Series I: COMMENTATIONES MATHEMATICAE X X (1977) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X X (1977)
An d r z e j So l t y s i a k (Poznan)
Some remarks on the joint capacities in Banach algebras
Abstract. In the first section of this paper the relationship between joint capa
cities of an w-tuple of elements of a commutative unital Banach algebra in the sense defined in [4] and [5] is established. The main result, contained in the second section, says that joint spectra of an те-tuple of mutually commuting operators on a complex Banach space have the same joint capacity in the sense of both [4] and [5].
Introduction. In [1] P. P. Halmos introduced the notion of capacity of an element of a unital Banach algebra. There appeared two general
izations of this notion to an тг-tuple of elements of a commutative unital Banach algebra. The generalizations in question were done by D.S.G. Stir
ling in [5] and independently by the author in [4].
In the first section a comparison between these two notions of the joint capacity is carried away. Namely, the following facts are proved:
(i) These two notions of a joint capacity coincide if and only if an algebra consists of quasi-algebraic elements (for a definition of a quasi- algebraic element see [1]).
(ii) The capacities in question have exactly the same properties.
In the second section the joint capacity of joint spectra of several mutually commuting operators on a complex Banach space X is studied.
The main purpose of this paper is the following theorem:
If A = ( At1 A n) is an тг-tuple of mutually commuting elements of L{ X), then the following sets:
an{A), ad(A), a n>k(A), ad>k{ A ), Oi(A)., ar{ A ), g(A), aT(A), o'(A), a" {A) and а[Л^{А)
have the same capacity in the sense of both [4] and [5] (for definitions of these spectra see [2], [3] and [6]).
I would like to express my gratitude to Dr. Z. Slodkowski for his help while working on this paper.
1. A relation between two definitions o f joint capacity. All Banach algebras will be assumed complex commutative and unital. For such an
algebra ЭД, ЭД1(ЭД) will denote the maximal ideal space with the Gelfand topology. An arbitrary polynomial w of degree к in n variables may be written in the form
w ( z x , z n) =
\i\<k
where we use the notation
i — (ii, in) an w-tuple of non-negative integers,
|i| = i x+ ... + in,
^ ... «j».
The coefficients щ are complex numbers.
Let v(w) and ju(w) be defined as
v ( w ) = J ? a it f i { w ) = J i ? \at \.
i» l= fc K i= fc
Let us denote by P k{n) ( Pk{n)) the set of all polynomials w of degree к in n variables such that v(w) = 1 (y,(w) = 1, respectively) and let
capk(al f . . . , a n) = inf {| И «Х, . .. , лп)||: weP\{n)}, сар(л1? ..., an) = lim(capfcK , ..., an))llk,
к
(cap^eq, . .. , an) = in f{\\w(au ..., ая)||: пе РЦп) }, cap{ax, . . . , a n) = liminf(eap*(ax, . . . , a n))llk
к
respectively), where a1} . . . , a n belong to a complex commutative unital Banach algebra. The quantities cap an) and cap (ax, ..., an) are the joint capacities of (alf . .., an) in the sense of [4] and [5], respectively;
for n — 1, these yield the same quantity as Halmos’ capacity.
In particular, let % = C{Q) be an algebra of all continuous complex
valued functions defined on the compact subset Q <= Cn. Let P; (z) = gf , where j = 1, ..., n , be the j-th projection. Then the quantities cap (Px, . . . . . . , P J = Cap/2 and cap(Px, . . . , P n) = CapQ are called the joint capa
cities of the set Q. The quantity Cap Q appears also in [7] and is called by the author the homogeneous Tchebycheff constant for a compact set Q.
Now we are going to see how these two notions of the joint capacities compare.
First, let us notice that the set P lk(n) in the definition of ca p («q ,. .. , an) may be replaced by the set P k(n) of all polynomials w of n variables of degree к such that ju(w) > 1. Namely, the inclusion P k{n) c P k(n) implies
the inequality
inf{|Heq, ..., an)||: wePk{n)} < inf {\\w(au «»)II: w e P lk{n)}, Let w be an arbitrary element of the set P k(n)', then y{w) > 1 and the polynomial v(z) — [y(w))~l w (z) belongs to the set Pl(n). This implies
..., an)\|: wePlin)} < \\v{ax, . .., an)||
1
fx{w) \\w{alt an)\\ < I\w{alf Since w is arbitrary, we have the following inequality
in fd lw K , ..., an)\\: wePft(w)}< i n f { | | w a n)||: w e P k(n)}.
