a Xi :1908.05032 3 [ma h.FA] 2 Jul 2020
OPERATOR INEQUALITIES, FUNCTIONAL MODELS AND
ERGODICITY
LUCIANO ABADIAS, GLENIER BELLO, AND DMITRY YAKUBOVICH
Abs ac . We discuss when an ope a o , subjec o a a he gene al inequali y in he edi a y
o m, admi s a uni a ily equi alen unc ional model o Agle ype in he ep oducing ke nel
Hilbe space associa ed o he inequali y. To he con a y o he p e ious wo k, he ke nel
need no be o Ne anlinna-Pick ype. We de i e some consequences conce ning he e godic
beha io o he ope a o .
1. In oduc ion
1.1. Mo i a ion. Le α( ) be a unc ion ep esen able by he powe se ies P∞
n=0 αn nin
D:= {| |<1}, whe e he coe icien s αna e eal numbe s, and le T∈L(H) be a bounded
linea ope a o on a Hilbe 1space H. Pu
(1.1) α(T∗, T) := ∞
X
n=0
αnT∗nTn,
whe e he se ies is assumed o con e ge in he s ong ope a o opology SOT in L(H). When
αis a polynomial, he se ies abo e is jus a ini e sum, and he e is no con e gence p oblem.
In pa icula , when α( ) = 1 − , he igh hand side o (1.1) is I−T∗T, so T∈L(H) is a
con ac ion i and only i (1 − )(T∗, T)≥0. In he 1960’s Sz.-Nagy and Foias de eloped a
beau i ul spec al heo y o con ac ions (see [62]) based on he cons uc ion o hei unc ional
model.
In his landma k pape [5], Agle showed ha i Thas spec um σ(T) con ained in he uni
disc Dand α(T∗, T)≥0, hen i is na u al o model Tby pa s o B⊗IE, whe e Bis a
sui able weigh ed backwa d shi and IEis he iden i y ope a o on some auxilia y Hilbe
space E. (By a pa o an ope a o we mean i s es ic ion o an in a ian subspace.) Mo e
gene ally, when σ(T)⊂D, i has been ound in a ious pa icula cases ha ins ead o B⊗IE
one should conside ope a o s o he o m (B⊗IE)⊕S, whe e Sis an isome y o a uni a y
ope a o . This ep esen a ion is called a coanaly ic model. As Agle p o ed in [6], i holds,
in pa icula , o m-hype con ac ions, i.e., ope a o s T∈L(H) such ha (1 − )j(T∗, T)≥0
o j= 1,...,m. Agle ’s heo em was gene alized in [46] by M¨ulle and Vasilescu o uples
Da e: July 6, 2020.
Key wo ds and ph ases. dila ion; unc ional model; ope a o inequali y; e godic p ope ies.
1All Hilbe spaces will be assumed o be sepa able
1
2L. Abadias, G. Bello, and D. Yakubo ich
o ope a o s. The i s esul s on Agle model echniques a e exposed in he book [7] by
Agle and McCa hy. In [48], Olo sson ob ained ope a o o mulas o wande ing subspaces,
ele an in he models o m-hype con ac ions. His esul s we e gene alized by Eschmeie in
[31] o uples o commu ing ope a o s, and by Ball and Bolo niko in [10] o wha hey call
β-hype con ac ions.
M¨ulle s udied he case whe e α=pis a polynomial in [45]. He conside s he class C(p)
o ope a o s T∈L(H) such ha p(T∗, T)≥0. He p o es ha any con ac ion T∈ C(p) has
a coanaly ic model whene e p(1) = 0,1/p( ) is analy ic in D, and 1/p( ¯wz) is a ep oducing
ke nel. This las condi ion is equi alen o he ac ha all Taylo coe icien s o 1/p( ) a he
o igin a e posi i e. M¨ulle also conside s some ope a o inequali ies o Twi h in ini ely many
e ms, wi h he same p ope y o posi i i y. This pe mi s him o show ha any ope a o T
is uni a ily equi alen o a pa o a backwa d weigh ed shi wi h he same spec al adius
(see [45, Co olla y 2.3]).
In [50], Olo sson deals wi h he case whe e αis no a polynomial. His assump ions a e
ha αis analy ic on D, does no anish on D, and 1/α has posi i e Taylo coe icien s a
he o igin. Unde his se ing, he s udies con ac ions Ton Hsuch ha α( T∗, T )≥0 o
e e y ∈[0,1). Wi h mo e assump ions, he ob ains he coanaly ic model o his class o
ope a o s.
In [11], he las wo au ho s conside ed unc ions αin he Wiene algeb a AWo analy ic
unc ions in he uni disc wi h summable sequence o Taylo coe icien s, subjec o ce ain
condi ions. I was assumed ha he se ies PαnT∗nTncon e ges in no m. The ope a o s
s udied he e u n ou o be simila o con ac ions (see [11, Theo em I]). This will no longe
be ue in he se ing o he p esen pape (see Example 7.3).
In [11], an explici model in he spi i o Sz.-Nagy and Foias model was cons uc ed o
he class o ope a o s conside ed he e. The oles o he de ec ope a o and he de ec space
we e played by
(1.2) D:= (α(T∗, T))1/2,D:= DH,
whe e he non-nega i e squa e oo is aken.
1.2. Ou se ing. He e he ope a o Dand he space D, de ined by (1.2) whene e α(T∗, T)≥
0, will also play an impo an ole. Recall ha now we conside he con e gence o (1.1) in
SOT. As i will be seen om Example 7.3, his is he app op ia e con e gence in his con ex .
Ou assump ions a e he ollowing.
Hypo heses 1.1. Suppose αis a unc ion in AWwhich does no anish on D. We pu
k( ) = 1/α( ) = ∞
X
n=0
kn n ∈D,
wi h α0=k0= 1, and assume ha kn>0 o e e y n≥1.
Ope a o Inequali ies, Func ional Models and E godici y 3
Unde Hypo heses 1.1, we deno e by Hk he weigh ed Hilbe space o powe se ies ( ) =
P∞
n=0 n nwi h ini e no m
k kHk:= ∞
X
n=0 | n|2kn1/2
.
Le Bkbe he backwa d shi on Hk, de ined by
(1.3) Bk ( ) = ( )− (0)
.
De ini ion 1.2. Fix a unc ion αsa is ying Hypo heses 1.1, and le Tbe an ope a o in
L(H). We say ha Tis α-modelable i Tis uni a ily equi alen o a pa o an ope a o o
he o m (Bk⊗IE)⊕S, whe e Sis an isome y.
We ema k ha Bk⊗IEac s on he Hilbe space Hk⊗ E, which can be iden i ied wi h
he weigh ed Hilbe space o E- alued powe se ies ( ) = P∞
n=0 n nwi h no m gi en by
k kHk⊗E =∞
X
n=0 k nk2
Ekn1/2
.
I ac s acco ding o he same o mula (1.3).
I is na u al o pose he ollowing ques ion.
Ques ion 1.3. Gi en a unc ion αsa is ying Hypo heses 1.1, gi e a good su icien condi ion
