Journal of Function Spaces
Volume 2018 (2018), Article ID 4732836, 5 pages
https://doi.org/10.1155/2018/4732836

Research Article

Boundedly Spaced Subsequences and Weak Dynamics

C. S. Kubrusly¹ and P. C. M. Vieira²

¹Applied Mathematics Department, Federal University, Rio de Janeiro, RJ, Brazil
²National Laboratory for Scientific Computation, Petrópolis, RJ, Brazil

Correspondence should be addressed to P. C. M. Vieira; paulocm@lncc.br

Received 14 March 2018; Accepted 11 July 2018; Published 5 September 2018

Academic Editor: John R. Akeroyd

Copyright © 2018 C. S. Kubrusly and P. C. M. Vieira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Weak supercyclicity is related to weak stability, which leads to the question that asks whether every weakly supercyclic power bounded operator is weakly stable This is approached here by investigating weak l-sequential supercyclicity for Hilbert-space contractions through Nagy–Foliaş–Langer decomposition, thus reducing the problem to the quest of conditions for a weakly l-sequentially supercyclic unitary operator to be weakly stable, and this is done in light of boundedly spaced subsequences.

1. Introduction

The purpose of this paper is to characterize weak supercyclicity for Hilbert-space contractions, which is shown to be equivalent to characterizing weak supercyclicity for unitary operators. This is naturally motivated by an open question that asks whether every weakly supercyclic power bounded operator is weakly stable (which in turn is naturally motivated by a result that asserts that every supercyclic power bounded operator is strongly stable). Precisely, weakly supercyclicity is investigated in light of boundedly spaced subsequences as discussed in Lemma 1. The main result in Theorem 2 characterizes weakly l-sequentially supercyclic unitary operators that are weakly unstable in terms of boundedly spaced subsequences of the power sequence Remark 3 shows that characterizing any form of weak supercyclicity for weakly unstable unitary operators is equivalent to characterizing any form of weak supercyclicity for weakly unstable contractions after the Nagy–Foiaş–Langer decomposition.

2. Notation and Terminology

Throughout this paper denotes a complex (infinite-dimensional) Hilbert space, and denotes the Banach algebra of all operators on (i.e., of all bounded linear transformations of into itself). An operator is power bounded if (same notation for norm in and for the induced uniform norm in ). An operator is unitary if , where stands for the identity in and denotes the adjoint of an operator An operator is strongly stable or weakly stable if the -valued power sequence converges strongly (i.e., in the norm topology) or weakly to zero for every In other words, if

for every , which means for every or for every , respectively (clearly, strong stability implies weak stability). Let

denote the orbit of a vector under an operator —we write for The orbit of a set under is likewise defined: In particular, the orbit of the one-dimensional space spanned by ,

is referred to as the projective orbit of a vector under an operator A nonzero vector is a supercyclic vector for an operator if the projective orbit of is dense in in the norm topology; that is, if

where the upper bar  −  stands for closure in the norm topology. Thus a nonzero is a supercyclic vector for if and only if for every there exists a -valued sequence (which depends on and and consists of nonzero numbers) such that

for some subsequence of If has a supercyclic vector, then it is a supercyclic operator. The weak counterpart of the above convergence criterion reads as follows. A nonzero vector is a weakly l-sequentially supercyclic vector for an operator if for every there exists a -valued sequence (which depends on and and consists of nonzero numbers) such that, for some subsequence of ,

An operator in is weakly l-sequentially supercyclic if it has a weakly l-sequentially supercyclic vector. An equivalent definition reads as follows. The weak limit set of is the set consisting of all weak limits of weakly convergent -valued sequences, and an operator is weakly l-sequentially supercyclic if there exists a vector (called a weakly l-sequentially supercyclic vector for ) for which the weak limit set of is equal to .

Several forms of weak supercyclicity, including weak l-sequential supercyclicity, have recently been investigated in [1–8]. An operator is weakly supercyclic if there exists a vector (called a weakly supercyclic vector for ) such that the projective orbit is weakly dense in (i.e., dense in the weak topology of ). A set is weakly sequentially closed if every -valued weakly convergent sequence has its limit in it; and the weak sequential closure of is the smallest weakly sequentially closed set (i.e., the intersection of all weakly sequentially closed sets) including An operator is weakly sequentially supercyclic if there exists a vector (called a weakly sequentially supercyclic vector for ) for which the weak sequential closure of is equal to Observe that

and the reverse implications fail (see, e.g., [5, pp.38,39], [9, pp.259,260]).

The notion of weak l-sequential supercyclicity was introduced explicitly in [10] and implicitly in [3] and investigated in [5] which introduced a terminology similar to the one adopted here (we use the letter “l” for “limit” instead of the numeral “1” used in [5]). Supercyclicity for an operator implies weak supercyclicity, which in turn implies the operator acts on a separable space (see, e.g., [7, Section 3]), and so separability for is a consequence of any form of supercyclicity for , including weak l-sequential supercyclicity.

3. An Auxiliary Result

Let denote the self-indexing of the set of all nonnegative integers equipped with the natural order. A subsequence (in fact, a subset) of is of bounded increments (or has bounded gaps) if (Integer sequences of bounded increments have been used in [11] towards weak stability.) We say a subsequence of any sequence is boundedly spaced if it is indexed by a subsequence of bounded increments (i.e., is boundedly spaced if .

