PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 134, Numbe 3, Pages 689–695
S 0002-9939(05)08338-3
A icle elec onically published on Oc obe 17, 2005
EQUICOMPACT SETS OF OPERATORS
DEFINED ON BANACH SPACES
E. SERRANO, C. PI ˜
NEIRO, AND J. M. DELGADO
(Communica ed by Jona han M. Bo wein)
Abs ac . Le Xand Ybe Banach spaces. We say ha a se M⊂K(X, Y )
(K(X, Y ) deno es he space o all compac ope a o s om Xin o Y)isequicom-
pac i he e exis s a null sequence (x∗
n)nin X∗such ha Tx≤supn|x∗
n(x)|
o all x∈Xand all T∈M. I is easy o show ha collec i ely com-
pac ness and equicompac ness a e dual concep s in he ollowing sense: Mis
equicompac iff M∗={T∗:T∈M}is collec i ely compac . We s udy some
p ope ies o equicompac se s and, among o he esul s, we p o e: 1) a se
M⊂K(X, Y ) is equicompac iff each bounded sequence (xn)nin Xhas a
subsequence (xk(n))nsuch ha (Tx
k(n))nis a con e ging sequence uni o mly
o T∈M;2)i Ydoes no ha e fini e co ype and M⊂K(X, Y ) is a maximal
equicompac se , hen, gi en ε>0 and a fini e se {x1,...,x
n}in X, he eis
an ope a o S∈Msuch ha Tx
i≤(1 + ε)Sxi o i=1,...,n and all
T∈M.
1. In oduc ion
Le us conside ( eal o complex) Banach spaces Xand Y.AsusualK(X,Y )
will deno e he ec o space o all compac linea maps endowed wi h he ope a o
no m. We say ha a se M⊂K(X,Y )isequicompac i he e exis s a null sequence
(x∗
n)nin X∗so ha
Tx≤sup
n|x∗
n(x)|
o all x∈Xand all T∈M. We ecall ha a se M⊂K(X,Y ) is called collec i ely
compac iff T∈MT(BX) has compac closu e. A s anda d p oo using sepa a ions
heo ems and he well-known ac ha a compac se in a Banach space is con ained
in he closu e o he con ex hull o a null sequence o he space allows us o s a e
ha Mis equicompac iff M∗={T∗∈K(Y∗,X∗): T∈M}is collec i ely compac .
We s udy some p ope ies o equicompac se s. Among o he esul s we ha e p o ed
he ollowing:
(1) A se M⊂K(X,Y ) is equicompac iff Msa isfies he nex p ope y (in-
oked as p ope y (P)):
(P) “Fo e e y bounded sequence (xn)nin X he e exis s a
subsequence (xk(n))nso ha (Txk(n))nis a con e ging
sequence uni o mly o T∈M.”
Recei ed by he edi o s Ap il 20, 2004.
2000 Ma hema ics Subjec Classifica ion. P ima y 47B07.
Key wo ds and ph ases. Compac ope a o s, equicompac se , collec i ely compac se .
689
690 E. SERRANO, C. PI ˜
NEIRO, AND J. M. DELGADO
(2) Gi en a null sequence (x∗
n)nin X∗, we will deno e by M((x∗
n)n) he equicom-
pac se o all compac ope a o s T∈K(X,Y ) sa is ying Tx≤
supn|x∗
n(x)| o all x∈X. These se s a e absolu ely con ex and closed
in he s ong ope a o opology. I Ydoes no ha e fini e co ype we p o e
ha , gi en ε>0 and a fini e se {x1,...,x
n}in X, he e exis s an ope -
a o Q∈M((x∗
n)n)so ha Txi≤(1 + ε)Qxi o i=1,...,n and all
T∈M((x∗
n)n).
We also ha e ob ained a pa ial con e se o (2): i M⊂K(X,Y ) is coun ably
compac o he s ong ope a o opology ( o sho , SOT), hen Mis equicompac
when i e ifies he ollowing p ope y (in oked as p ope y (F)):
(F) The e exis s a posi i e cons an Csuch ha , o e e y
fini e se {x1,...,x
n}⊂X he e is an ope a o Qin he
closed absolu ely con ex hull o Msa is ying Txi≤
CQxi o i=1,...,n and all T∈M.