o an ope a o T∈L(H) o be α-modelable.
One o he s onges esul s in his di ec ion is con ained in he ecen pape s by Bickel,
Ha z and McCa hy [17] and by Clouˆa e and Ha z [21]. I is s a ed o sphe ically symme ic
uples o ope a o s. Fo he case o a single ope a o , hei esul can be o mula ed as ollows.
Theo em 1.4 ([21, Theo em 1.3]).Le αbe a unc ion wi h α0= 1 and αn≤0 o all n≥1.
Suppose ha k= 1/α has adius o con e gence 1,kn>0 o e e y n≥0and
(1.4) lim
n→∞
kn
kn+1
= 1.
Then Bkis bounded, and a Hilbe space ope a o Tis α-modelable i and only i α(T∗, T )≥0.
I is easy o see ha he hypo heses o Theo em 1.4 imply Hypo heses 1.1. This heo em
conce ns he Ne anlinna-Pick case, ha is, when α0= 1 and αn≤0 o n≥1. Al e na i ely,
we say ha kis a Ne anlinna-Pick ke nel. In he ecen wo k [22], Clouˆa e, Ha z and
Schillo es ablish a Beu ling–Lax–Halmos heo em o ep oducing ke nel Hilbe spaces in
he Ne anlinna-Pick con ex . We e e he eade o [26, 49, 56, 57] o mo e esul s in he
Ne anlinna-Pick case. In he ecen p ep in [32], Eschmeie and To h ex end p e ious esul s
by Eschmeie [31] o all comple e Ne anlinna-Pick ke nels, in he con ex o ope a o uples.
1.3. Main esul s. The ollowing esul gi es a new answe o Ques ion 1.3.
4L. Abadias, G. Bello, and D. Yakubo ich
Theo em 1.5. Assume Hypo heses 1.1. I k∈AW, and i s Taylo coe icien s {kn}sa is y
k1/n
n→1,sup kn/kn+1 <∞and
(1.5) lim
m→∞ sup
n≥2mX
m≤j≤n/2
kjkn−j
kn
= 0,
hen Bkis bounded, and he ope a o T∈L(H)is a pa o Bk⊗IE( o some Hilbe space
E) i and only i bo h P|αn|T∗nTnand PknT∗nTncon e ge in SOT and α(T∗, T)≥0.
Mo eo e , in his case one can ake E=D.
As i will be seen la e , he SOT-con e gence o P|αn|T∗nTnimplies he he SOT-
con e gence o PαnT∗nTn.
No ice ha in Theo em 1.5, he isome ic pa Sis unnecessa y (see Theo em 1.12 (ii)
below o mo e in o ma ion). This heo em shows ha Tis α-modelable in many cases when
kis no a Ne anlinna-Pick ke nel, and so Theo em 1.4 does no apply. No much abou hese
ke nels has been known p e iously. Gi en an in ege N≥2, he e a e examples o unc ions
ksa is ying he hypo heses o Theo em 1.5 wi h wha e e p esc ibed signs o he coe icien s
α2,...,αN(see Example 5.1). No e ha α1=−k1is always nega i e.
Rema k 1.6. Suppose ha k∈AWand he sequence
kn
kn+1 1 + 1
n+ 1a
is inc easing o some a > 1. Then (1.5) holds. This is close o [60, P oposi ion 34]. Indeed,
pu k∗
j:= (j+ 1)−a, and de ine ρj:= kj/k∗
j. Then ou condi ion educes o he condi ion
ρn+2/ρn+1 ≥ρn+1/ρn, o all n, which implies ha ρjρn−j/ρn≤C, o 0 ≤j≤n. Since
kjkn−j
kn
=ρjρn−j
ρn
k∗
jk∗
n−j
k∗
n
and {k∗
n}sa is ies (1.5), i ollows ha {kn}also sa is ies (1.5).
Hence, o su icien ly egula sequences {kn}, he condi ion (1.5) is a he close o he
condi ion Pkn<∞. I can be added ha , in ac , in Theo em 1.5 {kn}need no be egula ;
mo eo e , he quo ien s kn/kn+1 need no con e ge (see Rema k 5.2).
The echniques employed in he p oo a e di e en om [21]. We use, basically, a combi-
na ion o M¨ulle ’s a gumen s in [45] and Banach algeb as echniques.
The abo e heo ems open he ques ion o desc ibing in a ian subspaces o Bk⊗IEand
o cons uc ing a unc ional model o ope a o s unde he s udy, which ce ainly would be
in e es ing. We do no add ess his ques ion in his pape .
Gi en an ope a o C:H→ E, whe e Eis an auxilia y Hilbe space, we de ine
(1.6) VCx(z) = C(IH−zT)−1x, x ∈H, z ∈D.
Ope a o Inequali ies, Func ional Models and E godici y 5
The nex esul shows ha whene e Tis α-modelable, he ope a o VD:H→ Hk⊗Dis
a con ac ion, and we can gi e an explici model o T( ha is, gi e explici ly E,Sand he
ans o m which sends he ini ial space in o he model space). Fi s we need o s a e one
mo e echnical hypo hesis, whose meaning will be clea la e .
Hypo heses 1.7. Le αbe a unc ion sa is ying Hypo hesis 1.1. Pu
(1.7) β( ) = X
n≥0
βn n,whe e βn=|αn|,
and γ( ) = β( )k( ). We assume ha kn/kn+1 ≤C′and γn≤C′′kn o all n≥0.
The condi ion kn/kn+1 ≤C′is equi alen o boundedness o Bk. I i holds, hen he
second condi ion is sa is ied whene e he e is some Nsuch ha ei he αn≥0 o n≥N, o
αn≤0 o n≥N.
Theo em 1.8 (Explici model).Assume Hypo heses 1.1 and 1.7. Le Tbe α-modelable.
Then α(T∗, T)≥0,VDis a con ac ion, and hence we can de ine
W= (IH−V∗
DVD)1/2,W=WH.
Mo eo e , S:W → W, gi en by SW x := WTx, is an isome y and he ope a o
(VD, W) : H→(Hk⊗D)⊕W,(VD, W )h= (VDh, Wh)
p o ides a model o T, in he sense ha (VD, W)is isome ic and
((Bk⊗ID)⊕S)·(VD, W) = (VD, W)·T.
Rema k 1.9. Suppose αsa is ies he abo e wo hypo heses, and suppose ha Tis an α-
modelable ope a o , which is gi en al eady by i s model wi hou he isome ic pa . Tha is,
he e is an in a ian subspace Lo an ope a o Bk⊗IE, ac ing on Hk⊗ E, such ha Tis
he es ic ion o his ope a o o L. Then D=E(iden i ied wi h he cons an unc ions in
Hk⊗E), and VDis he iden i y ope a o on L. This ollows om Co olla y 2.13 below.
Simila ly, in he gene al case, i Tis a pa o an ope a o (Bk⊗IE)⊕S, whe e Sis an
isome y, he e is a uni a y ope a o usuch ha he ans o m (VD, uW ) is jus he iden i y.
I i is known ha Tis α-modelable, one can ask abou he uniqueness o he model. Fo
answe ing his ques ion, we need he ollowing de ini ions.
De ini ion 1.10. Le Lbe an in a ian subspace o (Bk⊗IE)⊕S, whe e S:W → W is an
isome y. We will say ha he co esponding model ope a o
(Bk⊗IE)⊕S|L
is minimal i he ollowing wo condi ions hold.
(i) Lis no con ained in (Hk⊗E′)⊕W o any E′$E.

