CONTRACTIONS ON `
p- SPACES by
Sophie Grivaux & Étienne Matheron
Abstract. — We study some local spectral properties of contraction operators on `p, 1 ă p ă 8 from a Baire category point of view, with respect to the Strong˚ Operator Topology. In particular, we show that a typical contraction on`p has Dunford’s Property (C) but neither Bishop’s Propertypβqnor the Decomposition Propertypδq, and is completely indecomposable. We also obtain some results regarding the asymptotic behavior of orbits of typical contractions on`p.
To the memory of Jörg Eschmeier
1. Introduction
In this note, we shall be interested in the typicality, in the sense of Baire Category, of some natural properties of Banach space operators pertaining tolocal spectral theory.
In the whole paper, the letterX will denote an infinite-dimensional complex (separable) Banach space with separable dual. Most of the time, X will be in fact `p :“ `ppZ`q for some 1 ă p ă 8. We denote by BpXq the algebra of bounded operators on X, and by B1pXq the set of contraction operators onX:
B1pXq “ T PBpXq; ||T|| ď1( .
There are several natural topologies onB1pXq which may turn it into a Polish (i.e. sepa- rable and completely metrizable) space, thus allowing for a study of “large” sets of contrac- tions onXin the sense of Baire Category. Some well-known such topologies are the Weak Operator Topology (WOT), the Strong Operator Topology (SOT), and the Strong˚ Operator Topology (SOT˚). In this note, we shall almost exclusively be concerned with the topology SOT˚, which is defined as follows: a net pTiq ĎBpXq converges toT PBpXq with respect to SOT˚ if and only if Tix tends to T x in norm for every xPX and Ti˚x˚ tends to T˚x˚ in norm for every x˚ PX˚. In other words, Ti Ñ T for SOT˚ if and only if Ti Ñ T and Ti˚ Ñ T˚ for SOT. Under our assumptions on X, it is well-known that B1pXq is a Polish
2000 Mathematics Subject Classification. — 47A15, 47A16, 54E52.
Key words and phrases. — Polish topologies, `p- spaces, typical properties of operators, local spec- trum, orbits of operators.
This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).
space when endowed with SOT˚. In addition to its Polishness, one pleasant feature of the topology SOT˚ is that when X is reflexive, the map T ÞÑ T˚ is a homeomorphism from B1pXq ontoB1pX˚q.
In this paper, the word typical will always refer to the topology SOT˚, unless the con- trary is explicitly mentioned. Thus, we will say that a given property (P) of opera- tors T P B1pXq is typical, or that a typical T P B1pXq has Property (P), if the set
T P B1pXq ; T has property (P)(
is comeager in pB1pXq,SOT˚q, i.e. contains a dense Gδ subset ofpB1pXq,SOT˚q. In other words, a property (P) of contractions onX is typical if the set of all T PB1pXq enjoying it is very large in the sense of Baire Category.
Typical properties ofHilbert space contractions with respect to the topologiesWOT,SOT and SOT˚ are investigated in depth in [12] and [13]. The main conclusions drawn from these works are that the generic situation is rather well understood in the case of WOT (a typical contraction is unitary); that it is “trivial” in the case ofSOT(a typical contraction is unitarily equivalent to the backward shift with infinite multiplicity, so there is generically just one Hilbert space contraction); and that things are more complicated for the topology SOT˚. This line of thought is pursued in [18] and [19]. The setting of [18] is mostly Hilbertian, although many of the proofs work as well on `p- spaces; and for that reason, emphasis is mostly put on the topology SOT˚. In [19], the setting is explicitly that of
`p- spaces, and the typicality of certain properties of operators pertaining to the study of the Invariant Subspace Problem is considered for the topologies SOT andSOT˚.
This note may be viewed as anaddendumto [19]. Our aim will be to determine whether certain properties of operators related to local spectral theory are typical in B1p`pq, 1 ă p ă 8. As will be explained below, our study was in part motivated by results from [8], [14], and [15] showing the existence of non-trivial invariant subspaces under suitable spectral assumptions.
Throughout the paper, we denote by D the open unit disk in C, and by D the closed unit disk. Also, we denote byσpTq the spectrum of an operatorT PBpXq. Thus, we have σpTq Ď Dfor every T PB1pXq.
2. Background from local spectral theory
In order to be as self-contained as possible, we recall in this section some basic definitions as well as some important results from local spectral theory. We follow the terminology and notations of [20].
2a. Decomposable operators. — A large part of local spectral theory is motivated by the study of so-called decomposable operators. An operator T PBpXq is decomposable if for any covering pU1, U2q of C by two open subsets, there exist two closed T- invariant subspaces E1 and E2 such that σpT|E1q Ď U1, σpT|E2q Ď U2, and X “ E1 `E2. For instance, it follows from the Riesz Decomposition Theorem that any operator with totally disconnected spectrum is decomposable. Also, any surjective isometry is decomposable (see [20, Proposition 1.6.7]). Two related notions introduced and studied in [4] are strong indecomposability, and complete indecomposability: an operator T PBpXq is strongly in- decomposable if its spectrum σpTq is not a singleton and is included in the spectrum of the restriction ofT to any of its (closed) non-zero invariant subspaces; andT is completely indecomposable if both T andT˚ are strongly indecomposable.
2b. Local spectra. — Thelocal resolvent setof an operatorT PBpXqat a vectorxPX is the open subset ρxpTq of Cdefined in the following way: a complex number λ belongs to ρxpTqif there exists a holomorphic function f :V ÑX defined on some neighborhood V of λ such that pT ´zqfpzq “ x for every z P V. Note that the usual resolvent set ρpTq :“ CzσpTq is contained in ρxpTq: for any λ P ρpTq, we may take V :“ ρpTq and we must take fpzq :“ pT ´zq´1x for every z P ρpTq. The local spectrum of T at x is the compact set σxpTq “ CzρxpTq, which is contained in σpTq. Obviously σxpTq “ H if x“0; but it may happen thatσxpTq “ Hfor some non-zero vectorx.
