye) + & y(a4)Ỏ)
<i,
and éy(aệ)"? < 1 Then from Lemma 2.6 and the relation (2.4) we obtain that 5(N(a), N(Bồ)) < 1, therefore dim N(ồ) = dim N(aồ) (see [6] or [3]) If we take
e, satisfying (2.9) for p > 1 then we have H?(X, Ữ) = 0 by Lemma 2.3, hence the assertion is valid in this case
Suppose now that the assertion is true for m(X, ề) = m 2 1 and let us obtain it for m(X,a) = m-+1 We have therefore H?(Xầ,0)=0 ặ p>m+ 1 and
dim A"(X, a) = n,, < oo Let us write R(vỖỢ 4) + M = NM(aỢ), where dim M = n,, We define the space X"-1 = X"-1 ệ M and the operator
wx @ vy =a" (x) +, xeD@"Ỏ), vem
It is clear that R(ẽ "9 = NỂ@"), hence if ầ? = X? and =o? forp #m-ỞI1 then (X, & = (X?, #3 Ư has the property m(, 8) Ở m Let {ễ pipzo be the sequence given by the induction hypothesis for (X, %) By changing, if necessary, ZẤ, ZẤ+;
with smaller positive numbers, we may assume that there exists 6 > 0 with the pro- perties
(2.10) 5 2Ở (1 + ấẤ }y(@9-9(1 + ấẤƯ¡ y(Ợ*)Ở9) Sn HY + Eves 10H)
andn,, 5 < #Ấ_¡ We defne then eẤ = ế, (pm Ở 1) and take eẤ_¡ S(S, ỞHmĐ5)3,
Consider now y? : D(a?) > X??! with i|y?|| < sẤ and 8? = Ủ? + y?.We shall
construct a map #"~+on Đ(zỢ~?)@ M such that if "4 == 8" Ở a" then |!""4 | <Z,,4 For, take a basis {v,, ., 0n,$ of M with the property that if v= Ỳ,,2,ặ; then
Trang 11llt; Ở 3,1 <5(j = 1, .,7,,), which is possible according to (2.10), (2.4) and Lemma
2.6 Then for all x Ạ D(@Ợ"}) and ve M, v = 2 v,, we define J1?
Bre @ 0) = BP") + ầ AG, j=l
Note that we can write
"4 @ ụ)|| < lIyỢ'x|| + +f <-
< Emal|X |) + nmổ|||| < (62_Ở+ -E n2đệ) (|X ệ v|| < Enallx ệ oll
If we put B= Ữ? For p # m Ở ỳ then, by the induction hypothesis, the complex X, f) = = (X?, 5 o Satisfies dim Hồ(X, B) < dim H(A, #)for every pandind(X, ự)= (= ind x, %) Since by Lemmas 2.1 and 2.7
dim N(#")/R(6"~9) = dim RB" )/RB"Ỏ) <n,
and by the induction hypothesis
dim N(B"-)/R(B""2) < dim N(B"Y RB") <
< dim NỂG""ĐJR("?) = dim N(um-DJR(v"ệ),
Ổwe obtain dim HỖ(X, B) < dim HỖ(X, ề) for any p> 0
From Proposition 2.9 we infer the relations
ind (X, &) = ind (X, 0) + (Ở1)"7'n m
and
ind (X, B) = ind (X, B) + (-1)"~"n,, ,
therefore ind (X, B) = ind (X, ề) and the proof of Theorem 2.12 1s complete Theorem 2.11 is a particular case of Theorem 2.12, with X? = 0forp> ụ + Ì In this case we may take
e(X, a) = min {e,;0 <p <n}
Trang 123 THE STABILITY UNDER COMPACT PERTURBATIONS
A notion analogous to relative boundedness (Definition 2.4) is that of relative
compactness
3.1 DEFINITION Consider Se @(X, Y), Te @(Y, Z) and Aa linear operator with D(A) > D(S) and R(A) c Đ(T) We say that A is (S, T)-compact if for every
sequence {x,}, < D(S) with both {x,}, and {Sx,}, bounded, the sequences {Ax,}, and {TAx,}, contain convergent subsequences
Note that A is (S, 0)-compact if and only if A is S-compact in the sense of [3,
Ch IV]
3.2 LEMMA If A is (S, T)-compact then A is (S, T)-bounded
Proof Indeed, if A is not (S, T)-bounded then there is a sequence {x,}, < D(S)
such that ||x,|| + ||Sx,|] <1 and ||Ax;l| + ||7'Ax,|| => &, therefore {Ax,}, and
{TAx,}, cannot contain convergent subsequences
Let us remark that if A is (S, T)-compact, x = D(S) is endowed with the
norm ||x|ls = ||x|l + || Sxl] (xX), Y= D(T) is endowed with the norm |ly|l) =:
= |lyll + |#y|Ì (ye Y) and A is the operator from Xx into Y induced by A then
Ae BX ; Y ) and A is compact in the usual sense, as follows from Definition 3.1 and
Lemma 3.2 Conversely, the compact operators that we work with are relatively compact in the sense of Definition 3.1 (see Lemma 3.4 below), hence it is enough, from our standpoint, to consider only compact perturbations
3.3 LEMMA Consider SẠ @(X, Y) and Te @(Y, Z) with R(S) < N(T) and R(S) closed We have dim N(T)/R(S) < 00 if and only if for every bounded sequence
{W}, & N(T) there exists a sequence {x,}, < D(S) with the property that {y, Ở Sx,}y
contains a convergent subsequence
Proof If dim N(T)/R(S) < oo then we can write M(T) = R(S) + M, where dim M < Ủ and M n R(S) = 0 Since both M and R(S) are closed, the projection
P of NT) onto M parallel to R(S) is continuous If {y,}, ằ M(T) is a bounded sequence then y, = Sx, + w,, with {w,}, c AM As ||w;|| < |LP|[ llỪ,ll, the sequence {w,}, = (% Ở Sx,}, contains a convergent subsequence
Conversely, let us assume that dim N(T)/R(S) = oo Then we can construct a