6L. Abadias, G. Bello, and D. Yakubo ich
(ii) Lis no con ained in (Hk⊗E)⊕W′ o any W′$Win a ian by S.
In Rema k 3.5 we show ha he explici model ob ained in Theo em 1.8 is indeed minimal.
No e ha unde Hypo heses 1.1, αis de ined on he closed uni disc Dand does no
anish on he in e al [0,1). Since α(0) = α0= 1, we ob ain ha α(1) ≥0. We dis inguish
he ollowing wo cases. This dis inc ion appea s al eady in [21, Subsec ion 2.3] o he
Ne anlinna-Pick case.
De ini ion 1.11. Suppose ha αmee s Hypo heses 1.1. We will say ha αis o c i ical ype
(o , al e na i ely, ha we ha e he c i ical case) i α(1) = 0. I α(1) >0, we will say ha α
is o subc i ical ype (o , al e na i ely, ha we ha e he subc i ical case).
Theo em 1.12 (Uniqueness o he minimal model).Suppose ha αmee s Hypo heses 1.1
and 1.7. Le Tbe an α-modelable ope a o .
(i) In he c i ical case, he minimal model o Tis unique. Mo e p ecisely, he pai o
ans o ms (VD, W0), whe e W0= (I−V∗
DVD) : H→ W0and W0:= Ran(I−V∗
DVD),
gi es ise o a minimal model, and any minimal model is p o ided by (VC, W), whe e
C= D,W=wW0:H→ W and , w a e uni a y isomo phisms.
(ii) In he subc i ical case, he minimal model o Tis no unique, in gene al. Howe e ,
he e always exis s a minimal model gi en by V=VD, in he sense ha VD:H→
Hk⊗IDis an isome y such ha (Bk⊗ID)VD=VDT. No e ha in his case he
isome y Sis absen .
We ema k ha he e a e o he wo ks ha gi e answe s o he abo e Ques ion 1.3. In
pa icula , Po [52] ga e a model o ope a o s sa is ying wo inequali ies (1 −p)(T∗, T)≥0
and (1 −p)m(T∗, T)≥0, whe e pis a polynomial wi h nonnega i e coe icien s, m≥1 and
p(0) = 0 ( his class is a gene aliza ion o m-hype con ac ions). In ac , she ea s uples o
ope a o s. In [10], Ball and Bolo niko conside a unc ion α( ) in he Wiene algeb a such
ha k= 1/α has posi i e coe icien s sa is ying 0 < ε ≤kn/kn+1 ≤1 o all n(so ha Bk
is a con ac ion). They show ha an ope a o Tis α-modelable, wi h absen isome ic pa ,
i and only i i α(T∗, T)≥0 as well as in ini ely many addi ional inequali ies hold (Tis β-
hype con ac i e, whe e βn= 1/kn), and Tis wha hey call β-s able. See [10], Theo em 4.3.
In [10], Theo em 7.2, Ball and Bolo niko gi e a model o Tin e ms o hei gene aliza ion
o he cha ac e is ic unc ion, which is an in ini e amily o ope a o - alued unc ions.
Whe eas hese au ho s ea bo h subc i ical and c i ical cases, Theo em 1.5 only conce ns
he subc i ical case (because o he condi ion k∈AW).
1.4. Consequences o he model. I an ope a o Tis α-modelable, i is na u al o s udy
wha consequences can be de i ed om he model. He e we ob ain wo ypes o consequences:
(1) when he de ec ope a o Dhas ini e ank ( ha is, dim D<∞), and
(2) e godic consequences when α( ) = (1 − )awi h 0 < a < 1.
Ope a o Inequali ies, Func ional Models and E godici y 7
We will use he space Rk=H˜
k, whe e ˜
kn= 1/kn. I is easy o see ha i is he ep o-
ducing ke nel Hilbe space, co esponding o he posi i e de ini e ke nel k(z, w) := k( ¯wz).
The pai ing h , gi=P n¯gn( ∈ Hk, g ∈ Rk) makes Rkna u ally dual o Hκ. In his
in e p e a ion, he adjoin ope a o o Bkis he ope a o g(z)7→ zg(z), ac ing on Rk.
I αis o subc i ical ype, we ha e he ollowing esul ela ed o he Ca leson condi ion.
Theo em 1.13. Le Tbe an ope a o simila o a pa o Bk⊗ID, ac ing on he space Hk⊗D,
whe e Rkis a Banach algeb a and Dis ini e dimensional. Suppose ha
lim
n→∞in
j≥0
kj
kn+j1/n
= lim
n→∞k1/n
n= 1,
and also ha
(1.8) ∞
X
n=N
kn≤CN−ε∀N≥0,
o some posi i e cons an s Cand εwhich do no depend on N. Suppose ha he spec um
σ(T)does no co e D. Pu
E:= (σ(T)∩D)∩T
and le {lν}deno e he leng hs o he ini e complemen a y in e als o E(in T). Then he
Lebesgue measu e o Eis 0, and he Ca leson condi ion holds:
X
ν
lνlog 2π
lν
<∞.
Some o he a gumen s employed in he p oo o his heo em a e ela ed wi h he so-called
index o an in a ian subspace o Rk⊗E; see Sec ion 6 o mo e de ails.
In he c i ical case, an impo an amily o unc ions αa e hose o he o m α( ) := (1− )a,
o a > 0. No e ha hey sa is y Hypo heses 1.1. When a=mis a posi i e in ege , i is said
ha T∈L(H) is an m-con ac ion i (1 − )m(T∗, T)≥0, and ha Tis an m-isome y i
(1 − )m(T∗, T) = 0. The pape s [13, 15, 16, 35, 55] (among o he s) s udy m-isome ies. The
pape [43] is dedica ed o a p o ound s udy o 2-isome ies. In [36], Gu ea s a mo e gene al
class o (m, p)-isome ies on Banach spaces, and in [37], he discusses m-isome ic uples o
ope a o s on a Hilbe space. In [20], Cha an and Sholapu ka s udy ano he in e es ing class
o ope a o s: Tis a join comple e hype expansion o o de mi (1 − )n(T∗, T)≤0 o e e y
in ege n≥m. Tha wo k, in ac , is de o ed o uples o commu ing ope a o s.
He e we in oduce he case when he exponen ais no an in ege . The de ini ions o
a-con ac ion and a-isome ies a e he na u al ones: we say ha Tis an a-con ac ion i
(1 − )a(T∗, T)≥0, and Tis an a-isome y i (1 − )a(T∗, T) = 0.
No e ha α( ) := (1 − )ais o Ne anlinna-Pick ype when 0 < a < 1. In his case, wi h
he help o he model gi en by Theo em 1.4, we will ge he ollowing wo e godic esul s.
8L. Abadias, G. Bello, and D. Yakubo ich
Theo em 1.14. I Tis an a-con ac ion, wi h 0< a < 1, hen Tis quad a ically (C, b)-
bounded o any b > 1−a.
Tha Tis quad a ically (C, b)-bounded (whe e he le e C s ands o Ces`a o) means ha
he e exis s a cons an c > 0 such ha
sup
n≥0
1
kb+1(n)
n
X
j=0
kb(n−j)kTjxk2≤ckxk2(∀x∈H),
whe e he numbe s k−s(n), called Ces`a o numbe s, a e de ined by
(1 − )s=: ∞
X
n=0
k−s(n) n.
As we will show (see Example 7.3), o any a∈(0,1), he class o a-con ac ions on His
s ic ly la ge han he class o con ac ions. I is ob ious ha any con ac ion is quad a ically
(C, b)-bounded (due o he equali y Pn
j=0 kb(n−j) = kb+1(n) o any b > 0). The meaning o
he abo e ac is ha some e godic p ope ies o con ac ions s ill hold ue o a-con ac ions.
Theo em 1.15. Le Tbe an a-con ac ion wi h 0< a < 1and le b > 1−a. Then he
ollowing s a emen s a e equi alen .
(i) The isome y Sdoes no appea in he (1 − )a-model o T.
(ii) Fo e e y x∈H,
(1.9) ∃lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)kTjxk2= 0.
(iii) Fo e e y x∈H,
lim in
n→∞ kTnxk= 0.
Rema k 1.16. Fo any a∈(0,1), he e a e a-con ac ions which a e no con ac ions. This
ollows om Theo em 7.2 below. The same holds o a > 1. Indeed, i m < a ≤m+ 1, whe e
mis an in ege , hen i is easy o ge (see ou o hcoming pape [1]) ha any (m+1)-isome y
Tis also an a-isome y, which means ha (1 − )a(T∗, T) = 0. The e a e (m+ 1)-isome ies
ha a e no con ac ions, and each o hem is an example o his ype.
1.5. Con en s. The pape is o ganized as ollows. In Sec ion 2 we in oduce wo amilies
o ope a o s in L(H) depending on a ixed unc ion α( ) = Pn≥0αn n: Admw
αand Cw
α.
Essen ially, Admw
αis he amily o ope a o s T o which we can de ine α(T∗, T), and i s
sub amily Cw
αconsis s o hose T o which α(T∗, T )≥0. We use he supe sc ip no a ion
“w” in Admw
αand Cw
α o make i easie o compa e he esul s om [11] and om he p esen
pape . No ice ha in [11], only he con e gence o
We ob ain some in e es ing p ope ies o hese amilies and cha ac e ize he membe ship o
backwa d and o wa d weigh ed shi s o hem. In Sec ion 3, we p o e Theo ems 1.8 and 1.12.
Ope a o Inequali ies, Func ional Models and E godici y 9
The p oo o Theo em 1.5 is gi en in Sec ion 4. In Sec ion 5 we s udy he scope o Theo em 1.5.
The e we p esen examples sa is ying he hypo hesis o Theo em 1.5, whe e Theo em 1.4 does
no apply. In Sec ion 6 we p o e Theo em 1.13. The p oo s o Theo ems 1.14 and 1.15 a e
gi en in Sec ion 7.
In ou o hcoming pape [1], we will s udy models up o simila i y (ins ead o uni a y
equi alence). The e we will conside unc ions α ha may ha e ze oes in D. We will p o e
ha unde ce ain hypo heses, any ope a o in Cw
αis simila o an a-con ac ion i α( )
“beha es like” (1 − )ain a neighbo hood o 1. We will also s udy a-con ac ions in mo e
de ail.
2. P elimina ies on classes de ined by ope a o inequali ies
In his sec ion we in oduce he ope a o classes Admw
αand Cw
αassocia ed o a unc ion
α( ) = Pn≥0αn n, wi h αn∈R. A e s udying hem, we analyze why Hypo heses 1.1 a e
na u al. Finally, a he end o he sec ion we discuss he membe ship o weigh ed shi s in
he classes Admw
αand Cw
α.
2.1. The classes Admw
αand Cw
α.Be o e en e ing in o he de ini ions and basic p ope ies
o hese classes, le us men ion he ollowing well known esul ha will be used epea edly.
Lemma 2.1 (see [38, P oblem 120]).I an inc easing sequence {An}o sel adjoin Hilbe
space ope a o s sa is ies An≤CI o all n, whe e Cis a cons an , hen {An}con e ges in
he s ong ope a o opology.
De ini ion 2.2. Gi en a unc ion α( ) = Pn≥0αn nwi h αn∈R, we pu
(2.1) Admw
α:= T∈L(H) : ∞
X
n=0 |αn|kTnxk2<∞ o e e y x∈H.
No e ha his class o ope a o s is no a ec ed i we change he signs o some coe icien s
αn’s.
I Xand Ya e wo quan i ies ( ypically non-nega i e), hen X.Y(o Y&X) will
mean ha X≤CY o some absolu e cons an C > 0. I he cons an Cdepends on some
pa ame e p, hen we w i e X.pY. We w i e X≍Ywhen bo h X.Yand Y.X.
P oposi ion 2.3. The ollowing s a emen s a e equi alen .
(i) T∈Admw
α.
(ii) P∞
n=0 |αn|kTnxk2.kxk2 o e e y x∈H.
(iii) The se ies P∞
n=0 |αn|T∗nTncon e ges in he s ong ope a o opology in L(H).
16 L. Abadias, G. Bello, and D. Yakubo ich
o e e y x∈Hand e e y non-nega i e in ege n. Fix a posi i e in ege N. Then
N
X
n=0
knkDTnxk2=
N
X
n=0
kn
∞
X
m=0
αmTm+nx2
=∞
X
j=0 
X
n+m=j, n≤N
knαm
Tjx2=: ∞
X
j=0
τjTjx2.
Since αk = 1 we ge τ0= 1 and τ1=···=τN= 0. Mo eo e ,
τN+i=k0αN+i+···+kNαi<0
o e e y i≥1, because all he αj’s abo e a e nega i e o ze o and he kj’s a e posi i e.
The e o e N
X
n=0
knkDT nxk2≤ kxk2
o e e y Nand hence he se ies PknkDTnxk2con e ges o e e y x∈H. This gi es
kVDxk2=∞
X
n=0
knkDT nxk2≤ kxk2,
as we wan ed o p o e. 
The ollowing ac is simple and well-known .
P oposi ion 3.2. Le T∈L(H)wi h σ(T)⊂D, and le Ebe a Hilbe space. A bounded
ans o m V:H→ Hκ⊗E sa is ies
(3.1) V T = (Bκ⊗IE)V
i and only i he e is a bounded linea ope a o C:H→ E such ha V=VC(see (1.6)).
P oo . I is well-known (and s aigh o wa d) ha any bounded ans o m VCsa is ies (3.1).
Con e sely, suppose ha V T = (Bκ⊗IE)V. De ine an(x) by
V x(z) := ∞
X
n=0
an(x)zn, x ∈H.
Then ∞
X
n=0
an(Tx)zn=V Tx = (Bκ⊗IE)V x =∞
X
n=0
an+1(x)zn.
The e o e an+1(x) = an(Tx). The s a emen ollows, pu ing C:= a0, which has o be a
bounded linea ope a o . 
P oposi ion 3.3. Le C:H→ E be a bounded ope a o and le T∈ Cw
α. Then he e exis s
a bounded ope a o W:H→ W such ha he ope a o (VC, W )is isome ic and ans o ms