Ou no a ion is s anda d. I Xis a Banach space, BXwill deno e i s closed uni
ball. I Iis an a bi a y index se , we will w i e 1
a(I,X) ( espec i ely, ∞(I,X))
o he Banach space o all absolu ely summable ( espec i ely, bounded) X- alued
unc ions defined on I, endowed wi h he no m ξ=i∈Iξ(i)( espec i ely,
ξ=sup{ξ(i):i∈I}) o eachξ∈1
a(I,X) ( espec i ely, ξ∈∞(I,X)).
2. Equicompac se s
I M⊂K(X,Y ) is bounded, we can conside he con inuous linea map
U:1
a(M,X)−→ Ydefined by U(ξ)=T∈MT(ξ(T)) o all ξ∈1
a(M,X).
P oposi ion 2.1. The ollowing s a emen s a e equi alen o a se M⊂K(X, Y ):
a) Mis collec i ely compac .
b) The ope a o Uis compac .
P oo . Gi en S∈Mand x∈BX,deno ebyξS,x he elemen o 1
a(M,X) defined
by
ξS,x(T)=0i T=S,
xi T=S.
I is clea ha U(ξS,x)=Sx; henH=U{ξS,x :S∈M,x∈BX}=T∈MT(BX).
On he o he hand, o e e y ξ∈B1
a(M,X)we ha e:
U(ξ)= 
T∈M
T(ξ(T)) = 
T∈M
ξ(T)=0
ξ(T)Tξ(T)
ξ(T).
The e o e, U(B1
a(M,X))⊂co (H). So we ha e ob ained H⊂U(B1
a(M,X))⊂co (H)
and his concludes he p oo . 
Now we conside he ope a o V:X−→ ∞(M,Y) defined by (Vx)(T)=Tx
o all T∈Mand x∈X. Fo he p oo o he ollowing p oposi ion, no ice ha
he equi alence a)⇔b)isob iousandb)⇔c) is s aigh o wa d.
P oposi ion 2.2. The ollowing s a emen s a e equi alen o a se M⊂K(X, Y ):
a) Mis equicompac .
b) The ope a o Vis compac .
c) Mhas p ope y (P).
EQUICOMPACT SETS OF OPERATORS DEFINED ON BANACH SPACES 691
Now we a e eady o ace ou main esul :
Theo em 2.3. I M⊂K(X,Y )is bounded, he ollowing s a emen s a e equi alen :
a) Mis collec i ely compac .
b) M∗has p ope y (P).
P oo . The adjoin map o U:1
a(M,X)−→ Yis he ope a o U∗:Y∗−→ ∞(M,X∗)
defined by (U∗y∗)(T)=T∗y∗, o all T∈Mand y∗∈Y∗.In ac ,weha e:
ξ,U∗y∗=Uξ,y∗=
T∈MT(ξ(T)),y∗=
T∈Mξ(T),T∗y∗
o all ξ∈1
a(M,X)andy∗∈Y∗. A call o P oposi ions 2.1 and 2.2 concludes he
p oo . 
The nex lemma easily yields he dual equi alence
Mhas p ope y (P)⇐⇒ M∗is collec i ely compac .
Lemma 2.4. I (x∗
n)nis a null sequence in X∗and Mis a subse o K(X, Y )such
ha
Tx≤sup
n|x∗
n(x)|
o all x∈Xand all T∈M, hen
T∗∗x∗∗≤sup
n|x∗∗(x∗
n)|
o all x∗∗ ∈X∗∗ and T∈M.
P oo . Gi en T∈M,ε>0andx∗∗ ∈BX∗∗ ,choosey∗∈BY∗so ha T∗∗x∗∗=
|y∗(T∗∗x∗∗)|. By hypo hesis, he e exis s n0∈Nsuch ha x∗
n<ε/4 o all
n≥n0. Now we conside he weak∗neighbo hood W=W(x∗
1,...,x
∗
n0,T∗y∗;ε/2)
o 0; he e exis s x∈BXsa is ying x∈x∗∗ +W.Thenweha e:
T∗∗x∗∗=|T∗y∗,x
∗∗|
≤|T∗y∗,x
∗∗−T∗y∗,x|+|T∗y∗,x|
<ε
2+sup
n|x∗
n,x|
≤ε
2+sup
n|x∗
n,x−x∗
n,x
∗∗|+sup
n|x∗
n,x
∗∗|
<ε+sup
n|x∗
n,x
∗∗|
o all ε>0. Le ing ε→0, we ob ain T∗∗x∗∗≤supn|x∗∗(x∗
n)| o all x∗∗ ∈X∗∗
and all T∈M.