sequence {y,}, < M(T) such that ||y,|| = 1, dQy, RGS)) > 1/2 and
1
đỢy, sp{ R(S), Vi wa}) > 2 , k >2,
by a well-known lemma of Riesz [3], where ỘỘspỢỢ stands for the expression ỘỘthe linear-
space spanned byỢ In this case for each {x,}, c D(S) the sequence {y, Ở Sx,},
Trang 133.4 Lemma Consider Sé@(X, Y) and Te @(Y, Z) with R(S) < N(T), R(F) closed and dim N(T)/R(S) < 00 Take the compact operators Ae MX, Y) and
Be BY, Z) with the properties R(S) c MT) and RS) closed, where S= S+ A and T = T + B Then A is (S, T)-compact, dim N(T)/R(S) < Ủ and R(T) is closed
Proof We show first that A is (S, T)-compact Indeed, if {x,}, <ằ D(S) and
{Sx,}, are bounded sequence then, by the equality TAx, = Ở(BS + BA)x, for
all k, we infer that both {Ax,}, and {TAx,}, contain convergent subsequences The other assertions are consequences of the following fact: If {y,}, < D(T)
is a bounded sequence with Ty, Ở 0 as & > o then there exists a sequence {x;}ẤC c D(S) such that {y, + Sx}, contains a convergent subsequence Let us prove this
statement Since Ty, + By, > 0 as k > 00, we may suppose that {By,},, hence {Ty,},, is a convergent sequence As R(T) is closed, we can find ve D(7) and a sequence {v,}, c N(T) with y, +Ừ, + ò > 0 as kỞ Ủ,
Now, let us write M(T) = R(S) + M, where M n R(S) = Oanddim M < Ủ Denote by P the projection of M(T) onto M parallel to R(S) Then v, = Sx, + My with w, Ạ M for all kK The vectors x, can be chosen such that
lx,ll < rl Sxl] < rill Ở Pll [eal
where r > y(S)71 is fixed Since {v,}, is bounded, we may suppose that the sequences
{My}, ẹ M and {Ax,}, are convergent Then we have
Yyt 0% + 0 = yy + Sxy Ở Ax, + wy + 0 0, ko Ủ,
hence {y, + Sx,}, is convergent
In particular, if {y,}, < = NT) i is a bounded sequence then we can find {x,}, < c D(S)_ such that {y, + Sx,}, contains a convergent subsequence, hence
dim NỂ)/R(S < Ủ, by the previous lemma
Assume now that R(T ) is not closed Let T, ọ be the (closed) operator induced by T in Y, = Y/N(T) Then T, is injective and R(T) = = R(T) Since R(T) is not closed, we can find a sequence {n,}, ằ Y, with ||7,|| = 1 and Ton >0 as k-o Let us choose a bounded sequence {y,},, with y, representing y, for each k Then
Ty, > 0 as k > oo, hence there exists a sequence {xy}Ư = D(5) with {ye + Sx
containing a convergent subsequence In this way the sequence {y,}, may be supposed convergent to a certain 4 and ||a|| == 1 Moreover, Tot = 0, hence 4, is an eigen-
vector of 7) This contradiction shows that R(7Ỗ) must be closed
Trang 14Proof If (X, ề) is semi-Fredholm, but not Fredholm, with no loss of generality we may suppose that dim HỢ(X, ề) = 00 If p <n and R($?~}) is supposed closed then we obtain that dim H?(X, B) < co and R(B?) is closed, by Lemma 3.4 As
R(B-) = 0 is closed, the property is true for every p < n, by induction In particular, R(B"~4) is closed In this case we cannot have dim H"(X, B) < 00, by the same Lemma 3.4
From this argument, the case (X, ề) Fredholm is clear
Corollary 3.5 shows that in order to investigate the stability of the index under compact perturbations, only the case of Fredholm complexes must be took into consideration
3.6 LEMMA Let (X, a) = (X?, a?)p_o, (Y,) =(Y?,P?);~o and (Z, y) =
= (Z?, y)_9 be complexes of Banach spaces Assume that the sequence
0 D(a)Ỗ D(B) > Diy?) + 0
ds exact and uta? = BPv?, vP+ipe = yPv?, for every p If any two of the complexes (X, z), (Y, B), (Z, y) are Fredholm then the third is also Fredholm and we have the equality
ind (Y, B) = ind (X, ề) + ind (Z, y)
Proof The hypothesis implies the existence of a long exact sequence of coho-
mology
G1) + HX, ot) Ộs Hecy, py) 2S HZ, y) > HOU a) >
where #ồ and 0? are induced by u? and v? respectively, while w? is a connecting homo- morphism (see [7] for details) From the exactness of (3.1) it follows that if any two
of the complexes (X, a), CY, 8), (Z, y) are Fredholm then the third is Fredholm as well In this case (3.1) is a complex of finite dimensional spaces, whose index must
be zero on account of its exactness On the other hand, by the formula (1.5),
ind (X, a) Ở ind (Y, B) + ind (Z, y) = 0
3.7 THEOREM Assume that (X, 8) = (X?, ề?)%_y is a Fredholm complex of Banach spaces Take y? e @(X?, XỖ+1) compact for each p, such that (X, B) = (X", ?)7 g