Ope a o Inequali ies, Func ional Models and E godici y 17
Tin o a pa o he ope a o (Bk⊗IE)⊕S, whe e S∈L(W)is an isome y, i and only i
he ollowing condi ions hold.
(i) VC:H→ Hk⊗E is a con ac ion.
(ii) Fo e e y x∈H,
kxk2−kVCxk2=kTxk2−kVCTxk2.
P oo . Le us suppose i s he exis ence o such ope a o W. Since (VC, W ) is an isome y,
(i) holds. No ice ha (ii) is equi alen o p o ing ha kWxk2=kWTxk2 o e e y x∈H.
Bu his is also immedia e since SWx =W T x and Sis an isome y.
Con e sely, suppose now ha (i) and (ii) a e ue. By (i), we can pu W:= (I−V∗
CVC)1/2
and W:= Ran W. Using (ii) we ha e
(3.2) kW xk2=kxk2−kVCxk2=kTxk2−kVCTxk2=kWT xk2.
We de ine
S(Wx) := W T x,
o e e y x∈H. No e ha Sis well de ined, since kSW xk=kWxkby (3.2). Since WH is
dense in W,Scan be ex ended o an isome y on W. By he de ini ion o W, we know ha
(VC, W) is an isome y and i is immedia e ha
(Bk⊗ID)VC=VCTand SW =WT.
This comple es he con e se implica ion. 
P oposi ion 3.4. Le T∈ Cw
α. Assume ha C:H→ E and W:H→ W a e any bounded
ope a o s such ha (VC, W )is isome ic on (Hk⊗E)⊕W and ans o ms Tin o a pa o
(Bk⊗IE)⊕S, whe e S∈B(W)is an isome y. Then Cand Da e ela ed by
(3.3) kDxk2=kCxk2+α(1)kWxk2,∀x∈H.
P oo . Since (VC, W) is isome ic, we ha e
(3.4) kxk2=kVCxk2+kW xk2=∞
X
n=0
knkCTnxk2+kWxk2,
o e e y x∈H. Subs i u ing xby Tjxabo e and mul iplying by αj, we ob ain ha
αjTjx2=∞
X
n=0
αjknCTn+jx2+αjWTjx2
=∞
X
n=0
αjknCTn+jx2+αjkWxk2,
18 L. Abadias, G. Bello, and D. Yakubo ich
whe e we ha e used ha kWxk2=kWTxk2. The e o e
kDxk2=∞
X
j=0
αjTjx2=∞
X
j=0
∞
X
n=0
αjknCTj+nx2+
∞
X
j=0
αj
kWxk2
(⋆)
=∞
X
m=0 
X
j+n=m
αjkn
kCTmxk2+α(1) kWxk2.
Since αk = 1, he only non- anishing summand in he las se ies abo e is o m= 0 and we
ob ain (3.3). Finally, no e ha he ea angemen in (⋆) is co ec as
∞
X
j=0
∞
X
n=0 |αj|knCTn+jx2≤∞
X
j=0 |αj|Tjx2<∞,
whe e we ha e used (3.4) and ha T∈ Cw
α.
Recall he de ini ion o he minimal model (De ini ion 1.10).
Rema k 3.5. Suppose ha Tis α-modelable. Then Tis uni a ily equi alen o (Bk⊗IE)⊕
S|L, whe e L=Ran (VC, W). This model is minimal i and only i
(a) Ran C=E; and
(b) Ran W=W.
Indeed, in his case, i is easy o see ha (a) is equi alen o (i), and (b) is equi alen o (ii)
in De ini ion 1.10.
P oo o Theo em 1.12. Suppose ha he hypo heses a e sa is ied. Fi s we no ice ha
α(T∗, T)≥0, as i ollows om Co olla y 2.13 and P oposi ion 2.6. The e o e Dis well-
de ined.
(i) In he c i ical case (i.e., α(1) = 0), (3.3) gi es
kDxk=kCxk ∀x∈H,
so he e exis s a uni a y ope a o such ha C= D. This implies he s a emen .
(ii) Suppose we a e in he subc i ical case (i.e., α(1) >0). Fi s , we ema k ha he
model is no unique in gene al. Fo ins ance, ake T=Uany uni a y ope a o . Using
P oposi ion 2.3 and ha α∈AW, we ob ain ha T∈Admw
α. Since
∞
X
n=0
αnkTnxk2= ∞
X
n=0
αn!kxk2∀x∈H,
we ge ha α(T∗, T) = α(1)I≥0. Ob iously, T=Uis a minimal model o T(whe e E= 0,
and W=H). Mo eo e , i k= 1/α i s (1.5), hen Theo em 1.5 (which is p o ed in he nex
sec ion, bu i s p oo is comple ely independen ) gi es ano he model o T. (See Example 5.1
and Rema k 5.2.)
Ope a o Inequali ies, Func ional Models and E godici y 19
Now suppose ha Tis any α-modelable ope a o and (VC, W ) p o ides i s model. Le us
see ha he e exis s a minimal model o Twi h V=VDand Wabsen . Changing xby Tnx
in (3.3) we ob ain
kDT nxk2=kCT nxk2+α(1) kWxk2,
whe e we ha e used ha kWT xk=kW xk. The e o e
kVDxk2=∞
X
n=0
knkDT nxk2=∞
X
n=0
knkCTnxk2+k(1)α(1) kW xk2
=kVCxk2+kWxk2=kxk2,
so VD:H→ Hk⊗E is an isome y and he e o e p o ides a model o T. The space Lis jus
Ran VDin Hk⊗D(which is closed). This model is minimal, because Ran Dis dense in D.
(See Rema k 3.5.) This gi es all s a emen s o (ii). 
P oo o Theo em 1.8. I is an immedia e consequence o Theo em 1.12 and P oposi ion 3.3
(i) ha VDis a con ac ion. Finally, o p o ing ha (VD, W) gi es a model, we jus need o
use he same a gumen employed in he ecip ocal implica ion o P oposi ion 3.3. 
Resul s close o Theo ems 1.8 and 1.12 appea in Schillo’s PhD hesis [58]. He deals wi h he
gene ali y o uples o commu ing ope a o s, bu o he case o one ope a o , he hypo heses
needed he e a e mo e es ic i e han ou s.
Fo example, in [58, Theo em 5.16], he uniqueness o he coex ension is p o ed when T
is wha he calls a s ong k-con ac ion. Fo one single ope a o Tand using ou no a ions,
hese a e ope a o s such ha α(T∗, T)≥0, he limi
Σ(T) := IH−lim
N→∞
N
X
n=0
knT∗nα(T∗, T)Tn
exis s (in SOT), Σ(T)≥0, and Σ(T) = T∗Σ(T)T. In [58, Co olla y 5.17], he gi es an explici
model in ol ing he de ec space DT. His assump ions a e somewha echnical (see [58,
Assump ion 5.8]). He also assumes he exis ence o α(B∗
k, Bk), o which [58, P oposi ion 2.10]
says ha a su icien condi ion is ha he coe icien s {αn}o he unc ion αha e e en ually
he same sign.
Recall ha ou Theo em 1.12 (ii) says ha o he subc i ical case he model is no unique
in gene al. The e o e, since Schillo ob ains uniqueness o he coex ension, i seems ha his
assump ions exclude he subc i ical case.
Schillo’s hesis also con ains a esul on he desc ip ion o in a ian subspaces o a backwa d
shi , analogous o Bk⊗IE, in his se ing o ope a o uples.
No ice ha in Theo ems 1.8 and 1.12 we a e only assuming ha Tis α-modelable. In
pa icula , we do no impose any es ic ion abou he signs o he Taylo coe icien s o he
unc ion α.
20 L. Abadias, G. Bello, and D. Yakubo ich
4. P oo o Theo em 1.5
In his sec ion we p o e Theo em 1.5. Fo ha , we need o ci e some esul s conce ning
Banach algeb as.
Fo any sequence ω={ωn}∞
n=0 o posi i e weigh s, de ine he weigh ed space
ℓ∞(ω) := ( ( ) = ∞
X
n=0
n n: sup
n≥0| n|ωn<∞).
In gene al, i s elemen s a e o mal powe se ies. We will also use he sepa able e sion o his
space:
ℓ∞
0(ω) := ( ( ) = ∞
X
n=0
n n: lim
n→∞| n|ωn= 0).
P oposi ion 4.1 (see [47]).ℓ∞(ω)is a Banach algeb a (wi h espec o he o mal mul ipli-
ca ion o powe se ies) i and only i
(4.1) sup
n≥0
n
X
j=0
ωn
ωjωn−j
<∞.
Theo em 4.2. Le ωn>0and ω1/n
n→1. I supnωn+1/ωn<∞and
(4.2) lim
m→∞ sup
n≥2mX
m≤j≤n/2
ωn
ωjωn−j
= 0,
hen he ollowing is ue.
(i) ℓ∞(ω)is a Banach algeb a.
(ii) I ∈ℓ∞(ω)does no anish on D, hen 1/ ∈ℓ∞(ω).
P oo . The hypo heses imply (4.1), so ha (i) ollows om P oposi ion 4.1. To ge (ii), we ap-
ply he esul s o he pape [29] by El-Fallah, Nikolski and Za abi. We use he no a ion o his
pape . Pu ω′(n) = ω(n)/(n+ 1), A=ℓ∞(ω) and A0=ℓ∞
0(ω). The hypo heses imply ha A
(and hence A0) is compac ly embedded in o he mul iplie con olu ion algeb a mul (ℓ∞(ω′)),
see [29, Lemma 3.6.3]. Hence, by [29, Theo em 3.4.1], o any ∈A0,δ1(A0,M(A0)) = 0,
see [29, Subsec ion 0.2.3] o he de ini ion o his quan i y. This means ha o any δ > 0
he e is a cons an c1(δ)<∞such ha he condi ions ∈A0,k kA= 1 and | |> δ on D
imply ha 1/ ∈A0and k1/ kA≤c1(δ). In pa icula , (ii) holds o in A0. To ge (ii) in
he gene al case, suppose ha ∈Aand | |> δ > 0 on D. Since ( )∈A0 o all < 1,
we ge ha he no ms o he unc ions 1/ ( ) in Aa e uni o mly bounded by c1(δ) o all
< 1. When →1−, each Taylo coe icien o 1/ ( ) ends o he co esponding Taylo
coe icien o 1/ ( ). I ollows ha 1/ is in A(and k1/ kA≤c1(δ)). 
P oo o Theo em 1.5. Pu
ωn:= 1/kn.
Ope a o Inequali ies, Func ional Models and E godici y 21
The i s pa o Theo em 1.5 ( ha Bkis bounded) is s aigh o wa d. Also, by Theo-
em 4.2 (i), ℓ∞(ω) is an algeb a.
Fi s suppose ha Tis a pa o Bk⊗IE, and le us p o e ha Bk∈ Cw
α∩Admw
k.
By Theo em 2.12 (i), we know ha Bk∈Admw
ki and only i
m
X
j=0
kjkm−j.km,
which ollows om Theo em 4.2 (i) and P oposi ion 4.1.
Now le us see ha Bk∈ Cw
α. By Theo em 4.2 (ii), α= 1/k belongs o ℓ∞(ω), and he e o e
|αn|.kn. Then, since Bk∈Admw
k, we ob ain ha Bk∈Admw
α. Finally, Theo em 2.12 (ii)
gi es ha Bk∈ Cw
α(because αk = 1 has non-nega i e Taylo coe icien s). Hence Talso is in
Cw
α∩Admw
k.
Con e sely, le us assume now ha T∈ Cw
α∩Admw
k. We wan o p o e ha Tis a pa o
Bk⊗IE. We adap he a gumen o [45, Theo em 2.2] (whe e he con e gence o he se ies o
ope a o s is in he uni o m ope a o opology).
By P oposi ion 3.2,
(4.3) (Bk⊗ID)VD=VDT.
Mo eo e ,
kVDxk2=∞
X
n=0
knkDTnxk2=∞
X
n=0
kn
∞
X
m=0
αmTn+mx2
=∞
X
j=0 X
n+m=j
knαmTjx2=kxk2,
whe e we ha e used ha Pn+m=jknαmis equal o 1 i j= 0 and is equal o 0 i j≥1. The
e-a angemen o he se ies is co ec since, using ha T∈Admw
α∩Admw
k, we ha e
∞
X
n=0
kn
∞
X
m=0 |αm|Tn+mx2.∞
X
n=0
knkTnxk2.kxk2
and he se ies con e ges absolu ely.
Hence VDis an isome y. Joined o (4.3), his p o es ha Tis uni a ily equi alen o a
pa o Bk⊗ID.
No ice ha in pa icula , we showed ha he hypo heses o Theo em 1.5 imply Hypo he-
ses 1.7.
5. Discussion o Theo em 1.5
In his sec ion we discuss he scope o Theo em 1.5 and gi e a se ies o examples whe e i
applies, whe eas Theo em 1.4 does no . We also will gi e a di ec p oo o a pa icula case
o Theo em 4.2, which does no use he esul s o [29].