2c. Eigenvector fields, and SVEP. — LetT PBpXq. A holomorphicT- eigenvector field is a holomorphic function E :V Ñ X defined on some open set V ĎC, such that pT ´zqEpzq “ 0 for every z P V. (The terminology is slightly inaccurate since Epzq is allowed to be 0.) The operatorT has theSingle Valued Extension Property (SVEP) if for any open set V ĎC, the only holomorphic T- eigenvector field E :V Ñ X is E “ 0. It is rather clear that if T has SVEP then, for any x P X, there is a unique holomorphic functionf :ρxpTq ÑXsolving the equationpT´λqfpλq ”xonρxpTq, which is called the local resolvent function forT atx. Moreover, it can be shown thatT has SVEP if and only ifσxpTq ‰ Hfor everyx‰0(see [20, Proposition 1.2.16]). Obvious examples of operators with SVEP are all operators whose set of eigenvalues has empty interior. In particular, any isometry has SVEP. On the other hand, the usual backward shiftB on`p lacks SVEP in a strong way, since it has a spanning holomorphic eigenvector field defined in the unit diskD,i.e. a holomorphic eigenvector fieldE:DÑ`p such that spantEpzq; zPDu “`p; namely, Epzq :“ ř8
j“0zjej, where pejqjě0 is the canonical basis of `p. More generally, by a classical result of Finch [16] (see also [20, Proposition 1.2.10]), any surjective but non-invertible operator lacks SVEP. At the other extreme, the usual forward shift S on
`p has “maximal local spectra”, i.e. it satisfies σxpSq “ D for every x ‰ 0. In fact, the Remark after [28, Theorem 1.5] shows the following: if T PBpXq is such that its Banach space adjoint T˚ P BpX˚q admits a spanning holomorphic eigenvector field E defined on some connected open setΩĎC, thenσxpTq contains Ωfor every x‰0.
2d. Local spectral radii. — IfT PBpXq and x PX, the local spectral radius of T at x is the number
rxpTq:“lim sup
nÑ8
||Tnx||1{n.
It is not difficult to see that rxpTq ě supt|λ| ; λP σxpTqu; and a littler harder to show that if T has SVEP, then rxpTq “ supt|λ|; λPσxpTqufor every x‰0. This is thelocal spectral radius formula, see [20, Proposition 3.3.13].
2e. Spectral subspaces. — LetT PBpXq. For any subsetF of C, let XTpFq:“ txPX; σxpTq ĎFu.
With this notation, we see from2c that the operatorT has SVEP if and only ifXTpHq “ t0u; and it turns out that this holds if and only ifXTpHq is closed inX. Also, it can be shown thatXTpFqis always aT- hyperinvariant linear subspace ofX. See [20, Proposition 1.2.16]. Moreover, if T has SVEP and ifF is a closed set such thatXTpFqis closed, then σ`
T|XTpFq˘
ĎσpTq XF (see [20, Proposition 1.2.20]).
The subspaces XTpFq may be called local spectral subspaces for T, since x P XTpFq means that for anyλPCzF, there is a holomorphic solution of the equationpT´zqfpzq ”x defined in a neighborhood of λ. When F is closed, one can also define a global spectral
subspace, denoted byXTpFq: a vectorxbelongs toXTpFqif there is a holomorphic solution of pT ´zqfpzq ”x globally defined on CzF. Obviously XTpFq Ď XTpFq; and if T has SVEP then XTpFq “XTpFq for any closed setF ĎC.
2f. Property (C). — An operator T P BpXq is said to have Dunford’s Property pCq if XTpFq is closed in X for every closed set F Ď C. By what has been said just after the definition of XTpFq, Property (C) implies SVEP. Moreover, it can be shown that decomposable operators have Property (C), and in fact that an operator T P BpXq is decomposable if and only if (i)T has Property (C) and (ii) for every open coveringpU1, U2q of C, it holds thatX “XTpU1q `XTpU2q. See [20, Theorem 1.2.23].
2g. Propertypδq. — The global spectral subspacesXTpFqare involved in the definition of the important Decomposition Property pδq: an operator T P BpXq has Property pδq if for any open coveringpU1, U2q ofC, we haveXTpU1q `XTpU2q “X. Every decomposable operator has property pδq; and more precisely, an operator is decomposable if and only if it has both properties (C) and pδq. See [20, Proposition 1.2.29].
2h. Property pβq. — An operatorT PBpXqhasBishop’s Property pβq if the following holds true: for any open setV ĎCand any sequence of holomorphic functionsφn:V ÑX, if pT ´λqφnpλq Ñ 0 uniformly on compact subsets of V, then φnpλq Ñ 0 uniformly on compact sets. Obviously, Property pβq implies SVEP; and in fact, Property pβq implies Property (C) (see [20, Proposition 1.2.19]). That the converse is not true is a classical result due to Miller and Miller [22] (see also [20, Theorem 1.6.16]). On the other hand, decomposable operators have Propertypβq (see [20, Theorem 1.2.7]); and the link between decomposability and Propertypβqis actually much deeper: by a famous result of Albrecht and Eschmeier [1] (see also [20, Theorem 2.4.4]), an operator T PBpXq has Property pβq if and only if it is similar to the restriction of some decomposable operator to one of its invariant subspaces.
Since property pβq implies property (C), and since T is decomposable if and only if it has both properties (C) and pδq, it follows that T is decomposable if and only if it has both properties pβq and pδq.
Also, since invertible isometries are decomposable and since every (not necessarily in- vertible) isometry is the restriction of some invertible isometry on a larger Banach space by a classical result of Douglas [11] (see also [20, Proposition 1.6.6]), we see that every isometry has Property pβq, and hence Property (C). (Incidentally, this shows that an op- erator T may have Property (C) whereasT˚ does not and even lacks SVEP: consider the usual forward shiftS on`p, for which XTpFq “X if F Ě DandXTpFq “ t0uotherwise.) Finally, there is a quite remarkable duality between properties pβq and pδq, due to Albrecht-Eschmeier [1] (see also [20, Theorem 2.5.18]): an operator T has Property pβq if and only if T˚ has Property pδq, and T has Property pδq if and only if T˚ has Property pβq. In particular, T is decomposable if and only ifT˚ is decomposable.
2i. Invariant subspaces. — Local spectral theory provides sophisticated and extremely efficient tools for extending to the Banach space setting some Hilbertian invariant subspaces results concerning operators with thick spectrum. Recall that a compact subset K of C is said to be thick if there exists a non-empty bounded open set U Ď C in which K is dominating, which means that supzPU|fpzq| “ supzPUXK|fpzq| for every bounded holomorphic function f on U.
A well-known result of Brown [8] states that any hyponormal operator with thick spec- trum on a Hilbert space admits a non-trivial invariant closed subspace. This result admits several extensions to the Banach space setting, perhaps the ultimate one being due to Eschmeier and Prunaru [15]. It involves the notion oflocalizable spectrum of an operator.
IfT PBpXq, the localizable spectrumσlocpTqof T is the closed subset of Cdefined as σlocpTq:“ λPC; XTpVq ‰ t0u for any neighborhood V ofλ(
.
It is not difficult to see, using Liouville’s Theorem, that σlocpTq ĎσpTq; and it is shown in [23, Theorem 2] that equality holds if T has Property pδq. Regarding the existence of invariant subspaces, the main result of [15] runs as follows: If T P BpXq is such that eitherσlocpTqorσlocpT˚qis thick, thenT has a non-trivial closed invariant subspace. Since σlocpTq “ σpTq if T has Property pδq, it follows that any operator with Property pδq or Property pβq and with thick spectrum has a non-trivial invariant subspace ([14], see also [20, Theorem 2.6.12]). As any hyponormal operator on a Hilbert space has propertypβq, this extends indeed the Hilbertian result from [8].