Rema k 2.5.Ase M⊂L(X, Y )issequen ially weak–no m equicon inuous (o
uni o mly comple ely con inuous) i , o e e y weakly null sequence (xn)nin X,
limnTxn= 0 uni o mly o T∈M. I is ob ious ha e e y equicompac se is
uni o mly comple ely con inuous ( o sho , u.c.c.). I Xdoes no con ain a copy
o 1, hen Rosen hal’s heo em abou 1 ells us ha each bounded sequence in X
has a weakly Cauchy subsequence. So, in his case, e e y u.c.c. se has p ope y
(P) and, he e o e, i is equicompac . Tha is o say, i X1, hen he ollowing
692 E. SERRANO, C. PI ˜
NEIRO, AND J. M. DELGADO
s a emen s a e equi alen o a bounded se M⊂K(X,Y ):
a) Mis equicompac .
b) Mis u.c.c.
c) M∗is collec i ely compac .
The equi alence s a ed in Rema k 2.5 and [4, heo em 2.2] yields di ec ly he
ecen Mayo al’s heo em [3]:
Theo em (F. Mayo al, 2001).I Xdoes no con ain a copy o 1,ase M⊂
K(X,Y )is ela i ely compac iff Mis u.c.c. and, o e e y x∈X, hese M(x)=
{Tx:T∈M}is ela i ely compac in Y.
Ne e heless, o an a bi a y Banach space X, a u.c.c. se M⊂K(X,Y )
is equicompac i , in addi ion, e e y 1-sequence (xn)nin Xhas a subsequence
(xk(n))nsuch ha (Txk(n))nis uni o mly con e gen o T∈M.
3. Domina ed se s o ope a o s
The simples examples o equicompac se s a e he se s domina ed by a compac
ope a o , ha is o say, he se s o which he e exis s an ope a o S∈K(X,Y )
such ha Tx≤Sx o all x∈Xand all T∈M. We a e going o p o e ha a
maximal equicompac se Mis domina ed by an ope a o QH∈Mon e e y fini e
se H={x1,...,x
n}⊂X. Bu a mo e gene al class o se s M⊂L(X,Y )enjoys
his p ope y. Tha is why we now conside he class o (Z, S)-domina ed se s.
Gi en a Banach space Zand a linea map S:X−→ Z,wesay ha ase
M⊂L(X,Y )is(Z, S)-domina ed i Tx≤Sx o all x∈Xand all T∈M.
Examples.
(1) I M⊂K(X,Y ) is an equicompac se sa is ying Tx≤supn|x∗
n(x)| o
all x∈Xand all T∈M,wi h(x∗
n)na null sequence in X∗, conside he
map S:X−→ c0defined by Sx =(x∗
n(x))n. Then, Mis (c0,S)-domina ed.
(2) Le Πp(X,Y )be hespaceo p-summing ope a o s om Xin o Yendowed
wi h he no m πp(T)=sup{(nTxnp)1/p :(xn)n∈Bp
w(X)},whe e
p
w(X) is he Banach space o he weakly p-summable sequences in X.A
se M⊂Πp(X,Y ) is said o be uni o mly p-domina ed i he e exis s a
posi i e Radon measu e µon BX∗such ha
Txp≤BX∗|x∗(x)|pdµ(x∗)
o all x∈Xand all T∈M.Pu Z=Lp(µ, BX∗) and define S:X−→ Z
by (Sx)(x∗)=x∗(x) o all x∗∈BX∗and all x∈X. Then e e y uni o mly
p-domina ed se is (Z, S)-domina ed.
I Xand Ya e Banach spaces, we will deno e by M(Z, S) he se o all ope a o s
T∈L(X,Y ) sa is ying Tx≤Sx o all x∈X.No e ha M(Z, S) is absolu ely
con ex and closed in K(X,Y ) o he SOT opology. The nex heo em shows ha
M=M(Z, S) is domina ed by an ope a o Q∈Mon e e y fini e se {x1,...,x
n}⊂
Xwhen Ydoes no ha e fini e co ype.