22 L. Abadias, G. Bello, and D. Yakubo ich
Gi en an analy ic unc ion ( ) = P n n, we deno e by [ ]Ni s unca ed polynomial o
deg ee N, ha is,
[ ]N:= 0+ 1 +...+ N N.
Example 5.1. Le σ2,...,σNbe an a bi a y sequence o signs ( ha is, a sequence o numbe s
±1). We asse ha he e a e unc ions α, k mee ing all he hypo heses o Theo em 1.5 such
ha sign(αn) = σn, o n= 2,...,N. This is in con as wi h Theo em 1.4, whe e he
Ne anlinna-Pick condi ion was assumed: αn≤0 o n≥2.
To p o e he exis ence o αand kas abo e, ake a polynomial eαo deg ee Nsuch ha
eα0= 1,eα1<0. Fo n= 2,...,N, we se eαn<0 i σn=−1 and eαn= 0 i σn= 1. Pu
ek:= [1/eα]N. The o mula
(5.1) ekn=X
s≥1
n1+···+ns=n
(−1)seαn1···eαns
shows ha all he coe icien s o eka e posi i e. We also equi e ha nei he eαno he
polynomial ek anish on D. I is so i , o ins ance, |αn|a e su icien ly small o n= 2,...,N.
Now pe u b he coe icien s eαj ha a e equal o ze o, ob aining a new polynomial bαsuch
ha
bαj:= (εi σj= 1
eαjo he wise (2 ≤j≤N).
By con inui y, i ε > 0 is small enough, we can gua an ee ha he polynomial bk= [1/bα]N
also has posi i e Taylo coe icien s, and we can also gua an ee ha bk(which is a sligh
pe u ba ion o ek) does no anish on D.
Finally, ake as kany unc ion in AWwi h eal Taylo coe icien s such ha he i s ones
a e
k0=bk0= 1, k1=bk1, . . . , kN=bkN,
and kn−j
kn≤C0(∀n≥2j),
o some cons an C0. Fo ins ance, one can pu kn=An−b o n > N, wi h A > 0 (small
enough) and b > 1. Then k∈AWdoes no anish on D.
Then ob iously ksa is ies (1.5) and hence all he hypo heses o Theo em 1.5. The unc ion
α:= 1/k in AWhas he desi ed pa e n o signs.
Finally, i is impo an o no e ha α1=−k1is always nega i e.
Rema k 5.2. I is also easy o see ha whene e {kn}sa is ies (1.5), any o he sequence
{˜
kn}wi h k0= 1 and c < ˜
kn/kn< C o n > 1, whe e c, C a e posi i e cons an s, also
sa is ies his condi ion. In pa icula , i {kn}sa is ies (1.5) and {˜
kn}is as abo e, whe e C
is su icien ly small, hen k( ) is in e ible in AW, so ha all hypo heses o Theo em 1.5 a e
Ope a o Inequali ies, Func ional Models and E godici y 23
ul illed. So he e a e many examples o unc ions k( ) mee ing hese hypo heses, such ha
he quo ien s kn/kn+1 do no con e ge.
Le us men ion now some ema ks on Theo em 4.2.
Rema k 5.3. I is immedia e ha he condi ion
(5.2) ωn
ωjωn−j≤τj(∀n≥2j),whe e ∞
X
j=0
τj<∞,
implies (4.2) and (4.1) (in pa icula , supnωn+1/ωn<∞). Le us gi e a di ec p oo o
Theo em 4.2 o his pa icula case.
S a emen (i) ollows using P oposi ion 4.1.
(ii) Pu g:= 1/ . Suppose ha g6∈ ℓ∞(ω). This means ha
sup
n≥0|gn|ωn=∞.
Hence, i is clea ha he e exis s a sequence {ρ0
n}in [0,1] such ha ρ0
n→0 (slowly) and
(5.3) sup
n≥0|gn|ωnρ0
n=∞.
Claim. The e exis s a sequence {ρn}wi h
(5.4) ρ0
n≤ρn≤1 and ρn→0
such ha eωn:= ρnωnde ines a Banach algeb a ℓ∞(eω).
Indeed, since Pτj<∞, he e exis s a sequence o posi i e numbe s {cj}such ha cjր ∞
and s ill Pcjτj<∞. Take any sequence {ρn} ha dec eases, ends o ze o, and sa is ies
ρn≥max(ρ0
n,1/cn). Then, o eωn:= ρnωnwe ha e
eωn
eωjeωn−j
=ωn
ωjωn−j
ρn
ρjρn−j≤ωn
ωjωn−j
1
ρj≤τjcj(∀n≥2j).
Since Pτjcj<∞, P oposi ion 4.1 implies ha ℓ∞(eω) is a Banach algeb a, and he p oo o
he claim is comple ed.
Now ix {eωn}as in he claim. We may assume ha (ρ0
n)1/n →1 and he e o e (ρn)1/n →1.
Since he polynomials a e dense in he Banach algeb a ℓ∞
0(eω), any complex homomo phism
χon ℓ∞
0(eω) is de e mined by i s alue on he powe se ies . So he map χ7→ χ( ) is injec i e
and con inuous om he spec um ( he maximal ideal space) o ℓ∞
0(eω) o C. Since eω1/n
n→1,
i s image con ains Dand is con ained in D. Hence he spec um o ℓ∞
0(eω) is exac ly he se
{χλ:λ∈D}, whe e χλ( ) = (λ). (We bo ow his a gumen om [29].) As
neωn= ( nωn)ρn→0,
we ha e ∈ℓ∞
0(eω). Then, using he Gel and heo y (see, o ins ance, [54, Chap e 10]), we
ge ha g= 1/ ∈ℓ∞
0(eω), which con adic s (5.3). The e o e, he assump ion g /∈ℓ∞(ω) is
alse, as we wan ed o p o e.
24 L. Abadias, G. Bello, and D. Yakubo ich
Rema k 5.4. No ice ha he abo e cha ac e iza ion o he spec um o he algeb a ℓ∞
0(eω)
(see he abo e Rema k 5.3) implies he ollowing ac : he condi ions (4.1) and ω1/n
n→1
imply ha Pn1/ωn<∞. This can be p o ed in an elemen a y way, wi hou ecu ing o
he Gel and heo y.
Indeed, by (4.1), he e exis s a cons an C > 0 such ha
n
X
j=1
ωn
ωjωn−j≤C
o e e y n≥1. Fix a posi i e in ege L. Then ob iously, o e e y n≥L,
(5.5)
L
X
j=1
ωn
ωjωn−j≤C.
Le us see ha
(5.6) lim sup
n→∞ min
1≤j≤L
ωn
ωn−j≥1.
Indeed, i (5.6) we e alse, hen he e would exis some < 1 and a posi i e in ege Nsuch
ha
min
1≤j≤L
ωn
ωn−j≤ o n≥N.
F om his, i is easy o see ha
ωn≤ snmax
0≤k≤Nωk, sn:= n−N
L+ 1,
whe e [a] deno es he in ege pa o a. Since snbeha es asymp o ically as n/L, i ollows ha
lim supn→∞ ω1/n
n≤ 1/L <1, which con adic s he hypo hesis ha ω1/n
n→1. The e o e,
(5.6) is ue.
Now, using (5.5), i ollows ha
C≥
L
X
j=1
ωn
ωjωn−j≥min
1≤j≤L
ωn
ωn−jL
X
j=1
1
ωj
.
Taking lim sup when n→ ∞, and using (5.6), we ge ha P1/ωjcon e ges.
The ollowing s a emen shows ha in he subc i ical case, he hypo heses o Theo em 1.5
imply ha he adius o con e gence o he se ies o αis equal o one.
P oposi ion 5.5. I lim k1/n
n= 1 and αis o subc i ical ype, hen αdoes no con inue
analy ically o any disc RD, whe e R > 1.
P oo . Since k( ) has nonnega i e Taylo coe icien s, we ha e |k( )| ≤ k(1) o all ∈D.
Using ha k= 1/α, i ollows ha in he subc i ical case, |α( )| ≥ α(1) >0 o any ∈D.
So, αcanno con inue analy ically o any disc RD, whe e R > 1, because in his case, he
adius o con e gence o he Taylo se ies o kwould be g ea e han 1. 
Ope a o Inequali ies, Func ional Models and E godici y 25
6. Fini e De ec
I is well-known ha in he classical Sz.-Nagy-Foias model, he case o a ini e ank (o
Hilbe -Schmid ) de ec ope a o is an impo an one, whe e much mo e ools and esul s a e
a ailable. In his sec ion, we de i e some consequences o ou model heo ems o he case
when an ope a o T∈ Cw
αis α-modelable and he de ec ope a o D= (α(T∗, T))1/2is o
ini e ank.
We will assume ha he ep oducing ke nel Hilbe space Rkis a Banach algeb a wi h
espec o he mul iplica ion o powe se ies. By [60, P oposi ion 32], i su ices o assume
ha
sup
n
n
X
j=0
k2
jk2
n−j
k2
n
<∞;
compa e wi h he condi ion (4.1). Pu
mn= in
j
kj
kn+j
, 1= lim
n→∞m1/n
n.
This limi exis s, see [60, P oposi ion 12].
We will assume ha
(6.1) 1= lim
n→∞k1/n
n= 1.
Bo h equali ies hold, in pa icula , i lim kn+1/kn= 1. The same is ue i , o ins ance, he
las limi does no exis , bu 0 < σ < kn< C < ∞ o all nand he e is some m≥2 such
ha limnkn+m/kn= 1. We also a e assuming he e ha he isome ic pa Sis no p esen
in he model o T. Hence, Tis uni a ily equi alen o he es ic ion o he backwa d shi
Bk⊗IDon he space Hk⊗D o an in a ian subspace L. Mo e gene ally, his applies o
simila i y ins ead o he uni a y equi alence (we bea in mind models o linea ope a o s up
o simila i y, which a e es ablished in [1]).
He e we p o e he ollowing esul .
Theo em 6.1. Suppose ha Tis simila o a pa o Bk⊗ID, ac ing on he space Hk⊗D,
whe e Rkis a Banach algeb a and Dis ini e dimensional. I he spec um σ(T)does no
co e he open disc D, hen σ(T)∩Dis con ained in he ze o se o a non-ze o unc ion in Rk.
Le us s a wi h some p elimina y ema ks. Suppose ha Tis as in he abo e Theo em 6.1.
Tha is, Tis simila o (Bk⊗ID)|L, whe e L ⊂ Hk⊗Dis an in a ian subspace o Bk⊗ID.
By ixing a basis in D, we may assume ha D=Cd, whe e d= dim D. We will iden i y he
space Hk⊗Dwi h Hd
k=⊕d
1Hk, whose elemen s a e columns wi h en ies in Hk. The adjoin
o Bkon he space Hd
kis he mul iplica ion ope a o Mzon he space Rd
k; his la e space
can be seen as a Banach module o e he Banach algeb a Rk. Pu
J=L⊥⊂ Rk⊗D.
32 L. Abadias, G. Bello, and D. Yakubo ich
he Ces`a o means o o de ao T. When his amily o ope a o s is uni o mly bounded, ha
is,
sup
n≥0kMa
T(n)k<∞,
we say ha Tis (C, a)-bounded.
Rema ks 7.5.
(i) No e ha Pn
j=0 ka(j) = ka+1(n) o any a≥0. Also, i a≥0, hen ka(j)≥0 o e e y
j≥0.
(ii) I a= 0, hen M0
T(n) = Tn. Hence (C, 0)-boundedness is jus powe boundedness.
(iii) I a= 1, hen M1
T(n) = (n+ 1)−1Pn
j=0 Tj. Hence (C, 1)-boundedness is jus Ces`a o
boundedness.
(i ) I is well-known ha i 0 ≤a < b, hen (C, a)-boundedness implies (C, b)-boundedness.
The con e se is no ue in gene al. Fo example, he Assani ma ix
T= −1 2
0−1!
is (C, 1)-bounded, bu since
Tn= (−1)n(−1)n+12n
0 (−1)n!
i is no powe bounded (see [30, Sec ion 4.7]).
De ini ion 7.6. I he sequence o ope a o s {Ma
T(n)}n≥0gi en in De ini ion 7.4 con e ges
in he s ong ope a o opology, we say ha Tis (C, a)-mean e godic.
I Tis (C, 1)-mean e godic, i is con en ional jus o say ha Tis mean e godic.
The e is a well es ablished li e a u e on (C, a)-bounded ope a o s, which explo es qui e a
numbe o p ope ies and hei in e plays. P ope ies, cha ac e iza ion h ough unc ional
calculus and e godic esul s o (C, a)-bounded ope a o s can be ound in [4, 9, 25, 27, 28, 30,
41] and e e ences he ein. The connec ion o hese ope a o s and e godici y da es back o
he ou ies o las cen u y, see [24] and [40]. In he la e pape , E. Hille s udies (C, a)-mean
e godici y in e ms o Abel con e gence ( ha is, ia he esol en ope a o ). As applica ion,
he well known mean e godic on Neumann’s heo em o uni a y g oups on Hilbe spaces
is ex ended o (C, a)-mean e godici y o e e y a > 0 [40, p. 255]. Also, he (C, a)-e godici y
on L1(0,1) o ac ional (Riemann-Liou ille) in eg als is elucida ed in [40, Theo em 11]. In
pa icula , i Vis he Vol e a ope a o hen TV:= I−V, as ope a o on L1(0,1), is no
powe -bounded, and i is (C, a)-mean e godic i and only i a > 1/2 [40, Theo em 11]. This
esul can be ex ended o TVac ing on Lp(0,1), 1 < p < ∞, using es ima es gi en in [44], see
[3, Sec ion 10].