be a complex of Banach spaces, where B? = a? + y? If
(3.2) dim R(y?t! yP) < Ủ, p=0,1, mỞ 2
then ind (X, B) = ind (X, #)
Proof Let us denote by X? the finite dimensional space R(y?"1y?-*) for 0 <
Trang 15conti-nuous for every p Note that both M(w?) and R(#Ỗ) remain unchanged in the new
topology and that the index is preserved Moreover, as y? is (ề?, ề?+1)-compact by Lemma 3.4, the restriction of y? on D(a?) * will be still compact in the new topology Let & be the restriction of ề on X? We have R(ã?) c X?+!, Indeed, ỉf xe +, x = y? 1 +?~ồu, by the Identity 8??~!1 = 0 we can write
ỦPồx = Ở(y?aP~1 + pPyP LH) yP-2y =
= P(yP Tah +Ẩ yP "ty? #)u Ở yPyP 1y? Êu = yPyPỢ lạPỢ 8p,
Note also that RỂ?Ợ) < Xồ+1, where ầ is the restriction of y? on X? In this way
both (X, #) = (X?, Z?);Ởo and (X, 8) = (X?, B?)"_o, with B? = a? + 7, arecomplexes of finite dimensional Banach spaces, therefore by the formula (1.5) we obtain
(3.3) ind (X, &) = ind (X, )
Consider now the quotient space x? = XrJX? and denote by Ủồ and ?? the maps inducedin X? by ề? and y?, respectively, for all p From the equality
G291 +: 0yPặ)(g? + 0y) + O(L Ở 8) y?# yP = 0,
where 0 < 0 < 1, we infer that R(a? + 0y?) N(+1 + 0+0), therefore ỂỦ,â + + 0) = (xe, a? + Opry ois a complex of Banach spaces As CX, a) and (X, #) are Fredholm, by Lemma 3.6 it follows that (x, 3) is also Fredholm | and ind és Ộ) = = ind (X, #) + ind L(x, a) A similar property is also true for Ể, đỞ (x, a + ?), therefore if ind(X, 8) = ind (x, B) then, by (3.3), ind (X%, a) = ind (X, B) as well Indeed, by Theorem 2.11 we have that ind(X, 2) = ind (X, a+ 09) for small values of 0 By Corollary 3.5 (x ao + 0) 1s Fredholm for each @ Since the index is conti- nuous by Theorem 2.11 and its values are integers it must be constant, and the proof is complete
We think that Theorem 3.7 is true without the condition (3.2) Besides Corollary 3.5, one reason for this conjecture is a consequence of the following
3.8 THEOREM Assume that (X, #) = (Xồ, a?)%_9 is a complex of Hilbert spaces Then there exist two Hilbert spaces H, and H, and a closed operator T, from H, into H, with the properties:
(1) R(@?Ợ) is closed for all p if and only if R(T, is closed;
(2) (X, ề) is Fredholm if and only if T, is Fredholm and in this case indT, =
== ind (X, a);
Trang 16Proof With no loss of generality we may suppose that each ề? is densely defined, therefore the adjoint ề?* is also (densely) defined Let us set
Hy= @ X*, H= @ XE, k>0 k>o
and define the operator
(3.4) Tf 929 = 2 (a?* Xo, + okt 145 49),
>0
where xỪ, Ạ D(a?*) 1 D(Ủ3*~1*), for each k > 0 Plainly, 7Ư maps a subspace of H,
into Ay
Let us prove that 7, is closed Note that R(aệ*) c N(@?Ộ+) and R(ụ#^+1*)
c N(?*+1), and take Ạ,,= ẹ x3, Ạ D(T,) with {Ạ"},, and {7,é"},, convergent k>0
By the above remark we obtain that both {0x3}, and {a*+1*xỎ, \ are convergent sequences By using that ề?* and ề?+1* are closed we infer easily that 7, itself is closed
Let us prove the equality
(3.5) Nứ,) = BeeỢ) ẹ R@Ộ-1)
(where H ẹ K denotes the orthocomplement of K in H) Indeed, take @ xa, Ạ N(T,),
k>0
hence o?*x,, == 0 and a2*+1*x,,, = 0 by the orthogonality, for every k > 0 In this
way we have also x2, Ạ N(@*-!) = R(a*-4)1, hence x, Ạ N(e2") ẹ R(a?*), Con- versely, if x, Ạ N(ề*) ẹ R(a*-) then x, ằ D(o*) nN D(a?) and ẹ x, Ạ M(T,)
k>o0
We have also the equality
(3.6) H, ẹ R(T.) = ẹ (N@**) ẹ R(a**)) k>0
Indeed, if ệ Yoi,Ạ HW, ẹ R(T,) then yy,4, is orthogonal to both R(a#?*) and
k>0
R@*+1*), therefore yori Ạ N(a*+1) ẹ R(a*) for all k, which gives one inclusion
The other inclusion is similar
One more equality is needed Namely we have
(3.7) RT) = @ (R@?) @ R(@2*Ẩ1*)) >0
(extending the meaning of the direct sum for orthogonal not necessarily closed linear manifolds) It is clear that R(T,) is contained in the second side of (3.7) Conversely, take ệ yo.4, belonging to the second side of (3.7) Then we can write that you =
k>0
Trang 17@ R(Ộ*), we may suppose that xy, Ạ R(a**) and xij, Ạ N(a*), for all k In this way, setting Xo, = X4_ + x2% Ạ D(a) 1 D(oệ*-1*), we have ax, + oP*+*x, 9 =o x9,+ -+ 02k+1#x ta = JaƯ+¡,; therefore@ Yrs Ạ R(T,)
k>0
The assertion (1) results in the following way: If R(@Ợ) is closed for all p then R(ề?*) is closed for all p, hence (3.7) implies that R(T,) is closed Conversely, R(T.)