EQUICOMPACT SETS OF OPERATORS DEFINED ON BANACH SPACES 693
Theo em 3.1. Le Ybe a Banach space ha does no ha e fini e co ype. Gi en
ε>0, o e e y fini e se {x1,...,x
n}⊂X he e exis s Q∈M(Z, S)such ha
Txi≤(1 + ε)Qxi
o i=1,...,n and all T∈M(Z, S).
P oo . Since Ydoes no ha e fini e co ype, Ycon ains ∞
nuni o mly (∞
n=
(Rn,.∞)). By [2, heo em 14.1], o e e y ε>0andn∈N, he e is an iso-
mo phism Jn om ∞
non o a subspace o Ysa is ying J−1
n=1andJn≤1+ε
o all n∈N.
Gi en {x1,...,x
n}⊂X,choosez∗
i∈BZ∗so ha |z∗
i(Sxi)|=Sxi o i=
1,...,n.Pu yi=Jnei,(ei)n
i=1 being he uni basis o ∞
n. We define an ope a o
Q:X−→ Yby
Qx =1
1+εJn((Sx,z∗
i)n
1).
Then we ha e:
Qx≤(1 + ε)−1Jn(Sx,z∗
i)n
1∞≤Sxsup
iz∗
i≤Sx
and hisp o es ha Q∈M(Z, S).
Finally, we need o p o e ha Txi≤(1 + ε)Qxi o i=1,...,n and all
T∈M(Z, S). Pu y∗
i=e∗
i◦J−1
n,(e∗
i)n
i=1 being he uni basis o (∞
n)∗≃1
n.No e
ha y∗
i≤1 o i=1,...,n. Wealsodeno ebyy∗
ia Hahn-Banach ex ension o
e∗
i◦J−1
n o Y. I is easy o show ha yi,y∗
j=δij.Weha e:
Qxj≥|Qxj,y∗
j|
=(1+ε)−1
n

i=1Sxj,z∗
iyi,y∗
j
=(1+ε)−1Sxj,z∗
j
=(1+ε)−1Sxj
≥(1 + ε)−1Txj
o all T∈M(Z, S)andj=1,...,n.
Co olla y 3.2. I he Banach space Ydoes no ha e fini e co ype, e e y equicom-
pac subse o K(X,Y )may be uni o mly domina ed, on each fini e subse o X,by
a compac ope a o .
Now we gi e a pa ial con e se o he las heo em in case M⊂K(X,Y )isaSOT-
coun ably compac se . No e ha he absolu ely con ex hull o an equicompac se
is equicompac , oo. So, wi hou loss o gene ali y, we can suppose om now on
ha Mis absolu ely con ex.
Theo em 3.3. Le Mbe an absolu ely con ex subse o K(X, Y )enjoying p op-
e y (F).I Mis coun ably compac o he s ong ope a o opology, hen Mis
equicompac .
P oo . (a) Fi s , we p o e he heo em in case Xis a sepa able Banach space. Le
(xn)nbe a dense sequence in X. By hypo hesis, o e e y n∈N, he eexis s
Qn∈Mso ha
Txi≤CQnxi

694 E. SERRANO, C. PI ˜
NEIRO, AND J. M. DELGADO
o i=1,...,n and all T∈M. The sequence (Qn)nhas a clus e poin Q∈M o
he SOT. P oceeding by con adic ion, i is easy o show ha
(1) Txi≤CQxi
o i=1,...,n and all T∈M.Now,gi enx∈Xand ε>0, choose xisuch ha
x−xi<ε/K,whe eK=sup
T∈MT. Using (1), we ha e
Tx≤Tx−Txi+Txi
<ε+CQxi
≤ε+C(Qxi−Qx+Qx)
≤ε+Cε+CQx
o all T∈M. Le ing ε→0 we conclude Tx≤CQx o all T∈M.
(b) Conside now an a bi a y Banach space X. In o de o show ha Mis
equicompac , we will p o e ha Mhas p ope y (P). Le (xn)nbe a bounded
sequence in Xand pu H=span {xn:n∈N}.I iHdeno es he inclusion map
om Hin o X, hen pa (a) applied o he se N={T◦iH:T∈M}⊂K(H, Y )
yields he equicompac ness o Nand, he e o e, also o M.