Ope a o Inequali ies, Func ional Models and E godici y 33
In [42], Luo and Hou in oduced a new no ion o boundedness: a bounded linea ope a o
Ton a Banach space Xis said o be absolu ely Ces`a o bounded i
sup
n≥0
1
n+ 1
n
X
j=0 Tjx.kxk
o e e y x∈X. In [14], he au ho s s udy he e godic beha iou o his class o ope a o s.
The abo e de ini ion has been ex ended ecen ly by Abadias and Bonilla in [2]: Tis said o
be absolu ely (C, a)-Ces`a o bounded o some a > 0 i
sup
n≥0
1
ka+1(n)
n
X
j=0
ka(n−j)Tjx.kxk
o e e y x∈X. No e ha o a= 1 he de ini ion o Luo and Hou is eco e ed.
Rema k 7.7. I is well-known ha he ollowing implica ions hold:
Powe bounded ⇒Absolu ely (C, a)-bounded
⇒(C, a)-bounded ⇒ kTnk=O(na).
The i s wo implica ions a e s aigh o wa d. Fo he sake o comple eness, we gi e a p oo
o he las one. Suppose Tis (C, a)-bounded o some a≥0. We deno e by [a] he in ege
pa o a. Then, o n > [a], we ha e
kTnk=
n
X
j=0
k−a(j)
n−j
X
m=0
ka(n−j−m)Tm
.
n
X
j=0 |k−a(j)|ka+1(n−j)
=
[a]
X
j=0
(−1)jk−a(j)ka+1(n−j) +
n
X
j=[a]+1
(−1)[a]+1k−a(j)ka+1(n−j)
=
[a]
X
j=0 (−1)j+ (−1)[a]k−a(j)ka+1(n−j) + (−1)[a]+1
n
X
j=0
k−a(j)ka+1(n−j)
.
[a]
X
j=0 |k−a(j)|ka+1(n−j) + k1(n).ka+1(n)≍(n+ 1)a.
The ollowing ex ension o he abo e de ini ions will be impo an o us.
De ini ion 7.8. Le a > 0 and p≥1. We say ha a bounded linea ope a o Ton a Banach
space Xis (C, a, p)-bounded i
sup
n≥0
1
ka+1(n)
n
X
j=0
ka(n−j)kTjxkp.kxkp,
34 L. Abadias, G. Bello, and D. Yakubo ich
o all x∈X.
No e ha o p= 1 his de ini ion is jus he absolu e (C, a)-boundedness. The case a= 1
has been ecen ly conside ed in [23]. We will use he e m quad a ically (C, a)-bounded ins ead
o (C, a, 2)-bounded.
Using he asymp o ics ka(n)≍(n+1)a−1gi en in (7.3), i is easy o see ha Tis (C, a, p)-
bounded i and only i
(7.7) sup
n≥0
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Tjxp.kxkp(∀x∈X).
The ollowing obse a ion will be essen ial o he p oo o Theo em 1.14.
Lemma 7.9. The ollowing holds.
(i) I Tis (C, a, p)-bounded, hen any pa o Tis also (C, a, p)-bounded.
(ii) I T1and T2a e (C, a, p)-bounded, hen any di ec sum T1∔T2is also (C, a, p)-bounded.
(iii) Le Tbe a bounded linea ope a o on a Hilbe space. I Tis quad a ically (C, a)-
bounded, hen T⊗IEis also quad a ically (C, a)-bounded, whe e IEis he iden i y
ope a o on some Hilbe space E.
P oo . (i) and (ii) a e immedia e. Fo (iii) no e ha i d= dim E ≤ ∞, hen he o hogonal
sum o dcopies o Tis clea ly quad a ically (C, a)-bounded (by he Py hago as Theo em). 
The ollowing esul is e y use ul. I s p oo is simple, and we omi i .
Lemma 7.10. Le 0≤a < b. Then (C, a, p)-boundedness implies (C, b, p)-boundedness.
This lemma shows an inclusion o classes o ope a o s. By [2, Co olla ies 2.2 and 2.3], i
Tis (C, a, 1)-bounded hen kTnk=o(na) o 0 < a ≤1 and kTnk=O(n) o a > 1. The
ollowing esul explains why he case a= 1 is special.
Theo em 7.11. I a > 1and p≥1, hen (C, a, p)-boundedness is equi alen o (C, 1, p)-
boundedness.
P oo . Fix a > 1 and p≥1. By he abo e Lemma, we only need o p o e ha any (C, a, p)-
bounded ope a o Tis (C, 1, p)-bounded. Le Tis (C, a, p)-bounded. Then
(7.8) 1
ka+1(2n)
2n
X
j=0
ka(2n−j)Tjxp.kxkp,
o e e y n≥0, and e e y x∈X. Since a > 1, ka(m) is an inc easing unc ion o m. In
pa icula , ka(n)≤ka(2n−j) o j= 0,...,n. Hence
(7.9) ka(n)
n
X
j=0 Tjxp≤
2n
X
j=0
ka(2n−j)Tjxp,
Ope a o Inequali ies, Func ional Models and E godici y 35
By (7.9) and (7.8), n
X
j=0 Tjxp.ka+1(2n)
ka(n)kxkp.(n+ 1) kxkp,
which means ha Tis (C, 1, p)-bounded. 
Theo em 7.12. Le a > 0and 1≤q < p. I Tis (C, a, p)-bounded, hen i is also (C, b, q)-
bounded o each b > qa/p. In pa icula , (C, a, p)-boundedness implies (C, a, q)-boundedness.
P oo . Le us i s ecall ha i > −1, hen
(7.10)
m
X
j=1
j .m +1 (∀m≥1).
Le Tbe (C, a, p)-bounded and le b > qa/p. Suppose i s ha b6= 1, and pu
s:= p
p−q, s′:= p
q, γ := q(a−1)
p(b−1) .
No e ha sand s′a e posi i e and sa is y 1/s + 1/s′= 1. Since
(b−1)(1 −γ)s=pb −qa
p−q−1>−1 and (b−1)γs′=a−1,
using H¨olde ’s inequali y and (7.10) i ollows ha
1
(n+ 1)b
n
X
j=0
(n+ 1 −j)b−1Tjxq
≤1
(n+ 1)b