This result from [15] was the initial motivation for this work. Indeed, [19] left open the following basic question: does a typical operator on`p for1ăp‰2ă 8has a non-trivial invariant subspace? As a typical operatorT on `p satisfiesσpTq “ D by [19, Proposition 7.2], and hence has thick spectrum, it is natural, in view of [15] to wonder whether a typical T P B1p`pq has Property pδq or Property pβq, or has thick localizable spectrum.
We will unfortunately see that none of this happens. Thus, it seems that methods from local spectral theory are not likely to help much for showing that a typical contraction on `p, p‰ 2, has a non-trivial invariant subspace. For p “2, it follows from the Brown- Chevreau-Pearcy Theorem [9] and from the fact that a typical T is such thatσpTq “ D, that a typicalT PB1p`2q does have a non-trivial invariant subspace.
2j. Results. — The rest of the paper is organized as follows. In Section3, we show that typical operators on`phave maximal local spectra, and we draw several consequences from this. Most notably, we show that a typical T PB1pXq has Property (C) but has neither Property pδq nor Property pβq, is completely indecomposable and has empty localizable spectrum. In Section 4, we present some results regarding the asymptotic behavior of orbits of typical operators on`p. The general idea is that, for a typicalT PB1p`pq, no non- zero orbit can be “too small”, yet most orbits are “partly small”. One consequence of these results is that for anyM ą1, a typicalT PBMp`2qsatisfies a strong form ofdistributional chaos. In Section5, we deviate a little bit from our main topic: we state a simple condition ensuring that an operator lacks SVEP in a strong way, and we use this to give examples of operators admitting a mixing Gaussian measure with full support. Finally, we gather in Section6 some open questions and remarks.
2k. Duality. — The following fact will be used several times in the paper. Each time we will write “by duality”, this will refer implicitly to this fact.
Fact 2.1. — Let (P) be a property of Banach space operators. If, for any 1 ă p ă 8, a typical T P B1p`pq satisfies (P), then it is also true that for any 1 ă p ă 8, a typical T PB1p`pq is such thatT˚ satisfies (P).
Proof. — This is clear since the mapT ÞÑT˚is a homeomorphism fromB1p`pqontoB1p`qq, where q is the conjugate exponent of p. Of course, this argument works only because we
are using the topology SOT˚ (it would break down for the topologies WOT and SOT, for instance).
3. Operators with maximal local spectra
It is proved in [19, Proposition 3.9] that a typical T P pB1p`pq,SOTq,1ăpă 8 is such that σpTq “ D; and the proof given there works for the topology SOT˚ as well. Thus, typical contractions on`p have “maximal” spectrum. The following result strengthens this statement and shows that typical contractions on `p have maximal local spectra. This is to be compared with [28, Theorem 1.5], where it is shown that for any Banach space X and anyT PBpXq with SVEP, the set txPX; σxpTq “σpTquis residual inX.
Theorem 3.1. — Let X “`p, 1ăpă 8. A typical T PB1pXq is such that σxpTq “ D for every x ‰ 0. Equivalently, a typical T P B1pXq has the following property: for any closed, T- invariant subspace Z ĎX, it holds that σx`
T|Z˘
“ Dfor every x‰0 in Z. The equivalence of the two statements is easy to check. Indeed, it follows directly from the definition of the local spectra that ifT PBpXqand ifZis a closedT- invariant subspace of X, thenσx
`T|Z˘
ĚσxpTq for everyxPZ. Hence, if}T} ď1 and σxpTq “ Dfor some xPZ, thena fortiori σx
`T|Z˘
“ D.
Before giving the proof of Theorem3.1, we present some consequences.
Corollary 3.2. — Let X “ `p, 1 ă p ă 8. A typical T P B1pXq has the following properties : T´λis one-to-one with dense range for everyλPC, and T´λdoes not have closed range for anyλP D. In particular, the essential spectrum of a typical T PB1pXq is equal to D.
Proof. — These results are known; see [13, Proposition 6.3], [18, Proposition 2.24 and Remark 2.30] and [19, Proposition 7.1]. However, they can also be deduced from Theorem 3.1. Indeed, sinceσxpTq Ď tλuif T x“λx, it follows from Theorem3.1that a typical T P B1pXq has no eigenvalue. By duality (see Fact2.1), this implies that a typical T PB1pXq is such thatT´λhas dense range for everyλPC. Moreover, it also follows from Theorem 3.1 that a typical T PB1pXq is such thatσpTq “ D; and so for a typical T PB1pXq, the operator T´λcannot have closed range for any λP D.
Corollary 3.3. — Let X “`p, 1 ăpă 8. A typical T PB1pXq is completely indecom- posable.
Proof. — Since σx` T|Z˘
Ďσ` T|Z˘
for any T- invariant subspace Z ĎX and every xPZ, it is clear from Theorem 3.1 that a typical T P B1pXq is strongly indecomposable. By duality, it follows that a typicalT PB1pXq is such that T˚ is strongly indecomposable as well.
As another consequence of Theorem 3.1, we are able to determine whether the local spectral properties presented in Section2 hold, or not, for a typical T PB1p`pq:
Corollary 3.4. — Let X “ `p, 1 ă p ă 8. A typical T P B1pXq has the following properties.
(i) rxpTq “supt|λ|; λPσxpTqu “1 for every x‰0.
(ii) For any F ĎC, either XTpFq “ t0u or XTpFq “X; more precisely: XTpFq “X if F contains D, andXTpFq “ t0u otherwise.
(iii) T andT˚ have Property (C), and hence SVEP.
(iv) T andT˚ have neither Property pβq, nor Property pδq.
(v) T has empty localizable spectrum: σlocpTq “ H.
Proof. — Note that ifσxpTq “ Dfor allx‰0, then in particularT has SVEP and hence rxpTq “sup |λ|; λPσxpTq(
for every x‰0. This proves (i).
Part (ii) is clear.
From (ii), we deduce that a typical T P B1pXq has Property (C) and does not have Property pδq. Indeed, this is clear for (C). As for pδq, assume thatT satisfies (ii). Setting U1 :“ Dp0,2{3q and U2 :“CzDp0,1{3q, we get an open covering pU1, U2q of C such that neither U1 nor U2 contains D; so we have XTpU1q “ t0u “XTpU2q, which shows that T does not have Property pδq.
Thus, for any 1 ă p ă 8, a typical T P Bp`pq has property (C) and does not have Property pδq. By duality, and since an operator T has Property pβq if and only if T˚ has Propertypδq, this proves (iii) and (iv). Part (v) follows immediately from (ii).
The next corollary shows that the algebraic core and the analytic core of a typical T P B1pXq are far from being equal. Recall that if T P BpXq, the algebraic core of T, denoted by CpTq, is the set of all x PX admitting a backward T- orbit, i.e. a sequence pxnqně0 such thatx0 “x and T xn“xn´1 for all ně1; and that the analytic core ofT, denoted byKpTq, is the set of allxPX admitting a “controlled” backward T- orbit,i.e. a backwardT- orbitpxnqsuch that}xn} ďδn for some constantδ and allně0. Obviously, KpTq ĎCpTq ĎŞ
nPNTnpXq.