closed implies that R(ề2*) and R(o?*+1*) are closed for every k, hence R(ề?) is closed
for all p
From the equalities (3.5) and (3.6) we obtain that (X, ụ) is Fredholm if and only if 7, is Fredholm and in this case ind (X, ề) = ind T,, therefore (2) is true
Finally, if Hồ(X,ề) = 0 for all p then M(T,) = 0 by (3.5) and R(T,) = Ay by (3.6), thus 77} exists as a bounded operator from H, into Hy The converse asser-
tion is similar, showing that (3) is also true
Let us mention that the consideration of the spaces H, and H, has been suggest- ed by a similar construction in [9, Ch IV]
3.9 COROLLARY If (X, ề)=(X?, ềo_o is a Fredholm complex of Hilbert spaces, y? Ạ BUXỖ, X?*1) is compact, B? = a+? and (X, B)=(X?, BỖ)h_o is a complex of Hilbert spaces then (X, B) is Fredholm and ind (X, B) = ind (X, #)
Proof Let T, and T, be the corresponding operators given by Theorem 3.8 for (X,ụ), (X, ) respectively It is clear that T; Ở T, is compact, therefore ind T, = ind 7, by the classical theorem of stability of the index under compact perturbations [3] (which is also a consequence of our Theorem 3.7) By Theorem 3.8 we obtain that ind (X, 8) = ind (xX, a)
Let us remark that by using Theorem 3.8 one can define a more extensive concept of semi-Fredholm complex of Hilbert spaces, in connection with the same notion for the corresponding operator
One more remark In the proof of Theorem 3.8 we can consider another operator T# from H, into Hy, defined by
+ - _ _
Tổ '( ệ X41) = @ CP" 1 xy + aPk* X54 4 1)5 k>0 k>o
where xaẤẤ¡ Ạ (2#*1) n ử(z#*), for all k>0 Then M(TP), Hy OR(T#) and R(T)
satisfy some variants of the formulas (3.5), (3.6) and (3.7), respectively It is easy
to see that (7,6, n> = <é, Ta'ny for all Ạ e Đ(T,) and n Ạ D(Te*) One can see that
T# is the adjoint of T,
Trang 184 COMMUTING SYSTEMS OF LINEAR TRANSFORMATIONS
For finite commuting systems of linear continuous operators in Banach spaces there is an adequate concept of joint spectrum which is strongly related to the combin- ed action of the operators on the space, introduced and studied by J L Taylor [10] The purpose of this section is to present in this spirit some elements of spectral and Fredholm theory, valid for certain systems of linear transformations, not necessarily continuous (see also [11] for a slightly different approach to the spectral theory of the unbounded systems)
Let us recall some basic definitions and notations [10], [11], [12] Consider
a system of n indeterminates o = (o,, .,6,) and let A[o] be the exterior algebra
over C generated by o,, .,0, For any integer p, 0 < p <n, we denote by A?[a] the space of all homogeneous exterior forms of degree ụ in ơi, ., ơ, The space
A{o] has a natural structure of Hilbert space in which the elements
On A Ao, Ì Sj]ù< <j,<n"; p=l, n and 1eCẠ = A%o] form an orthonormal basis
An important role will be played by the operators
(4.1) Si =0; A Ế, ẠeAfo], j=l1, ,n
and by their adjoints
(4.2) SACi +o, AC) = 67, jHl, ,n,
where Ạ; + a; AẠjỖ is the canonical decomposition of an arbitrary element ẠằẠ A[o]
with ằ; and Ạ}Ỗ not containing o, Note the anticommutation relations Suy + S,S; == 0
(4-3) }Lk= 1, con,
where eẤ is the Kronecker symbol
Consider now a complex linear space L Then the tensor product L @ 4[ụ] will be always denoted by Alo, L] Analogously, A?[o, Ll) is L @ A?[ụ],0 < p <n If 4 is any endomorphism of L then the action of A is extended on Alfa, L] by the endomorphism 4 @ 1 The latter will be also denoted by / (as a rule, we omit the symbol Ổ@ỖỖ when representing elements and endomorphisms connected with Alo, L}) Analogously, if 6 is an endomorphism of A[o] then the endomorphism 1 @ @, acting on A[o, L], will be also denoted by 0
Trang 19defined on D(T) @ Alo] = Alo, D(T)], which is still a closed operator Clearly, for any endomorphism @ of A[a] the endomorphism 7ằ@ extends OT
Let X be a fixed complex Banach space and o = (o,, ., 6,) a fixed system of
indeterminates
4.1 DEFINITION We say that a= (a, .,a,) c G(X) is a D-commuting system if there exists a dense subspace D of X in (~) D(a,) with the properties:
jel
(1) The restriction 6, = (a@,S, + + a,S,)|A[c, D] is closable;
(2) If 6, is the canonical closure of 5, then R(6,) < N(6,)
Note that Definition 4.1 can be equivalently expressed in the following way: (1Ỗ) The restriction 5p = (&Ế + + a,Ế,)|A4P[ụ, D] is closable for every p, O<p<n;
(2Ỗ) If 62 is the canonical closure of 52 then R(62) < N(62+4), for p =0, 1, ., 7,
where 67+1 = 0