n
X
j=0
(n+ 1 −j)(b−1)(1−γ)s

1/s 

n
X
j=0
(n+ 1 −j)(b−1)γs′Tjxqs′

1/s′
.(n+ 1)−qa/p 

n
X
j=0
(n+ 1 −j)a−1Tjxp

q/p
=
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Tjxp

q/p
o e e y x∈Xand e e y non-nega i e in ege n. Hence he s a emen ollows using (7.7).
Now suppose ha b= 1. Take any b′∈(qa/p, 1). We ha e al eady p o ed ha T
is (C, b′, p)-bounded. Then, by Lemma 7.10, i ollows ha Tis (C, 1, p)-bounded. This
comple es he p oo . 
Lemma 7.13. Le a > 0and p≥1. Then e e y isome y Sis (C, a, p)-bounded.
36 L. Abadias, G. Bello, and D. Yakubo ich
P oo . This is immedia e, since indeed
(7.11) 1
ka+1(n)
n
X
j=0
ka(n−j)kSjxkp=1
ka+1(n)

n
X
j=0
ka(n−j)
kxkp=kxkp
o e e y x∈X.
Lemma 7.14. Le 0< s < 1and le a > 0. Then Bsis quad a ically (C, a)-bounded i and
only i 1−s < a. Mo eo e , o 1−s < a we ha e
(7.12) lim
n→∞
1
ka+1(n)
n
X
j=0
ka(n−j)kBj
sxk2= 0 (∀x∈ Hs).
P oo . Recall he no a ion en= n∈ Hk=Hs, whe e k( ) = (1− )−s. Suppose ha a= 1−s.
Then
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Bj
sen2&1
(n+ 1)1−s
n
X
j=0
(n+ 1 −j)−s(n+ 1 −j)s−1
=1
(n+ 1)1−s
n+1
X
j=1
j−1&log(n+ 2) kenk2
(7.13)
o e e y n. The e o e Bsis no quad a ically (C, 1−s)-bounded, and by Lemma 7.10 we
ob ain ha Bsis no quad a ically (C, a)-bounded o a < 1−s.
Le us assume now ha 1 −s < a ≤1 and ix x∈ Hs. W i e xin he o m x=Pxmem,
whe e xm∈C. Then
Bj
sx2=∞
X
m=j
ks(m−j)|xm|2.∞
X
m=j
(m+ 1 −j)s−1|xm|2,
o e e y j≥0. Hence
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Bj
sx2
.1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1∞
X
m=j
(m+ 1 −j)s−1|xm|2
=1
(n+ 1)a
n
X
m=0 |xm|2
m
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1
+1
(n+ 1)a
2n
X
m=n+1 |xm|2
n
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1
+1
(n+ 1)a
∞
X
m=2n+1 |xm|2
n
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1
=: (I) + (II) + (III).
Ope a o Inequali ies, Func ional Models and E godici y 37
In (I), no e ha since 1 −s < a ≤1, and m≤n, we ha e
(7.14)
m
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1≤
m+1
X
j=0
(m+ 1 −j)a+s−2.(m+ 1)a+s−1,
whe e in he las es ima e we used (7.10). The e o e
(I).1
(n+ 1)a
n
X
m=0 |xm|2(m+ 1)a+s−1=


[√n]
X
m=0
+
n
X
m=[√n]+1

|xm|2(m+ 1)a+s−1
(n+ 1)a
.kxk2
√na+
n
X
m=[√n]+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞).
In (II), using ha m > n and s−1<0, we ha e
n
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1≤
n
X
j=0
(n+ 1 −j)a+s−2.(n+ 1)a+s−1.
The e o e
(II).1
(n+ 1)a
2n
X
m=n+1 |xm|2(n+ 1)a+s−1= (n+ 1)s−1
2n
X
m=n+1 |xm|2
.
2n
X
m=n+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞).
Finally, in (III), since m > 2nwe ha e ha
n
X
j=0
(n+ 1 −j)a−1(m+ 1 −j)s−1.(m+ 1)s−1
n
X
j=0
(n+ 1 −j)a−1.(m+ 1)s−1(n+ 1)a.
The e o e
(III).∞
X
m=2n+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞).
Hence (7.12) ollows when 1 −s < a ≤1. Finally, suppose ha 1 < a. Then
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Bj
sx2≤1
n+ 1
n
X
j=0 Bj
sx2−→ 0 (as n→ ∞),
since his is he case o a= 1 in (7.12) (al eady p o ed). No e ha (7.12) implies quad a ical
(C, a)-boundedness, so he p oo is comple e. 
This lemma allows us o p o e he ollowing mo e gene al esul .