Corollary 3.5. — LetX “`p,1ăpă 8. A typicalT PB1pXq is such thatKpTq “ t0u whereas CpTq is dense in X.
Proof. — It is known that for any T PBpXq, we have KpTq “ tx PX; 0 R σxpTqu, see e.g. [21, Proposition 1.3]. So, by the previous corollary, a typical T PB1pXq is such that KpTq “ t0u. On the other hand, by Corollary 3.2, a typicalT PB1pXqis one-to-one with dense range. Now, if T is one-to-one then it is clear that CpTq “Ş
nPNTnpXq; and if T has dense range then, by the Bourbaki-Mittag-Leffler Theorem (see e.g. [26]),Ş
nPNTnpXq is dense in X. This concludes the proof.
Recall that ifT PBpXq, then aspectral maximal subspace for T is a closedT- invariant subspaceZ ĎXwith the following property: for any closed,T- invariant subspaceZ1 ĎX such that σ`
T|Z1˘ Ď σ`
T|Z˘
, it holds that Z1 Ď Z. (See [10] for more on this notion.) Obviously, t0u andX are spectral maximal subspaces forT.
Corollary 3.6. — LetX “`p,1ăpă 8. A typicalT PB1pXqis such thatσ` T|Z˘
“ D for every closed T- invariant subspace Z ‰ t0u. In particular, the only spectral maximal subspaces of a typicalT PB1pXq are t0u and X.
Proof. — The first part of the corollary follows immediately from Theorem 3.1 since σ`
T|Z˘
Ě σx` T|Z˘
for any T- invariant subspace Z and every x P Z. For the second part, apply the definition with a maximal spectral subspaceZ ‰ t0uand Z1 :“X.
WhenX “`2, we have at our disposal a very strong result for proving the existence of invariant subspaces, namely the Brown-Chevreau-Pearcy Theorem [9]. It states that any contraction on a Hilbert space whose spectrum contains the unit circleThas a non-trivial invariant subspace. Combining it with Corollary3.6 above, we obtain:
Corollary 3.7. — A typical T P B1p`2q has no minimal invariant subspace except t0u:
for any closed T- invariant subspace Z ‰ t0u, the operatorT|Z has a non-trivial invariant subspace.
We now move over to the proof of Theorem3.1. For any Banach space X, we introduce the set
G:“ T PB1pXq; @x‰0, the set tµPD; xRRanpT ´µquis dense in D( . We start with two general lemmas:
Lemma 3.8. — Assume that X is reflexive. Then, G is a Gδ subset of pB1pXq,SOT˚q. This relies on the following fact.
Fact 3.9. — For any µPCand any closed ball BĎX, the set Cµ,B :“ tpT, xq PB1pXq ˆX; xP pT ´µqpBqu is closed inpB1pXq,SOT˚q ˆ pX, wq.
Proof of Fact 3.9. — The setCµ,B is the projection alongB of
Cµ,B :“ pT, y, xq PB1pXq ˆBˆX; pT´µqy“x( .
Therefore, since the ball B is weakly compact, it is enough to check that Cµ,B is closed inpB1pXq,SOT˚q ˆ pB, wq ˆ pX, wq. Now, the map pT, yq ÞÑ pT ´µqy is continuous from pB1pXq,SOT˚q ˆ pB, wq intopX, wq, because we have xx˚,pT ´µqyy “ xpT˚´µqx˚, yyfor any x˚ P X˚ and the map py˚, yq ÞÑ xy˚, yy is continuous on pK,} ¨ }q ˆ pB, wq for any bounded set KĎX˚. SoCµ,B is indeed closed in pB1pXq,SOT˚q ˆ pX, wq ˆ pB, wq.
Proof of Lemma 3.8. — For anyN PN, let us denote byBN the closed ballBp0, Nq ĎX.
Let alsopViqiPN be a countable basis of (non-empty) open sets forD. Then, the following equivalence holds true for anyT PB1pXq:
T RG ðñ DxPXzt0u DN PNDiPN :
´
@µPVi : pT, xq PCµ,BN¯ .
Indeed, given x P Xzt0u, each set FN :“ tµ P D;x P pT ´µqBNu is easily seen to be closed inD, so thatΩN :“DzFN “ tµPD;xR pT ´µqBNu is open inD. Since
č
Ně1
ΩN “ tµPD;xPRanpT´µqu, the Baire Category Theorem implies thatŞ
Ně1ΩN is dense inDif and only ifΩN is dense inDfor each N ě1. Hence
T RG ðñ DxPXzt0u DN PN; tµPD;xR pT ´µqBNuis not dense in D;
from which it follows that
T RG ðñ DxPXzt0u DN PNDiPN :
´
@µPVi : pT, xq PCµ,BN¯ .
By Fact3.9, the condition under brackets defines a closed subset ofpB1pXq,SOT˚q ˆ pX, wq. SinceXzt0uis a Kσ subset of pX, wq, it follows thatB1pXqzG is Fσ inpB1pXq,SOT˚q.
Our second lemma provides a large class of operators belonging to the set G.
Lemma 3.10. — Let T PB1pXq. ThenT belongs to G provided the following holds true:
for every open setV ‰ H in D, spanw˚
˜ ď
µPV
kerpT´µq˚
¸
“X˚.
Proof. — This is essentially obvious from the definition of G. Indeed, an operator T P B1pXq does not belong toG if and only if one can find an open setV ‰ HinDsuch that Ş
µPV RanpT ´µq ‰ t0u; and if this holds then spanw˚
´Ť
µPV kerpT ´µq˚
¯
‰X˚ since spanw˚
´Ť
µPV kerpT ´µq˚
¯ Ď
´Ş
µPV RanpT ´µq
¯K
.
Remark 3.11. — The assumption of Lemma 3.10is satisfied as soon as the operatorT˚ admits aw˚- spanning family of holomorphic eigenvector fields defined on the unit diskD, i.e. there exists a familypEiqiPI of holomorphic eigenvector fields forT˚ defined onDsuch that spanw˚tEipλq; i PI, λ PDu “ X˚. This follows from the Hahn-Banach Theorem and the identity principle for holomorphic functions.
We are now ready for the proof of Theorem3.1.
Proof of Theorem 3.1. — By the definition ofG, it is rather clear that if T PG andx‰0, thenσxpTqcontainsD, and henceσxpTq “ DsinceσxpTqis a closed subset ofCcontained in D. Indeed, letλPD. SinceT PG, we see that for every open neighborhoodV ofλthere existsµPV such thatxRRanpT´µq. Hence there cannot existany function f :V ÑX such that pT ´zqfpzq ”x on V, and thusλRρxpTq. So, by Lemma 3.8, it is enough to show that the setG is SOT˚- dense inB1pXq.