From this equivalent form of Definition 4.1 we obtain that each D-commuting system can be associated with the complex of Banach spaces (A?[o, X], 62)%_, which makes the connection with the previous sections The cohomology corresponding to this complex will be denoted by H(X, a; D) = (Hồ(X, a; Đ))p-o
Plainly, if a = (a, ., a,) < B(X) is a D-commuting system then 6, is conti-
nuous and from 6,6, =0 we infer that (ay, ., a,) isa system of mutually commuting
operators
The concept given by Definition 4.1 is a notion of Ộstrong commutativityỖ We can give also a concept of ỔỘỔweak commutativityỢỖ
4.2 DEFINITION Consider a system a=(a,, .,a,) of densely defined operators in @(X) We say that a = (aq, .,4,) is a D,-weakly commuting system if a* =
= (a*, .,a*) c G(X") is a D,-commuting system
In Definition 4.1 the basic operator a,S, + + a,S, may be replaced with the operator a,S*+ .-+ a,S* In order to prove this assertion, let us introduce a ỔỘỔHodge typeỖ transformation of A[c, X] into itself [9]
Note that each Ạ Ạ A?[o, X] may be represented uniquely as
E = ` - Sj, eee SypXjreecip? Xj jẤẠ Ấ 1<?< </p<n"
Let us defne then in 44Ộ~7[ơ, X] the element
(4.4) BE apm D/2 ` SF tae Sj, Si wae NINH: Ỉ
1<71Ộềềe<jp<n"
Trang 204.3 LEMMA The map+ of Alo, X] into itself is an isomorphism whose square is the identity
When X is a Hilbert space then +is a unitary transformation
Proof If 1 <j, < <j, <n is a fixed system of indices then for every
xeX we have by (4.3)
#(Sj, 2 Sy, X) == MOVAỞAa-d (11 Sk, Steg %
where 1 < ky < < ky, <nand {f, .,j,} U {ky , kẤ Ấ} = (1, 2, ., mh
We can write then
+(C#ỂSj, 5Ấx)) =
= yer" 1) a} " (Ở1)#-1(Ở1)Rr1 Tu (Ở1)*-e-'S;, ể Si, x=
= rr D(Ở1)r"DP 5 Spx = Sy Six
From this calculation we obtain by linearity that the square of +is the identity on Afo, X], hence + is an isomorphism
When X is a Hilbert space then + is an isometry, hence + is actually a unitary
transformation
4.4 Lemma Consider a system a = (ay, .,a,) < @(X) and assume that there exists a subspace D < (-\ D(a;), D dense in X Let us denote by 5,(52) the restriction
j=1
ụf aẾ, + +ayS, on A[ơ, DỊ (AP[ụ, DỊ) and by },()2) the restriction of
aSệ + + a,S* on Ala, D] (A?[e, D]) Then we have the properties: (1) 5,(52) is closable if and only if },()"~) is closable;
(2) The system a = (a, .,4,) is D-commuting if and only if R(y,)< N(,): where y, is the canonical closure of Tạ
Proof Let 4+ be the map given by (4.4) Obviously, +A?[o, D] = A"?[ụ, ĐỊ Take now xe D and fix a system of indices 1 <j, < <j, <1 Then we can Ổwrite
+ ((a,S, + + 4,S,)Sj,.- Sj, x) =
= # (3 SỐ cài Sia = ầ aSẼ #(SẤ, Ếx)Ỉ k=I k=l
hence
Trang 21From (4.5) we obtain easily that 62 is closable if and only if 7 ~P is closable n thĩs case, if 6? is the closure of 62 and y? is the closure of yP we have also +6? = y2-P 4
In particular, a = (a,, .,a,) is D-commuting if and only if R(y,) < M(y,)
Let us remark that if a = (aq, .,a,) ằ @(X) is a D,-weakly commuting
system and X is reflexive then we may define 67= (y,+)*, where y, is given by Lemma
4.4 Then 67 has the property R(62) < N(6), which follows from the corresponding
property of y,ề It is easily seen that 67 is an extension of the operator
(a,S, + + 4,S,)|A |Ừ Dứ, : j=l
In particular, if a = (aq, .,4,) is D-commuting for a certain D then 6Ợ always
extends 6,
Let us illustrate the consistency of Definitions 4.1 and 4.2 with a significant
particular case Take again an arbitrary Banach space X
4.5 PROPOSITION Assume that a,, ,a, from @(X) are densely defined, b; = az! Ạ BX) (j= 1, .,n) and that b,, .,b, mutually commute Assume also
that aầ, ., aầ are densely defined Then we have the following properties : ( a = (@, ., a,) is a D-commuting system, where D = b, b,X;
(2) a* = (aầ, ,a*) is a D,-commuting system, where D,= b* b*X*; (3) R(ỏ,) == N(6,) and R(5,*) = N(ỏẤ)
Proof Since a,, ,a, are densely defined, the subspace D == b, b,X
s dense in X Analogously, D, = b* bầX* is dense in X* Moreover, De ựD@,)
j=l
and D,, < (-) D(a#) Consider the restriction 6, of a,S,+ + 4,S, on Alo, D] j=l
Let us show first that 6, is closable For, consider Ạ Ạ A[o, D] Then for any 0 Ạ Afo, Dx] we can write (6,6, 0) = <é, V0), where },ề is given by a*S* + + a*S* res-
tricted on A[o, DẤ] and <7, Ạ> is the form associated with the duality of A[o, X] and Alo, X*] naturally induced by the duality of ầ and X* As Af[o, D,]} is dense in Alo, X*] we infer that 5, is closable
A similar argument shows that 6, is closable in Alo, X*]
Let us prove the inclusion R(6,) < N(6,) Notice that every 7 ằ Afo, D] can be written as 47 = b, b,ằ and by the density of A[ằ, D] in A{o, Xầ] we have that &é=limd, 6,6, therefore