38 L. Abadias, G. Bello, and D. Yakubo ich
Theo em 7.15. Le 0< s < 1and 1≤q≤2. Then Bsis (C, b, q)-bounded i and only i
b > q(1 −s)/2. Mo eo e , o b > q(1 −s)/2we ha e
(7.15) lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)Bj
sxq= 0 (∀x∈H).
P oo . No e ha q= 2 is p ecisely Lemma 7.14. So we assume ha 1 ≤q < 2. I b=
q(1 −s)/2, aking x=en, we ge , as in (7.13), ha
1
kb+1(n)
n
X
j=0
kb(n−j)Bj
senq&log(n+ 2) kenk2
o e e y n. The e o e Bsis no (C, q(1 −s)/2, q)-bounded, and by Lemma 7.10 we ge ha
Bsis no (C, b, q)-bounded o b < q(1 −s)/2.
Now suppose ha b > q(1 −s)/2. Then b=qa/2 o some a > 1−s. Using H¨olde ’s
inequali y as in he p oo o Theo em 7.12, we ob ain
1
(n+ 1)b
n
X
j=0
(n+ 1 −j)b−1Bj
sxq.
1
(n+ 1)a
n
X
j=0
(n+ 1 −j)a−1Bj
sx2

q/2
−−−→
n→∞ 0,
by Lemma 7.14. Hence (7.15) ollows. 
P oo o Theo em 1.14. Le T∈ Cw
awi h 0 < a < 1 and le b > 1−a. By Theo em 1.4 and
Theo em 1.12 (i), Tis uni a ily equi alen o a pa o (Ba⊗ID)⊕S. Hence, by Lemma 7.9
(i), i is enough o p o e ha (Ba⊗ID)⊕Sis quad a ically (C, b)-bounded. Bu his is
immedia e using Lemma 7.9 (ii) and (iii), and Lemmas 7.13 and 7.14. 
Fo he p oo o Theo em 1.15 we need he ollowing lemma, which is in he spi i o
Lemma 7.9.
Lemma 7.16. The ollowing holds.
(i) I Tsa is ies (1.9), hen any pa o Talso sa is ies (1.9).
(ii) I T1and T2sa is y (1.9), hen any di ec sum T1∔T2also sa is ies (1.9).
(iii) Le Tbe a bounded linea ope a o on a Hilbe space. I Tsa is ies (1.9), hen he
ope a o T⊗IEalso sa is ies (1.9), whe e IEis he iden i y ope a o on some Hilbe
space E.
P oo . (i) and (ii) a e immedia e. Fo (iii) we use he same a gumen as in Lemma 7.9 (iii)
and a simple applica ion o Lebesgue’s Domina ed Con e gence Theo em. 
P oo o Theo em 1.15. As in he p oo o Theo em 1.14, we ha e ha Tis uni a ily equi a-
len o
(Ba⊗ID)⊕S|L,
Ope a o Inequali ies, Func ional Models and E godici y 39
whe e Lis a subspace o (Ha⊗D)⊕W in a ian by (Ba⊗ID)⊕S.
Le us p o e he ci cle o implica ions (i) ⇒(ii) ⇒(iii) ⇒(i).
Suppose ha (i) is ue. Tha is, Tis uni a ily equi alen o
(Ba⊗ID)|L,
whe e Lis a subspace o Ha⊗Din a ian by Ba⊗ID. Then (ii) ollows using Lemmas 7.14
and 7.16.
Suppose now ha
lim in
n→∞ kTnxk>0
o some x∈H. Then, ob iously, kTnxk> ε > 0 o e e y n≥0. Hence o his ec o x
(1.9) does no hold. The e o e we ha e p o ed ha (ii) ⇒(iii).
Finally, suppose ha he isome y Sappea s in he minimal model. Then o some ec o
ℓ= (ℓ1, ℓ2)∈ L, i s second componen ℓ2∈ W is no 0. The e o e
lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)((Ba⊗ID)⊕S)jℓ2
= lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)(Ba⊗ID)jℓ1⊕Sjℓ22
= lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)(Ba⊗ID)jℓ12+ lim
n→∞
1
kb+1(n)
n
X
j=0
kb(n−j)Sjℓ22.
The second limi is kℓ2k26= 0 because o (7.11). Hence we ob ain ha (iii) ⇒(i). 
Rema k 7.17. In he same way, we ge ha i Tis an a-con ac ion and 0 < a ≤1, hen
lim in
n→∞ kTnxk ≤ kxk.
In pa icula , his lowe limi is ini e o any x.
Since Cw
1is jus he se o all con ac ions on H,T∈ Cw
1i T∗∈ Cw
1. Howe e , his is no
longe ue o a∈(0,1).
P oposi ion 7.18. I a∈(0,1), hen he e is an ope a o T∈ Cw
asuch ha T∗/∈ Cw
a.
P oo . No e ha B∗
ais a o wa d weigh ed shi such ha kB∗n
a 0k → ∞ as ngoes o ∞. So
Ba∈ Cw
a, whe eas i s adjoin canno belong o Cw
a, because B∗
ais no quad a ically (C, b)-
bounded o any b(see Lemma 7.14). 
I is na u al o pose he ollowing ques ion.
Ques ion 7.19. Fo which unc ions α, sa is ying Hypo heses 1.1, is i ue ha T∈ Cw
α
implies T∗∈ Cw
α?
40 L. Abadias, G. Bello, and D. Yakubo ich
I is so o α( ) = 1 − and, mo e gene ally, o α( ) = 1 − n,n≥1. The au ho s do no
know o he examples.
Rema ks 7.20.
(i) I Tis an ope a o in Cw
awi h 0 < a < 1, and 0 < q < 2, hen by Theo em 1.14 and
Theo em 7.12, i ollows ha Tis (C, b, q)-bounded o all b > q(1−a)
2.
(ii) An m-isome y T, which is no an isome y, canno be (C, a, p)-bounded, because
he e a e ec o s xsuch ha he no ms kTnxkgo o in ini y. The possibili y o
hese ope a o s o ha e weake e godic p ope ies, such as he Ces`a o boundedness
and weak e godici y, ha e been s udied in [13].
(iii) Le Tbe an ope a o in Cw
awi h 0 < a < 1. Using Theo em 1.14 (i) and Theo em 7.12
(wi h p= 2 and q= 1) we ob ain ha Tis (C, b, 1)-bounded o e e y b > (1 −a)/2.
By [2, Co olla y 3.1], we ge ha Tis (C, b)-mean e godic, ha is, he e exis s
Pbx:= lim
n→∞Mb
T(n)x, x ∈H.
The e o e, by [3, Theo em 3.3], we ha e
H= Ke (I−T)⊕Ran(I−T).
In ac ,
Ke (I−T) = RanPband Ran(I−T) = Ke Pb.
Also no e ha
Mb
T(n)x=x o x∈Ke (I−T),and lim
n→∞Mb
T(n)x= 0 o x∈Ran(I−T).
Le now 0 < γ < 1,by [3, P oposi ion 4.8 and Rema k 4.9], one can de ine a
bounded ope a o (I−T)γby means o a ce ain unc ional calculus, and
Ke (I−T) = Ke (I−T)γ,Ran(I−T) = Ran(I−T)γ,
wi h Ran(I−T)⊆Ran(I−T)γ. Fu he mo e i γ < 1−b, o x∈Ran(I−T),
x∈Ran(I−T)γ⇐⇒ ∞
X
n=1
1
n1−γTnxcon e ges,
see [3, Theo em 9.2].
(i ) By [2, Theo em 3.1], i Tis an ope a o in Cw
awi h 0 < a < 1 and b > (1 −a)/2, hen
lim
n→∞kMb
T(n+ 1) −Mb
T(n)k= 0.
Acknowledgmen s
The au ho s hank T. Bha acha yya and N. Nikolski and D. Schillo o hei use ul e-
ma ks, and A. Bonilla o his ad ice conce ning Theo em 7.11. The i s au ho has been
Ope a o Inequali ies, Func ional Models and E godici y 41
pa ly suppo ed by P ojec MTM2016-77710-P, DGI-FEDER, o he MCYTS, P ojec E26-
17R, D.G. A ag´on, and P ojec o Young Resea che s, Fundaci´on Ibe caja and Uni e sidad de
Za agoza, Spain. The second au ho has been pa ially suppo ed by La Caixa-Se e o Ochoa
g an (ICMAT Se e o Ochoa p ojec SEV-2011-0087, MINECO). Bo h second and hi d au-
ho s acknowledge pa ial suppo by Spanish Minis y o Science, Inno a ion and Uni e si ies
(g an no. PGC2018-099124-B-I00) and he ICMAT Se e o Ochoa p ojec SEV-2015-0554 o
he Spanish Minis y o Economy and Compe i i eness o Spain and he Eu opean Regional
De elopmen Fund, h ough he “Se e o Ochoa P og amme o Cen es o Excellence in
R&D”. Bo h second and hi d au ho s also acknowledge inancial suppo om he Span-
ish Minis y o Science and Inno a ion, h ough he “Se e o Ochoa P og amme o Cen es
o Excellence in R&D” (SEV-2015-0554) and om he Spanish Na ional Resea ch Council,
h ough he “Ayuda ex ao dina ia a Cen os de Excelencia Se e o Ochoa” (20205CEX001).
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