Let us denote by pejqjě0 the canonical basis of X “ `p; and for each N P Z`, set EN :“ re0, . . . , eNs. To prove theSOT˚-denseness ofG, we show that any finite-dimensional operator A P BpENq such that }A} ă 1 (viewed as an operator on X “ `p) can be approximated in the topologySOT˚ by operators belonging toG.
Let M ąN be a large integer, let δ be a small positive number, and let T PBpXq be defined as follows:
T ej :“
"
Aej`δej`M`1 if 0ďjďM , ej`M`1 if jąM .
Since }A} ă 1, we have }T} ă 1 if δ is small enough. Moreover, if M is large enough and δ small enough, then T is as close as we wish to A in the topology SOT˚. So, by Remark 3.11, it is now enough to show that the operator T˚ admits a spanning family of holomorphic eigenvector fields defined on D. But this follows from the proof of [18, Proposition 2.10]: indeed, with the notations of [18, Proposition 2.10] and identifying A
withPMAPM, we haveT˚ “Bωωω,A, where the weight sequenceωωω“ pδ, . . . , δ,1,1,1. . .q is such thatlimkÑ8ωkM`l¨ ¨ ¨ωM`lωl“1ą }A}for every 0ďlďM.
4. Asymptotic behavior of orbits
The results in this section are concerned with the study of some properties of orbits of typical contractions on `p- spaces. Recall that if T P BpXq, the orbit of a vector x P X under the action of T is the set tTnx ; n P Z`u. The dynamics of typical operators T P pB1p`2q,SOT˚qare already studied in some detail in [18], where emphasis is put mostly on properties related to hypercyclicity. Here, we are interested in asymptotic properties of orbits of typical operators. Our results are strongly motivated by [24, Chapter V], where several striking results valid for every operatorT PBpXq are obtained.
4a. Not too small orbits. — We begin this section with a somewhat abstract statement which will be later on applied to several concrete situations.
Proposition 4.1. — Let X “ `p, 1 ă p ă 8. Let also G be a Gδ subset of p0,8qN. Assume that G contains all sequences ω “ pωnqnPN P p0,8qN such that infnPNωn ą 0, and that G is upward closed for the product ordering of p0,8qN, i.e. (ωPG and ωďω1) implies pω1 P Gq. Then, a typical T P B1pXq is such that for every x ‰ 0, the sequence
`}Tnx}˘
nPN belongs to G.
The proof of Proposition4.1relies on the next two lemmas.
Lemma 4.2. — Let X “ `p, 1 ă pă 8. The set of all T P B1pXq such that @x ‰ 0 : infnPN }Tnx} ą0 is SOT˚- dense in B1pXq.
Proof. — Let us denote byDthe considered set of operators. As in the proof of Theorem 3.1, given a finite-dimensionalAPBpENq, we consider the operator T defined by
T ej :“
"
Aej`δej`M`1 if 0ďjďM , ej`M`1 if jąM ,
whereM ąN and δ ą0. This operator T belongs to B1pXq and isSOT˚- close toA if M is large enough and ifδ is small enough. So it is enough to show thatT belongs toD,i.e.
thatinfnPN}Tnx} ą0for every x‰0.
LetxPXzt0u. If xe˚j
0, xy ‰0 for somej0 ąM, then, since xe˚j
0`npM`1q, Tnxy “ xe˚j
0, xy for every nPNby the definition of T, we see that infnPN }Tnx} ą0. Ifxe˚j, xy “0 for all j ąM, i.e. if x PEM, thenx1 :“ T x satisfiesxe˚j0, x1y ‰ 0 for some M ăj0 ď 2M `1 by the definition of T; so we have infnPN }Tnx1} ą 0, i.e. infně2 }Tnx} ą 0, and hence infnPN }Tnx} ą0 since T x“x1 ‰0. This concludes the proof of the lemma.
Lemma 4.3. — If HĎ p0,1sN is Gδ and upward closed, then Hcan be written as H“ Ş
kPNOk, where the sets Ok are open in p0,1sN and also upward closed.
Proof. — WriteH“Ş
kPNUk for some open setsUkĎ p0,1sN. For each kPN, set Ok:“ ωP p0,1sN; @ω1 ěω : ω1PUk(
.
The sets Ok are obviously upward closed, and since H is upward closed, we have H “ Ş
kPNOk. So we just need to check that each Ok is open in p0,1sN. Now, since r0,1sN is compact and p0,1sN is upward closed in r0,1sN, it is easily checked that p0,1sNzOk is closed inp0,1sN.
We are now ready for the proof of Proposition4.1.
Proof of Proposition 4.1. — By homogeneity and upward closedness ofG, it is enough to show that a typicalT PB1pXqis such that the sequencep}Tnx}qnPNbelongs toGfor every x ‰0 with }x} ď1. Moreover, if T PB1pXq and }x} ď1, then }Tnx} ď1 for allnPN. So, setting H:“ GX p0,1sN and denoting by BX the closed unit ball of X, we have to show that the set
H:“
!
T PB1pXq; @xPBXzt0u : `
}Tnx}˘
nPNPH )
is comeager in B1pXq.
By Lemma 4.2 and by assumption on G, we already know that the set H is dense in B1pXq. We show that His alsoGδ.
By Lemma 4.3, we may write H as H “ Ş
kPNOk, where the sets Ok are open and upward closed in p0,1sN. So it is enough to check that ifOĎ p0,1sN is open and upward closed, then the set
O:“
!
T PB1pXq; @xPBXzt0u : `
}Tnx}˘
nPNPO )
isGδ inB1pXq.
Let us denote by Σ the set of all finite sequences from p0,1s; and for any sequence σ“ ps1, . . . , srq PΣ, let us set
Vσ :“ tω P p0,1sN; ωnąsnfor n“1, . . . , ru.
Since Ois upward closed and open inp0,1sN, one can write O“ ď
σPI
Vσ,
for some set I Ď Σ. Indeed, O is a union of basic open sets of the form Şr
n“1Wn,sn,tn, where we setWn,s,t :“ tω; săωnătufor anynPNand any real numberss, tsuch that 0ăsď1 andsăt. Now, if t1, . . . , tr are given such that tnąsn for every n“1, . . . , r, then, for every ω P Vσ, one can find ω1 P Şr
n“1Wn,sn,tn such that ω ě ω1. Hence, by upward closedness, we have that ifŞr
n“1Wn,sn,tn ĎO, then in factVσ ĎO; which proves the claim.
So, for any T PB1pXq, we see that T RO ðñ DxPBXzt0u
@σ“ ps1, . . . , snq PI :
´
DnPJ1, rK : }Tnx} ďsn
¯
Now, observe that for any bounded set B ĎX, the map pA, xq ÞÑ }Ax} is lower semi- continuous on pB1pXq,SOT˚q ˆ pB, wq, because the map pA, xq ÞÑ Ax is continuous from pB1pXq,SOT˚q ˆ pB, wqintopX, wqand the norm is lower semi-continuous onpX, wq. Since BXzt0u isKσ inpX, wq, it follows easily that B1pXqzO isFσ inB1pXq.