k->00
5a = W bị by SE =
j=l
A
=lim J by .b; 0.2 By, By Sle s
Trang 22where the hat over b, means its deletion On the other hand, for every k we have
(3 by wea by - [yby bn Sita} =
f=
=W0, P, bub, .bụ b, Ở by bạ by by b,)S,S,ty=0, j<q
hence ban Ạ D(6,) and 5,6,n= 0 Since 6, is the canonical closure of b, we get actually R(6,) < N(6,), therefore a = (a,, .,a,) is a D-commuting system Analogously, a* = (aầ, , a*) is a D,-commuting system, hence the first and the second assertion are proved
In order to prove the third assertion we need the relation
(4.6) ÔẤyyẾ + yuồƯẾ = nệ, ằ Ạ Diô,),
where y, is given by Lemma 4.4 for (5,, , b,) Indeed, if Ạ Ạ A[o, D] then y,ẠằA[o, D] and we have by (4.3)
ÔẤyyẾ-+yuỷẤẾ= Ừ a;bjẾ = nễ,
=
whence we derive (4.6) In particular, if 7 ằ N(6,) then 4 = 6,ằ, where Ạ = n7y,y,
hence R(6,) = N(6,) Similarly, R(6,.) = N(6,ề), and the proof is complete
Let us remark that Proposition 4.5 applies to the case of the operators a; = =ằ;Ở2Z; (j= 1, ,m), where c, ,ằ, are unbounded self-adjoint operators whose spectral measures mutually commute, and z,, ., 2, are complex numbers whose imaginary part is non-null
Consider a D-commuting system a = (a, .,a,) < G(X) and a system of complex numbers z = (z,, ., Z,)Ạ CỢ It is easily seen that z Ở a = (z; Ở a,
+;2Ấ Ở đẤ) 1S also D-commuting
4.6 DEFINITION Suppose that a = (a,, ,4,) <@(X) is a D-commuting
system Then it is called nonsingular (singular) if R(6,) = N(6,)(R(6,) # N(6,)) The system a = (a,, .,4,) is said to be semi-Fredholm (Fredholm) if the associated complex of Banach spaces (A?[o, X], 62)f_9 is semi-Fredholm (Fredholm) For a D-commuting system a = (a, ., a,) < @(X) we can introduce now a notion of joint spectrum, denoted by op(a, X), consisting of those points z ằ CỢ such that zỞa is singular When a = (a, ., 4,) ằ A(X) this notion coincides with that of J L Taylor [10]
For a = (aq, .,a,) < @(X) semi-Fredholm we may define its index by the equality
Trang 23đ?)?-a-Similarly, for a D,-weakly commuting system a = (a), .,a,) in @(X) we
can introduce a notion of weak joint spectrum ụb (a, X) given by op,(a*, X*), as well as a notion of weak index defined by ind'},a =indp,a* Since both notions of weak joint spectrum and weak index are expressed in terms of the corresponding ỔỔstrongỢỖ ones, their properties can be easily derived from the properties of the other, there-
fore we shall not deal with the ỔỘỔweakỖỖ concepts in the sequel
Note that the system of operators a = (a, ., 4,) with the properties stated
in Proposition 4.5 is nonsingular In fact, a method used in Proposition 4.5 can
be adapted in order to obtain a more general criterion of nonsingularity (see also
[10, Lemma 1.1] for bounded systems)
4.7 LEMMA Assume that a = (a,, .,a,) ẹ @(X) is a D-commuting system
Assume also that there exists a system b = (b,, .,6,) in B(X) with the properties:
(1) 5;D < D and a,b;x = b,a,x for all j,k =1, ,n and xe D;
(2) ầ) ajbjx = x for every xe D
J=I1
Then a = (ứ, đụ) is nonsingular
Proof The assertion can be obtained by using an equality similar to (4.6) We omit the details
It is beyond our scope to make an extensive study of the notions of joint spectrum and index for commuting systems We shall restrict ourselves to some consequences of the previous sections
4.8 THEOREM Consider a D-commuting system a= (a,, ,@,) in @(X) which is semi-Fredholm, There exists an &, > 0 such that for each system (cy, .; Ạ,) ẹ
< BX) with |lc;|| < eg, if bj =a; +c; for j= 1, ,n and b = (bh, .,b,) is a
D-commuting system, then b = (b,, ,6,) ts semi-Fredholm, dim Hồ(X, 6; D) < < dim HỖ(X, a; D) for all p and indy b = indpa
If a=(a,, .,a,) is nonsingular then b = (b,, ,6,) is also nonsingular Proof Let &, > 0 be given by Theorem 2.11 applied to the complex (A?[o, X], 52)"_Ừ Take eẤƯ=n +ãẤ If b is as stated then (A?[o, X], 58)"_Ừ isa complex of Banach spaces with the property
155 Ở OE < || Mog Sill<@, j=l O<p<n,
since ||.S;|| = 1 for each j By Theorem 2.11 we obtain
dim Hệ(X, 6; D) < dim H?(X, a; D) and indp 6 = indy a
Trang 24The hypothesis b = (b,, ,5,) be a D-commuting system is redundant
As the operator 6, exists, it is enough to ask R(d,) < N(6,)
4.9 COROLLARY The joint spectrum of a D-commuting system a= (đụ, ., đ,)C
< @(X) is a closed set in C*
Similarly, the set of those points zẠ C" such that z Ở a is not semi-Fredholm
(Fredholm) is closed
The corresponding result concerning the invariance of the index under compact perturbations is given by the following