Therefore, we shall assume below that in the definition appears the set Pk(n) instead of Pl{n). By Lemma 7 from [7] we may also replace the limit inferior by the ordinary limit in this definition. Hence, in the sequel we shall use Stirling’s joint capacity in a modified version in the sense mentioned above.
Bern ark. If (ax, . .. , an) is an w-tuple of elements of a commutative unital Banach algebra, then
<*) cap ( « ! , . . . , » „ ) < cap (%,
This follows from the obvious inclusion P\{n) a P k(n).
Now, we are in a position to prove the following:
Theorem 1. For each n-tuple { n > 1) (< L ,...,a n) of elements of a commutative unital Banach algebra ЭД the following equality holds
cap(ax, . .. , an) = c a p K , ..., an)
if and only if 51 consists of quasi-algebraic elements, i.e., elements of zero capacity (see [1]).
P r o o f. Let us assume cap (al f . .., an) = cap ( % , . . . , an) for arbi
trary elements ax, . .. , an. Let a be an arbitrary element of this algebra.
Then b y Corollary 2 of [4] we have
(I) capu = cap(a, . .., a) — c a p (a ,..., a).
Now, let us take the sequence of the polynomials 1
P M = A
n- { A + . . . + A )j h = 1 , 2 , . . .
Since p k(a, . .., a) = 0 for 7c— 1 , 2 , . . . , we have cap (a, ..., a) = 0.
By (I) cap a = 0. So, we have proved that every element of this algebra is qnasi-algebraic.
Conversely, if 51 consists of elements of zero capacity, then by The
orem 3 of [4] and by the remark above the following inequalities hold cap (ax, . .., « » ) < cap(ax, ..., « „ ) < capax = 0
for arbitrary elements ax, ..., an. Hence, in this case we have cap (ax, ..., an) == cap {ax, ..., an) = 0.
This completes the proof. ,
This proof implies that if in an algebra there exists at least one non- qnasi-algebraic element, then there exists an w-tuple (ax, ..., an) such that in (*) inequality is essentially strong.
Now, we are going to examine a problem whether these capacities have the same properties? Answer to this question is positive. Namely, we can see that all theorems proved for one of these notions are also valid
for the other. *
We start with a theorem analogous to Theorem 1 in [4]. We define an ti-tuple (ax, . .. , an) of elements of a commutative unital Banach alge
bra to be quasi-algebraically dependent if there exists a sequence of poly
nomials wk€P d(k){n) such that
lim llî^ K , . .., an)\\mk) = 0.
к
Then we obtain the following
Th e o r e m 2 . An arbitrary n-tuple (ax, . .. , an) of elements of a commu
tative unital Banach algebra is quasi-algebraically dependent if and only if cap{ax, . . . , a n) = 0 .
P r o o f. It is exactly the same as the proof of Theorem 2 in [1].
It is worth noticing that this definition of quasi-algebraic dependence is not the same as in [4]; but it seems to be more natural.
Theorem 2 in [4] and Theorem 1 in [5] are in one-to-one correspondence.
Theorem 3 in [4] has the following analogue:
Th e o r e m 3. For arbitrary elements ax, . .. , an of a commutative unital Banach algebra and an arbitrary sequence 1 < i x < ... < im < n of indices the following inequality holds:
cap{ax, . . . , a n) ^ c a p , ..., aim) . P r o o f. It is analogous to that in [4].
Now, it is easy to observe that property (vi) of quasi-algebraically dependent elements given in [4] corresponds to Theorem 2 of [5].
Finally, we proceed to a perturbation theorem analogous to Theorem 6 in [5]. First of all we remind the definition of the derived spectrum of an element a of a commutative unital Banach algebra ЭД (see [5]). Namely it is the set
s(a) = {а {Ж): Ме<Ж(К)'},
where ЗЩЭДУ is the derived set of 9Jl(9I), i.e., the set of accumulation points (a~ (M) denotes the Gelfand transform of an element a; see [9]).
Let J .= {а<г91: s(a) c {0}}; J is a closed ideal of ЗД. Let n:
be the canonical epimorphism.
We have the following
Th e o r e m 4. For arbitrary elements ax, ane% the following equality, holds :
cap(u1? an) = с а р (я Ю , л{ап)).
P r o o f. First of all, let us notice that Lemma 4 of [5] is also true for the joint capacity in the sense of [4 ]. Now, an obvious modification of the proof of Theorem 5 in [5] gives the required result.