A first consequence of Proposition 4.1is that all non-zero orbits of a typicalT PB1p`pq are “not too small”.
Corollary 4.4. — Let X“`p,1ăpă 8.
(i) Given a sequence of positive real numberspanq such thatanÑ0, a typical T PB1pXq is such that, for everyx‰0, one has}Tnx} ąan for infinitely many nPN.
(ii) Given a sequence pφnq of increasing continuous functions from R` into itself such that ř8
n“0φnpεq “ 8 for every εą0, a typical T PB1pXq is such that
8
ÿ
n“0
φn
`}Tnx}˘
“ 8 for every x‰0.
Proof. — For (i), take G1 :“ ω P p0,8qN; ωn ą an for infinitely manyn(
, which is obviouslyGδand upward closed. For (ii), considerG2 :“ ωP p0,8qN; ř8
n“0φpωnq “ 8( , which is Gδ because ω PG2 ðñ @M PNDN : řN
n“0φnpωnq ąM, and upward closed because the functionsφn are increasing.
Remark 4.5. — Taking e.g.an:“ 1n in (i), one gets another proof that a typical operator T PB1pXqis such that rxpTq “1 for every x‰0.
4b. Nilpotent operators. — We now turn to a result that, sadly enough, we can prove only for X“`2. Let us denote by NpXq the set of all nilpotent operatorsT PBpXq, and setN1pXq:“NpXq XB1pXq.
Proposition 4.6. — If X “`2, then N1pXq is SOT˚- dense in B1pXq.
Before giving the proof of Proposition 4.6, we explore some of its consequences. First, it implies that a typical T PBp`2q has lots of “partly small” orbits (see Proposition 4.14 and Proposition4.15 below for concrete examples):
Corollary 4.7. — Let X “ `2. Let also G be a Gδ subset of r0,8qN. Assume that G contains all sequences ω “ pωnq P r0,8qN which are eventually 0. Then, a typical T P B1pXq has the following property: for a comeager set of vectors x PX, the sequence
`}Tnx}˘
nPN belongs to G.
Proof. — LetD be a countable dense subset ofX, and define G :“ T PB1pXq; @zPD : p}Tnz}qnPNPG(
.
Since G is Gδ, it is easily checked that G is SOT-Gδ, hence SOT˚-Gδ. Moreover, G contains all nilpotent operators inB1pXqby assumption onG, and henceGisSOT˚- dense inB1pXq by Proposition 4.6. Thus, we see that a typical T PB1pXq is such that the set DT :“ x PX; p}Tnx}qnPNPG(
contains D, and hence is dense in X. But for any fixed T PB1pXq, the setDT is clearly Gδ; so the proof is complete.
Remark 4.8. — By the Kuratowski-Ulam Theorem, it follows from Corollary 4.7 that the set pT, xq P B1pXq ˆX; p}Tnx}qnPN P G(
is comeager in `
B1pXq,SOT˚˘
ˆX, and also in `
B1pXq,SOT˘
ˆX. However this set has no reason for beingGδ. Let us point out another consequence of Proposition4.6.
Corollary 4.9. — Let X “ `2. For any a P D, the set of all T P B1pXq such that σpTq “ tau is SOT˚- dense in B1pXq.
Proof. — Let us denote byϕa the Möbius transformation associated with a, ϕapzq “ a´z
1´¯az¨
Sinceϕais holomorphic in a neighborhood of D, one can define ϕapTqfor anyT PB1pXq. Moreover, the mapT ÞÑϕapTqfromB1pXqinto itself isSOT˚- continuous because the power
series ofϕais absolutely convergent on D; and sinceϕa˝ϕapzq ”zin a neighborhood of D, we have ϕa`
ϕapTq˘
“T for every T PB1pXq. Finally, we also have that ϕapTq P B1pXq for any T P B1pXq, by von Neumann’s inequality. So we may conclude that the map T ÞÑϕapTqis an involutive homeomorphism of B1pXqonto itself. SinceσpTq “ tauif and only ifσ`
ϕapTq˘
“ t0uand since the set S PB1pXq; σpSq “ t0u(
isSOT˚- dense inB1pXq by Proposition4.6, the result follows.
The proof of Proposition4.6relies on the next two lemmas.
Lemma 4.10. — Let T be a bounded operator on a Banach space X. If E : ΩÑX is a holomorphic eigenvector field for T defined on some connected open set ΩĎCcontaining 0, then
span Epλq; λPΩ( Ď
ď
kPN
kerpTkq.
Proof. — Note first that sinceΩis connected, we have span Epλq; λPΩ(
“span Epnqp0q; ně0( .
Indeed, the inclusion Ě is clear; and the other inclusion follows from the Hahn-Banach Theorem and the identity principle for holomorphic functions. Next, differentiating the identityT Epλq ”λEpλq, we get thatT Epnqpλq “λEpnqpλq `nEpn´1qpλq for everyλPΩ and for all ně1. In particular, T Epnqp0q “nEpn´1qp0q for all ně1. Since T Ep0q “0, it follows that Tn`1Epnqp0q “ 0 for all n ě 0. So we see that span Epnqp0q; n ě 0(
Ď Ť
kPNkerpTkq, which concludes the proof of the lemma.
As an immediate corollary of Lemma4.10, we obtain
Corollary 4.11. — If an operator T P BpXq admits a spanning family of holomorphic eigenvector fields pEiqiPI defined on some connected open set Ω Ď C containing 0, i.e.
spantEipλq; iPI, λPΩu “X, then Ť
kPNkerpTkq is dense in X.
Our second lemma is specific to the Hilbertian setting.
Lemma 4.12. — Let X “ `2. If T P B1pXq is such that Ť
kPNkerpTkq is dense in X, then T belongs to the SOT˚- closure of N1pXq.
Proof. — For each k P N, let us denote by Qk P BpXq the orthogonal projection of X onto kerpTkq. Then }Qk} ď1 (we use specifically here the fact that X “ `2), and since the sequence of subspaces `
kerpTkq˘
kPNis increasing with Ť
kPNkerpTkq “X, we see that Qk ÝÝÑSOT I. Moreover, the projections Qk are self-adjoint, so in factQk SOT
˚
ÝÝÝÑI. Hence, if we setTk :“T Qk, thenTkPB1pXq for allkPNandTk SOT
˚
ÝÝÝÑT. Moreover, each operator Tk is nilpotent, in fact Tkk “ 0: indeed, since TpRanpQkqq “ TpkerpTkqq Ď kerpTkq “ RanpQkq, we have QkT Qk “ T Qk, so Tkn “ TnQk for all nP N and hence Tkk “ 0 since RanpQkq “kerpTkq. So we have found a sequencepTkq ĎN1pXqsuch thatTkÝSOTÝÝÑ˚ T.