4.10 THEOREM Consider a D-commuting system a= (a, .,4,) in G(X) and a system of compact operators c = (Cy, -,C,) ẹ BX) Suppose that b =
= (b,, ., 6,) is also D-commuting, where bị = dy + Cụ j = l, ,m JƑa=Ở (aụ ++,,) is semi-Fredholm (Fredholm) and dim R(cjcằ, Ở ằằ;) < 00 for all j and k then b = (by, .,b,) is semi-Fredholm (Fredholm) and indy b = indy a
Proof Note that 6, Ở6, = ằ,S, + + cằ,5S, is compact on A?[o, X], for each p Moreover,
(5 + + @S2)(6¡51 + + gS.) = ầ (eee Ở ces) SS
J<k
This equality shows that the condition dim R(cjc, Ở ằ,c;) < oo for all j and & implies the condition (3.2), therefore the conclusion can be derived from Theorem 3.7 4.11 THEOREM Consider a D-commuting system a= (a,, .,a,) in @(X), where X is a Hilbert space Then there are two Hilbert spaces H,, H, and T,Ạ @(A, H;) with the following properties:
(1) The system a = (a, .,4,) is Fredholm if and only if T, is a Fredholm operator and in this case indy a = ind T,;
(2) If (cy, -, C,) < BCX) is a system of compact operators and b =: (by, ., b,) is a D-commuting system, where b, = a; + ằ; (j= 1, .,m), when a = (ay, ., a,) is Fredholm then b = (b,, ., 6,) is Fredholm and indy b = indy a;
(3) The system a =(a,, .,4@,) is nonsingular if and only if Tz} ằ B(H,, H,) Proof We consider the associated complex of Hilbert spaces (A?[o, X], 62)}_o and apply Theorem 3.8 The spaces Hy, H, and the operator 7, correspond to this complex in the quoted theorem The assertion (1) follows from Thm 3.8 (2), the
assertion (2) is a consequence of Corollary 3.9 and the assertion (3) can be derived
from Thm 3.8 (3)
Note that Theorem 4.10 contains a characterization of the nonsingularity
in terms of invertibility Similar characterizations of the nonsingularity can be found
Trang 25Some results concerning almost commuting (i.e commuting modulo the
compacts) Fredholm systems of bounded operators in Hilbert spaces (and actually
an idea of Fredholm complex in this context) can be found in [2]
We end this section with the following question: Is there any reasonable connection between the ỖweakỖỖ and the ỖỖstrongỖỖ nonsingularity of a system of
closed operators in an arbitrary Banach space? 5 AN EXAMPLE
Let Q be a bounded open set in CỢ and A a finite dimensional Hilbert space (some of the assertions which follow can be obtained in an arbitrary Hilbert space) We denote by Hg the space L*(Q, H) of all (classes of) H-valued measurable functions on Q, whose norm is a square integrable function with respect to the Lebesgue
measure Consider also Cậồ(Q, H) (Cậồ(Q)), the space of all indefinitely differentiable #A-valued (complex-valued) functions on 2, whose support is compact
Notice that every areolar differential operator 0/02,, defined on C%ồ(Q, H), where Q contains the closure 8 of Q, is preclosed in H, and denote by a; its canonical
closure (j= 1, .,) We shall show that the system a = (a, .,a,) < @(Hog) is a D,-weakly commuting one, where D, ỞCS(Q, H) and that the 0-operator
[4], [12] is strongly connected with this property of (a,, ., 4,)
Let us fix a system of indeterminates f = (ặ,, ., Ạ,), which play the role of
the system of differentials dz = (dz,, ., dz,), but with no special meaning related
to the points of @ The operators given by (4.1) and (4.2), which correspond to ằ,, will be denoted by Z; and Z}, respectively
We recall that the ử-operator may be defined, as a A[f, H]-valued distribution, in the following way:
We say that ẠeẠ D@) c A[ễ, Họ] If there exists @ e A[Ế, Hạ] with the property
a Ấ _
5.1) [ụ@wG)d4@ = Ở{ ( SP (00+ + P @Ủ) En) A ÉG) dae), Oz, OZ,
for all o Ạ Cậồ(Q), where dd is the Lebesgue measure In this case we put dé = n {see [12] for details)
Each element Ạ Ạ A*[C, He] (0 < p <n) will be written in what follows as
Ế(z) = ` Ế, ,(Z) Ê, A wae Alin zc@
1&j< <jp<m
We remark also that the scalar product <x, y) of H combined with the Lebesgue integral defines naturally a scalar product on A?[f, H] by the formula
Trang 265.1 LEMMA Ée D(@) n A[ễ, Họ] ỮƑ_ and only tẶ there exists nạ AP*Ữ, Hạ}
uch that
\ <n(z), 0(2)) da(z) = (ce, 9.6(z)) da(z),
or all 0< AP*1[E, DẤ], where
s=-| 2+ + 528),
Oz, 02,
Proof Take ẠẠ D(0)n Al, Ho] and set 4=0Ạ Fix also l<&j¡<
Ừ <Jpsi <n and consider 0(z) = ụ(2)ẾẤ A AỌ,,,,, where @c C0, H) Since ụ 1s a ựnite linear combination of ựxed vectors of H whose coefficients are
functions from Cậồ(Q), the formula (5.1) appHes to ụ and we can write, by identi-
fication, the following equalities:
(6), @(z)) da (z) = \ Ề, Ấ.,), @Ể)) đà @) =
_ pti _ mà + a9 "
= \ x CD) m=] ÔZ/Ấ >) đá @) =