Examples. 1. A maximal equicompac se o ope a o s alued in 2 ha ails
p ope y (F).Le Mbe he se o all ope a o s T om c0in o 2sa is ying
Tα2≤sup
n
1
√ne∗
n(α)
o all α∈c0((e∗
n)ndeno es he uni basis o 1). By con adic ion, suppose he e
exis s a posi i e cons an Csuch ha , o e e y fini e se {α1,...,α
n}⊂c0, he eis
an ope a o Q∈Msuch ha Tαk2≤CQαk2 o k=1,...,n and all T∈M.
In pa icula , o e e y n∈N, he eexis sQn∈Mso ha
Tek2≤CQnek2
o k=1,...,n and all T∈M(he e (en)ndeno es he uni basis o c0). Then we
ha e
(2)
n

k=1 Tkek2
2≤C2
n

k=1 Qnek2
2
o all n∈Nand all {T1,...,T
n}⊂M. Now we conside he ope a o s Tk∈M
defined by Tkα=1
√ke∗
k(α)uk o all α∈c0,whe e(un)nis he uni basis o 2.
No ice ha Tkek2=1
√k; hen (2) yields
C−2
n

k=1
1
k≤
n

k=1 Qnek2
2
o all n∈N. Finally, ecall ha e e y bounded ope a o T om c0in o 2is
2-summing and π2(T)≤λT, o somecons an λ>0 (see [2, heo em 3.5]).
This ac allows us o ob ain
π2(Qn)2≥
n

k=1 Qnek2
2≥C−2
n

k=1
1
k
o all n∈N. This is a con adic ion because Mis bounded o he π2-no m.
2. I is in e es ing o show ha he coun able compac ness o Mcanno be
omi ed in Theo em 3.3. Fo example, conside he closed uni ball o L(c0,c
0).
EQUICOMPACT SETS OF OPERATORS DEFINED ON BANACH SPACES 695
This ball is he same ha M(c0,I), Ibeing he iden i y map on c0. The p oo o
Theo em 3.1 shows ha M(c0,I) has he ollowing p ope y:
“The e exis s a posi i e cons an Csuch ha , o e e y fini e se
{α1,...,α
n}⊂c0, he e is an ope a o o fini e ank Q∈M(c0,I)
sa is ying Tαk≤CQαk o k=1,...,nand all T∈M(c0,I).”
This implies ha he closed uni ball o K(c0,c
0), M, has p ope y (F). Ne -
e heless, we a e going o show ha Mis no equicompac . Fo each δ=(δn)n
belonging o he uni ball o c0,wedeno ebyTδ he ope a o defined by Tδ(αn)n=
(αn·δn)n o all (αn)n∈c0.I isob ious ha Tδ∈M o all δ∈Bc0. Fo e e y
n∈N,weha eTen=en=1. Since(en)nis weakly null in c0, his shows ha
Mis no u.c.c.
Acknowledgemen
The au ho s hank he e e ee o his in e es ing sugges ions.
Re e ences
[1] N. Dun o d, J. T. Schwa z, Linea ope a o s. Pa I: Gene al heo y, Wiley In e science, New
Yo k and London, 1958. MR0117523 (22:8302)
[2] J. Dies el, H. Ja chow, A. Tonge, Absolu ely summing ope a o s, Camb idge s udies in
ad anced Ma hema ics 43, Camb idge Uni e si y P ess, Camb idge, 1995. MR1342297
(96i:46001)
[3] F. Mayo al, Compac se s o compac ope a o s in absence o 1, P oc Ame . Ma h. Soc. 129
(2001), no. 1, 79–82. MR1784015 (2001e:46026)
[4]T.W.Palme ,To ally bounded se s o p ecompac ope a o s, P oc. Ame . Ma h. Soc. 20
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Depa amen o de Ma em´
a icas, Facul ad de Ciencias Expe imen ales, Campus Uni-
e si a io del Ca men, A da. de las Fue zas A madas s/n, 21071 Huel a, Spain
E-mail add ess:[email p o ec ed]
Depa amen o de Ma em´
a icas, Facul ad de Ciencias Expe imen ales, Campus Uni-
e si a io del Ca men, A da. de las Fue zas A madas s/n, 21071 Huel a, Spain
E-mail add ess:[email p o ec ed]
Depa amen o de Ma em´
a icas, Facul ad de Ciencias Expe imen ales, Campus Uni-
e si a io del Ca men, A da. de las Fue zas A madas s/n, 21071 Huel a, Spain
E-mail add ess:[email p o ec ed]