Proposition 4.6can now be readily deduced from Lemmas 4.10and 4.12.
Proof of Proposition 4.6. — By Lemmas 4.10and 4.12, it is enough to show that the set of allT PB1pXq admitting a spanning family of holomorphic eigenvector fields defined on DisSOT˚- dense inB1pXq. But this follows from the proof of [18, Corollary 2.12]. Indeed, this proof shows that for any M ą 1, the set of all T P BMpXq admitting a spanning family of holomorphic eigenvector fields defined on the open disk Dp0, Mq is SOT˚- dense inBMpXq; so we get the required result by homogeneity.
Remark 4.13. — Here is a completely different and highly non-elementary proof of Pro- position4.6. We prove in fact the following stronger result: A typical T PB1pXq belongs to the norm-closure ofN1pXq.
By a deep result of Apostol-Foias-Voiculescu [2], an operator T PBpXq belongs to the norm-closure of nilpotent operators if and only if it has the following properties:
(i) σpTq andσepTq are connected with0PσepTq.
(ii) indpT´λq “0 for anyλPCsuch thatT´λis a semi-Fredholm operator.
Now, by Corollary3.2, a typicalT PB1pXq certainly satisfies (i), and a typicalT PB1pXq also satisfies (ii) vacuously forλP D. SinceT´λis invertible if|λ| ą1, the result follows.
4c. Power-regular operators. — According to [3], an operatorT PBpXq is said to be power-regular if }Tnx}1{nÑ rxpTq asnÑ 8for everyx PX. It is shown in [3] that any decomposable operator is power-regular. Combining Theorem 3.1 and Corollary 4.7, we obtain the following result.
Proposition 4.14. — Let X “ `2. A typical T P B1pXq is not power-regular. More precisely, a typical T PB1pXq is such that rxpTq “ 1 for all x ‰ 0 yet lim}Tnx}1{n “ 0 for a comeager set of vectors xPX.
Proof. — By Theorem 3.1 (or Remark 4.5), a typical T PB1pXq is such that rxpTq “ 1 for everyx‰0. On the other hand, a typicalT PB1pXq is such thatlim}Tnx}1{n“0for a comeager set of vectorsxPX: this follows from Corollary4.7 applied to the set
G:“ ω“ pωnq P r0,8qN; limωn1{n“0( .
4d. Distributionally chaotic operators. — Let us recall that a vectorxPXis said to bedistributionally irregular for an operatorT PBpXqif there exist two setsA, BĎNboth having upper density equal to1, such that}Tnx} Ñ0asnÑ 8 alongAand }Tnx} Ñ 8 asnÑ 8 alongB. The operator T is said to be densely distributionally chaotic if it has a dense set of distributionally irregular vectors. We refer to [7] for more on these notions.
Note that forT to have any distributionally irregular vector, it is necessary that}T} ą1.
It is shown in [18] that for any M ą 1, a typical T P BMp`2q is densely distributionally chaotic. This result can be slightly improved, as follows:
Proposition 4.15. — Let X “ `2. For any M ą 1, a typical T P BMpXq has the following properties: for every x ‰ 0, there is a set A Ď N with denspAq “ 1 such that }Tnx} Ñ 8 as n Ñ 8 along A; and for a comeager set of vectors x PX, there is a set B ĎN with denspBq “1 such that }Tnx} Ñ0 as nÑ 8 along B.
Proof. — Let us first show that a typicalT PBMpXq is such that for everyx ‰0, there exists a set A ĎN with denspAq “ 1 such that }Tnx} Ñ 8 as nÑ 8 along A. Choose αP p0,1q such thatM αą1, and set
G:“
!
ω“ pωnq P p0,8qN; dens`
tnPN; ωnąαnu˘
“1 )
.
The set G is easily seen to be Gδ in p0,8qN. Moreover, Gis clearly upward closed, and sinceαă1it contains all sequencesωsuch thatinfnωną0. By Proposition4.1, it follows that a typical T P B1pXq is such that for every x ‰ 0, the set tn PN; }Tnx} ą αnu has upper density equal to 1. Hence, by homogeneity, a typical T P BMpXq is such that for everyx ‰0, the setAx,T :“ tnPN; }Tnx} ą pM αqnu has upper density equal to 1; and clearly}Tnx} Ñ 8 asnÑ 8 alongAx,T.
Now, let us show that a typical T PBMpXq is such that for a comeager set of vectors xPX, there exists a setB ĎNwith denspBq “1 such that }Tnx} Ñ0 asnÑ 8 along B. This follows in fact from [18, Proposition 2.40], but the proof we give here is rather different and more elementary. Chooseβ ą0 such that βM ă1. Applying Corollary 4.7 to the set
G:“
!
ω“ pωnq P r0,8qN; dens`
tnPN; ωnăβnu˘
“1 )
, we see that a typical T P B1pXq is such that dens`
tn P N; }Tnx} ă βnu˘
“ 1 for a comeager set of vectors xPX. By homogeneity, it follows that a typicalT PBMpXq has the following property: for a comeager set of vectors xPX, the set
BT ,x :“ tnPN; }Tnx} ă pβMqnu
has upper density equal to 1. Since obviously }Tnx} Ñ 0 as n Ñ 8 along BT,x, this concludes the proof of Proposition 4.15.
Remark 4.16. — A slight modification of the above proof gives the following result. Let pαnq be any sequence of positive real numbers tending to 0, and let pβnq be any sequence of positive real numbers. Then, a typical T P BMpXq has the following properties: for everyx‰0, there is a setAĎNwithdenspAq “1such that}Tnx} ąαnMnfor allnPA;
and for a comeager set of vectorsxPX, there is a set B ĎNwithdenspBq “1such that }Tnx} ăβn for all nPB.
Remark 4.17. — It is not true that given M ą 1, a typical T P BMpXq is such that everyx ‰0 is a distributionally irregular vector for T. Indeed, by [24, Corollary V.37.9], for any T P BpXq, there is a dense set of vectors x P X such that }Tnx}1{n Ñ rpTq as nÑ 8. Since a typicalT PBMpXqis such that its spectral radius is equal toM, it follows that for a typicalT PBMpXq, there is a dense set of vectors xPX such that}Tnx} Ñ 8, and hence a dense set of vectorsx which are not distributionally irregular forT.
5. Generalized kernels and Gaussian mixing As we have seen, sets of the form ker˚pTq :“ Ť
kPNkerpTkq, where T is a bounded operator on a complex separable Banach space X, play a prominent role in the proof of Proposition4.6. The setker˚pTqis usually called thegeneralized kernel of the operatorT. Operators with a 1- dimensional kernel and a dense generalized kernel are called general- ized backward shifts in [17], where it is shown that operators commuting with generalized backward shifts have remarkable dynamical properties. We would like to point out the fol- lowing kind of converse to Corollary4.11, which is very much in the spirit of [17, Theorem
3.6]. The proof is essentially the same as that of [16, Theorem 2], but we give the details for convenience of the reader.