Co r o l l a r y. Let Ьг, . .., bnçJ. Then for ax, ..., ane% the equality c a p K , . .., an) = cap{a1 + bx, ..., an + bn)
holds.
In conclusion, we can see that there is an entire analogy between these two concepts of the joint capacity.
2. The joint capacities of joint spectra. In this section we shall discuss the joint capacities of an n-tuple of mutually commuting operators acting on a complex Banach space.
Let X denote a complex Banach space and let A x, ..., A n be an n- tuple of mutually commuting bounded endomorphisms of X . For A = { Ax, A n) let us write
an(A) = {AeCn: there exists a net (xa) <= X, ||a?J = 1 for all a, with lim (^ — AiI ) x a = 0 for each i = 1, ..., n}
a
and П
O’d(^) = <*тЩ = {л e — ф x } ,
г = 1
where
П П
£ в {Х = \ ^ В ^ : x^ X , i = 1 , . . . , » } .
г=1 г'= 1
These are the approximate point joint spectrum and the defect spectrum of an w-tuple { Ax, ..., A n). For properties of these spectra see [3].
Let [A] be a commutative unital Banach algebra generated by an
^г-tuple A = ( Alf A n). By o[A](A) we denote the joint spectrum of the tuple ( Аг, . .., A n) in the algebra [A] and by Г ([A]) a Silov bound
ary of this algebra. Since a mapping M\-*[A^ ( M ) , A ~ ( M)) is a homeo- morphism of the maximal ideal space $t([J.]) onto o^Af A ) (see [9]), we may identify 5Ш([Л]) ( / ’([J.]) respectively) with a compact subset of Cn.
We also consider the following three groups of joint spectra of an w-tuple A = ( A1, . A n) of mutually commuting endomorphisms of X.
The first group consists of the spectra an,kiA )i adk(A)i & = 0 ,1
(for the definitions of these spectra see [2]), the second one of
left joint spectrum ог(А), right joint spectrum or(A) and joint spec
trum a {A),
(for these definitions see [3])
and the third one consists of the Taylor spectrum aT{ A ),
commutant joint spectrum o' {A), and bicommutant joint spectrum a"{A) (see [6]).
Now we give a proof of the following:
Theobem 5. For an arbitrary n-tuple A = (A x, ..., A n) of mutually commuting operators on X the following spectra
<r»(A), od( A ) , a„tk(A), odk(A), ог(А), or{A), a {A) oT( A ), o'(A), o"{A) and o^Af A )
have the same joint capacity (in the se nse of both [4] and [5]).
P r o o f. Let us notice that the following inclusions hold:
(I) to к 3, u
<*iiA ) c O(A) <=a [ A] i A )?
(П) ° A A ) c or(A) c= a (A) <=a [A] {A ) i
(III) o„(A) = ол>0(А) c: <Уп,ЛА ) cz . . . cz &Я,п{А ) ~= oT( A ) 1 (IV) ° d ( A ) = <td,o(A ) <= Cd,i(A ) c . . . cz <*d,n(A ) =- ат(А ) and
(V) oT(A) c о'(А.) с o"( A) C: a [A](A ) ‘
The proof of (I) and (II) is contained in [3], the proof of (III) and (IV) in [2], and the proof of (V) in [6].
Now we prove the following
r ([J .]) c o „ ( A ) n o d(A).
The inclusion / ’([^4]) <= on(A) was established in the proof of Theorem 1.11, p. 134 in [3]. So, we must prove that the inclusion jP ([J L ]) c od(A) holds.
Let us assume Ле.Г([А]); then — XtI e M, i = 1, where M is a maximal ideal corresponding to X under the identification 9ft ([J.])
^ <У[Л](А). By Zelazko’s Theorem (see [8]) M consists of joint topo
logical divisors of zero, thus, there exists a net (Ba) c [A], ||BJ = 1 for all a, such that lim (J.i — XiI ) B a = 0 for each i = 1, ..., n. Now, we take
a
a net (<pa) <=. X* (the dual space) such that ||ç?a|| < 2 and ||ç>ao B J > 1 for each a. Then tpa = cpao B a are also continuous linear functionals on X and we have
{**) Ит\\у)а(А { — Х{1)\\ < 2Пт\\(А{ - Х {1)В а\\ = 0
a a
for each г = 1, ..., n.