pei _, 99 5 2 3
=Ở W CY HDS WE Nae Mig Moe Ai, ) 446) =
mol Zin
= \ <E(z), 90(z)) da (2),
where the hat means deletion For an arbitrary 0 Ạ A?+1[f, D,,] we obtain the conclu- sion by linearity
The converse implication is similar and we omit it
5.2 LEMMA a = (a), .,4,) is a D,-weakly commuting system
Proof For any 9 Ạ D,, we have aj@ =Ở(09/0z,;) (j =1, .,n), therefore, with the notations from Lemma 4.4,
(5.2) 7ẤỞ= (aẨZỲ + + a223)|AE, Dz] = SAIL, Dy)
The operator Yor is preclosed since 0, which is the formal adjoint of 9, is densely defined Plainly, 3s 7= 0, implying the same relation for its closure
Trang 275.3 COROLLARY The operator 0 is equal to đ}
Proof The assertion follows from Lemma 5.{ and the equality (5.2) From now on we assume that Q is a strongly pseudoconvex domain in the sense of [4] Denote also by Q an open set containing 2
5.4, Lemma For every ằ Ạ D(@) there is a sequence &,ằ A{Z, C,(Q, H)] such that &, Ở Ế and 0É, > ÔẾ (k > co) in A[E, Hg] In particular, a= (ay, ., a,) is a
D-commuting system, where D = Ciồ(Q, A)
Proof Such an approximation result is known for Hilbert-Sobolev spaces on
bounded domains with smooth boundary (see, for instance, [8, Ch 3, ậ4}) and the
methods can be adapted to this case too However, we shall sketch the proof of this
result on a somewhat different line Assume first that U c Qis an analytic coordinate neighbourhood which is star-shaped, i.e
{tz = (tz,, ,tz,); zeQnU$}canu
for every t <1, t > 0 It is known that @ can be covered with a finite family of such coordinate neighbourhoods [4] If 6ằ A[f, Hp] and supp @ < U (where ỘsuppỢ stands for the support) then lim @, = @ in A[f, Ho], where ử,(z) = 0z) [4] More-
fol
over, if 9Ạ D(ô) then 0, e D(0) for every t and 0,= 1(00),, as a consequence of the formula (5.1)
Let us fix Ạ Ạ D(A) with supp Ạ c Qn Uandt <1 Then Ạ, can be naturally extended in the set V, = {z Ạ U; tz Ạ Q}, and keep the same notation for this exten- sion We assume also Ạ,=0 in U\V, Consider then a function ye C#ồ(CỢ) such that supp ầ = {23 |z))?-+ +]z,/7 < 1},0 <x < 1,x(Ở2) = x2) and | z(2)42 (z) = 1 For ằ > 0 we set x,(z) = e-?"y(z/e) If we define the convolution product
Zed) = (xe @ Ở #) Ew) d20W),
then we have lim ế, Ấ = Ạ, in A[f, Hy] (see, for instance, [8]) When ze V,, where
cể0
t <s <1 then by (5.1) and a change of variables we infer that
FE (2) = [ aalz Ở 9) EW) A2 (09,
for a sufficiently small ằ > 0, therefore the assertion of the lemma can be obtained
in star-shaped coordinate neighbourhoods The general assertion follows by an
Trang 28Lemma 5.4 shows that the definition (5.1) of 0 and the definition of @ in [4] (which is actually the stated property) are equivalent in domains with smooth
boundary
Let us consider the Cauchy-Riemann complex
(5.3) 05 AYZ, Hp] > > AMZ, Hg] > 0 which is semi-Fredholm and for which
dim N(@?)/R(8?) < co, p=1, ,7,
where 0Ợ is the restriction of ử on 4Ợ[Ế, Hg] This assertion is a consequence of the theory developed by J J Kohn in [4] In fact, in this case we have actually N(0?) = == R(9?"1) for p > 1, via the Grauert theorem asserting that strongly pseudoconvex
domains are holomorphically convex (see also [4]) and the well-known Theorem
B of Cartan We shall combine this property of 0 with our statements from the pre- vious section in order to obtain some significant results
Consider the functions c, :QỞ @(H), which are analytic in Q, such that C(Z)e,(z) = (z)e,(z), for all ze Q and j,k = 1, ,Ợ Define then on AỖ(Z, Ho] the continuous operator
(52E) (2) = (e(z) Z, + + ằ,(2Z,)E)
As 0?+1!ậPỆ == Ở 6P+19PE for Ee D(a"), we have that
(5.4) (APE, Hol, 0? + d2)2 6
is a complex of Hilbert spaces
5.5 PROPOSITION There is an sạ > 0 such that if max sup ||c,(z)|| < &
1<i<nzc@
then the complex (5.4) is semi- Fredholm In particular, the equation (214 SPO)E=y
has a solution Ế 6 ÁP T[Ệ, Họ], for every n Ạ N(O? + 6?) and p> 1
Proof The assertion follows from Lemma 5.4, Theorem 4.8 and from the mentioned properties of the complex (5.3)
Taking instead of A[f, Hp_] = Hy the orthocomplement in Hg of the space
Trang 29Added in proof A variant of Lemma 2.1, stated for bounded operators and with a different proof, has been obtained also by V Ptak (Commentationes Mathematicae, 21(1978), 343 Ở348)
Gr Segal has defined a concept of Fredholm complex in the context of vector-bundles
(Quart J Math Oxford, 21(1970), 385Ở402) One can see that, locally, the concept given
by our Definition 1.1 on Banach spaces is more general
10 11
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