Proposition 5.1. — Let X be a complex separable Banach space, and let T PBpXq. As- sume thatT is onto and that there exists an operatorB with dense generalized kernel such that T B “ BT and kerpBq ĎkerpTq. Then, T admits a spanning family of holomorphic eigenvector fields defined on the disk Dp0, cTq, where cT :“inft}T˚x˚}; }x˚} “1u. Proof. — The quantity cT is positive becauseT is onto; and we know that for anyxPX and anycąc´1T , one can find x1 PX such that}x1} ďc}x}and T x1 “x.
Note also that the assumptions imply that kerpBkq Ď kerpTkq for every k P N. So T itself has a dense generalized kernel.
Let Z :“ ker˚pTq. For any z PZ, let us denote by kz the smallest integer k ě1 such that Tkz“0. By the definition of cT, one can find a sequence pzjqjěkz´1 of vectors of Z with zkz´1 “ z such that T zj “ zj´1 and }zj} ď p1`2´jqc´1T }zj´1} for all j ěkz. Set also zj :“Tkz´1´jz for 0ďjăkz´1, so thatT zj “zj´1 for every ją0. Then, for any λPDp0, cTq, the series ř
jě0λjzj is convergent; so the formula Ezpλq:“ ÿ
jě0
λjzj
defines a holomorphic functionEz :Dp0, cTq ÑX. Note that T z0 “0since z0 “Tkz´1z.
By the choice of the sequencepzjq, it follows thatT Ezpλq “λEzpλqfor everyλPDp0, cTq. So we have defined on Dp0, cTq a family of holomorphic eigenvector fields pEzqzPZ for T. Moreover, this family is spanning. Indeed, if x˚ P X˚ is such that xx˚, Ezpλqy “ 0 for every z PZ and λPDp0, cTq, then xx˚, zy “ xx˚, zkz´1y “ 0 for every z P Z, and hence x˚ “0 since Z is dense in X.
Remark 5.2. — Combining Lemma4.10with the proof of Proposition5.1, we obtain the following result: ifT PBpXqis onto, then there exists a family of holomorphic eigenvector fields pEiqiPI defined on Dp0, cTqsuch thatspantEipλq;iPI, λPDp0, cTqu “ker˚pTq.
We now use Proposition5.1to provide a condition under which an operatorT PBpXqis mixing in the Gaussian sense, which means thatT admits an invariant Gaussian probability measure with full support with respect to which it is a strongly mixing transformation. (We refer to [6, Chapter 5] and [5] for unexplained terminology and for more about the ergodic theory of linear dynamical systems.) This result is essentially proved in [25, Corollary 9], but our formulation is slightly more general.
Corollary 5.3. — LetX be a complex separable Banach space, and letT PBpXq. Assume that T has the form T “fpBq, where B PBpXq is onto with a dense generalized kernel, andf is a function holomorphic on an open setΩĚσpBq, not constant on any connected component ofΩ, and such that|fpz0q| “1 for some z0 PΩXDp0, cBq. Then, the operator T is mixing in the Gaussian sense.
Proof. — LetV ĎΩXDp0, cBqbe a connected open neighborhood ofz0. By assumption, the function f is not constant on V, so fpVq XT is a non-empty open subset of T. In particularfpVq XTis uncountable, soVXf´1pTqis uncountable. On the other hand, the settzPV; f1pzq “0uis countable. So W :“V X tzPV; f1pzq ‰0u is a non-empty open set such that fpWq XT‰ H. By the implicit function Theorem, it follows that one can
find a non-trivial closed arc ΛĎTand a (one-to-one) Lipschitz map φ: ΛÑV such that fpφpλqq “λfor every λPΛ.
By Proposition5.1, the operatorB admits a spanning family of holomorphic eigenvector fields pEiqiPI defined on Dp0, cBq. If we set ĂEipλq :“ Eipφpλqq, we obtain a family of Lipschitz eigenvector fields for T “ fpBq defined on the arc Λ. Moreover, the family pEĂiqiPIis spanning by the Hahn-Banach Theorem and the identity principle for holomorphic functions, because EĂipΛq “EipφpΛqq for all iPI and φpΛq is an infinite compact subset of Dp0, cBq. By [5, Theorem 3.4], it follows that T is mixing in the Gaussian sense.
Example 5.4. — LetXbe a complex separable Banach space, and letBPBpXqbe onto with a dense generalized kernel. For anyλ0 PCsuch that distpλ0,Tq ăcB (in particular, for any λ0 PT) the operatorT :“λ0`B is mixing in the Gaussian sense.
Proof. — Apply Corollary5.3withfpzq:“λ0`z.
Remark 5.5. — If one only assumes thatker˚pBq is dense inX andker˚pBq ĎRanpBq, then one can conclude thatλ0`B istopologically mixing for any λ0 PT(see [6, Corollary 2.3]). But these weaker assumptions do not entail mixing in the Gaussian sense. For example, if B is a compact weighted backward shift on c0 or `p, then λ0`B is not even frequently hypercyclic by [27, Theorem 1.2], since its spectrum is countable.
6. Some further comments and questions
We collect in this last section some natural questions which arise in connection with this work.
To begin with, observe that since the space c0 :“c0pZ`q has separable dual, the ball B1pc0qendowed with the topologySOT˚is a Polish space. So it makes sense to ask whether the results from Sections 3and 4can be extended to typical contractions of c0.
Question 6.1. — Are Theorem3.1 and Proposition4.1still valid for X“c0?
Observe that the proofs of Theorem 3.1 and Proposition 4.1 use in a crucial way the weak compactness of closed balls of`p,1ăpă 8.
We are able to prove theSOT˚- density ofN1pXqinB1pXqonly in the case whereX“`2, and, as a consequence, Corollaries 4.9 and 4.11 as well as Propositions 4.14 and 4.15 are proved only in the Hilbertian setting. It would thus be interesting to be able to answer the following:
Question 6.2. — Let X “ `p,1 ă p ‰ 2 ă 8. Is it true thatN1pXq is SOT˚- dense in B1pXq?
Corollary 4.9and its proof motivate the next question:
Question 6.3. — Let X “`2 (or even X “`p, 1 ăp ă 8). Let aPT. Is it true that the set of all T PB1pXq such thatσpTq “ tau isSOT˚- dense inX?
In relation to this question, it is shown in [19, Proposition 3.10] that ifX “`p for some 1ďpă 8, the set of allT PB1pXqsuch thatσpTq ĎTisSOT- dense inB1pXq. The proof given there shows that for1ăpă 8, this set of operators isSOT˚- dense inB1pXqas well.
The results of this note show that essentially no known criterion of a spectral flavor can be brought to use to show that a typical T P B1p`pq, 1 ă p ‰ 2 ă 8, has a non-trivial