* 11
Now, let us assume Xiad{A). Then £ (А{ — Х{1 ) Х = X.
i —1
We define the continuous linear surjection C : Y
П
by the formula 0 { х г, . .. , xn) = ( А ^ Х{1)х{ .
г = 1
By the open mapping theorem the image of the unit ball B Y = {;у : ||y|| < 1} is a set having non-empty interior, this implies, there exists e > 0 such that eBx с C [ B 7]. Now, we have
\\y>a oC\\ = mip{\{y)doC)y\: ||y|| < 1} = sup{|ya(Cy)|: ||y||< 1}
> esup{|t/>a(a?)| : ||æ|| < 1} = e||^J > s > 0 for each a.
We have obtained a contradiction, because lim||y>e o(7|| = 0 by (**).
a
Therefore Xead{A) and the proof of the required inclusion is com
pleted.
We have
r ([J .]) c a„{A)ruxd{A) cz а[Л]{А).
Now, for a polynomial w in n variables, the following equalities hold su p {H /*)[: {1еГ([А])} = sup{|w(J.)A (Ж)|: ЛГ«Г([>1])}
= sup{|w(J.f(iH)|: J f € 9[R([^L])}
= s u p j j w ( A ;( J l f ) , .. ., < ( l f ) ) j : ЛГ«ЯИ([А])|
= sup {|to (/»)!:
These imply the following equality for a polynomially convex hull of Ч Ш )
-Р(Д[Л-])) = |w(A)|< sup{|w(//)|: (леа[Л]{А)}\
W
= P { a [ A ] ( A ) ) =
since a[A](A) is polynomially convex (see [9]). If Û is a compact subset of Cn, then it is obvious that Cap Q = CapP(Q) and also Cap Ü = CapP(Q).
By this CapjT([J.]) = Сароу^(А) (respectively С арГ([А]) = Capcr^j(A)) and the proof is completed.
Co r o l l a r y 1. For an arbitrary n-tuple of mutually commuting oper
ators A = ( Ax, ..., A n) acting on a complex Banach space X the following relations hold:
(i) cap A = CapaT(A),
(ii) cap A = 0 if and only if Capoy(A) = 0.
If A — ( Ax, . .. , A n) is an w-tuple of pairwise commuting operators in L( X), then A* — (A*, . A*) is the w-tuple of mutually commuting operators in L(X*) {A* denotes the conjugate operator defined on the conjugate Banach space X*) and by Lemma 2.10 of [3] we have ad(A)
= ° „ ( A %
Applying now Theorem 5 we obtain the following corollary.
Co r o l l a r y 2 . For an arbitrary n-tuple A = (Alt . .. , A n) of mutually commuting endomorphisms of a complex Banach space X the sets
<гя(А), <*AA), a d , k ( A ) , ог(А), <rr(A), <y(A), to o'(A), o " ( A ) , a[A] {A ) J О’л О О , od(A*) ? <*n,k(A ), ad,k(A ),
or(A*), *(A*), aT(A*), o'{A*), cr"(A*) and a [ A*\( A ) have the same joint capacity (in the sense of both [4] and [5]).
References
[1] P. R. H a lm o s , Capacity in Banach algebras, Indiana Univ. Math. J. 20, 9 (1971), p. 855-863.
[2] Z. S lo d k o w s k i, A n infinite family of joint spectra, Studia Math, (to appear).
[3] *— and W . Z e la z k o , On joint spectra of commuting families of operators, ibidem 50 (1974), p. 127-148.
[4] A . S o lt y s i a k , Capacity of finite systems of elements in Banach algebras, Comm.
Math. 19 (1977), p. 405-411.
[5] D . S. G. S t ir li n g , The joint capacity of elements of Banach algebras, J. London Math. Soc. (2), 10 (1975), p. 212-218.
[6] J. L. T a y lo r , A joint spectrum for several commuting operators, J. Funct. A n a
lysis 6 (1970), p. 172-191.
[7] В. П. З а х а р ю т а , Трансфинитный диаметр, постояные Чебышева и емкость для компакта в С п, Мат. Сборник т. 96 (138): 3 (1975), р. 374-389.
[8] W . Z e la z k o , On a certain class of non-removable ideals in Banach algebras, Stndia Math. 44 (1972), p. 87-92.
[9] — Banach algebras, Amsterdam 1973.
INSTYTUT MATEMATYKI UNIWERSYTETU IM. A. MICKIEWICZA INSTITUTE OF MATHEMATICS, A. MICKIEWICZ UNIVERSITY