A vector is a collapsing vector for an operator if the orbit of under meets the origin (i.e., if for some Otherwise we say is a noncollapsing vector for (i.e., if for every If there exists a collapsing vector for an operator, then we say the operator has a collapsing orbit. If there exists a noncollapsing vector for an operator, then we say the operator has a noncollapsing orbit. It is clear that a collapsing orbit is a finite set, and so a weakly supercyclic operator has a noncollapsing orbit. Moreover, has a noncollapsing orbit if and only if its adjoint has. (Indeed, every vector is collapsing for if and only if every vector is collapsing for ; that is, for every there exists an such that if and only if for every for some , which means for every for some , which is equivalent to saying for every for some .)

Lemma 1. Take an arbitrary operator and an arbitrary nonzero vector Let be any subsequence of the power sequence and let and for stand for the following properties:

    for some   for some

Moreover, if has a noncollapsing orbit, then consider the following additional properties which hold whenever is a power bounded operator.

  for some for every noncollapsing vector for   for some for every noncollapsing vector for

(where and may depend on ) and if is a contraction.
Claim. For each the assertions below are pairwise equivalent.
(a) holds
(b) holds for some boundedly spaced subsequence of
(c) holds for every boundedly spaced subsequence of

Proof. We split the proof into 2 parts.
Part 1. Let be a boundedly spaced (infinite) subsequence of an (infinite) sequence of complex numbers , let be an arbitrary complex number, and let and be arbitrary nonnegative and positive real numbers, respectively.

(a₁) as for every if and only if as (b₁) for every if and only if (c₁) for every if and only if

Proof of Part 1. This is an easy consequence of the elementary fact that if is of bounded increments with , then .
Part 2. Take any and let be a boundedly spaced subsequence of .

(a₂) for every if and only if for every (i.e., if and only if .(b₂) for some if and only if for some .

If is power bounded and has a noncollapsing orbit, then

(c₂) for every noncollapsing vector for if and only if for every noncollapsing vector for , for some where if is a contraction.

Proof of Part 2. Let be a boundedly spaced subsequence of so that is a boundedly spaced subsequence of for Recall that for every Take arbitrarily so that for each

Let be an arbitrary vector in .
() Suppose for every In particular, for every for every This means for every and every ; that is, for every and every This is equivalent to for every by () in Part 1. The converse is trivial.
() We prove () contrapositively. Suppose for every in Then for every in and every in , which means for every and every By () in Part 1 this is equivalent to for every The converse is trivial.
() Let be a power bounded operator with a noncollapsing orbit (and so its adjoint has a noncollapsing orbit as well). Take an arbitrary positive number Suppose for every noncollapsing vector for (i.e., for every vector such that for every —such vectors do exist since has a noncollapsing orbit). Take an arbitrary noncollapsing vector for and an arbitrary so that for some and hence for every , which means is a noncollapsing vector for as well. Thus for every and every noncollapsing vector for This implies for every whenever is an arbitrary noncollapsing vector for (since every ) with (since is power bounded—if is a contraction so that then we may take Therefore for every noncollapsing vector for by () in Part 1. Again the converse is trivial.
Thus the claimed equivalences in (a), (b), and (c) follow from Part 2 since was taken to be an arbitrary boundedly spaced subsequence of .

Lemma 1 is naturally extended to normed spaces by replacing inner product with dual pairs and Hilbert-space adjoints with normed-space adjoints.

4. Weak Supercyclicity and Weak Stability

It was shown in [12, Theorem 2.2] that if a power bounded operator on a Banach space is supercyclic, then it is strongly stable. Such a result naturally prompts the question: does weak supercyclicity imply weak stability for power bounded operators? The question was posed and investigated in [7] and remains unanswered even if Banach-space power bounded operators are restricted to Hilbert-space contractions, where the problem is equivalently stated for unitary operators (see Remark 3 below), and also if weak supercyclicity is strengthened to weak l-sequential supercyclicity: does there exist a weakly unstable and weakly l-sequentially supercyclic unitary operator?

No unitary operator is supercyclic (reason: no Banach-space isometry is supercyclic [12, Proof of Theorem 2.1]) and, in addition to this, no Hilbert-space hyponormal operator is supercyclic [13, Theorem 3.1]. But the existence of weakly supercyclic (weakly l-sequentially supercyclic, actually) unitary operators was shown in [3, Example 3.6, pp.10,12] (also see [5, Question 1]), and the existence of weakly supercyclic unitary operators that are not l-sequentially supercyclic was shown in [5, Proposition 1.1 and Theorem 1.2]. Next we consider in Theorem 2 the case of weakly l-sequentially supercyclic unitary operators that are not weakly stable in light of boundedly spaced subsequences of the power sequence , whose proof applies [7, Theorem 6.2] and Lemma 1.

Theorem 2. If a unitary operator on a Hilbert space is weakly l-sequentially supercyclic but not weakly stable, then there exists a weakly l-sequentially supercyclic vector for such that Moreover, for every weakly l-sequentially supercyclic vector for either

(a) for every equivalently, for some boundedly spaced subsequence of or(b) for some subsequence of .

Furthermore, if (a) fails and if the subsequence in (b) is boundedly spaced, then

Proof. We begin with a definition. A normed space is said to be of type 1 if strong convergence (i.e., convergence in the norm topology) for an arbitrary -valued sequence coincides with weak convergence plus convergence of the norm sequence (i.e., and —also called Radon–Riesz space and the Radon–Riesz property, respectively). Every Hilbert space is a Banach space of type 1.
Part 1. If a power bounded operator on a type 1 normed space is weakly l-sequentially supercyclic, then either

(i) is weakly stable or(ii) the set

 is nonempty, and if is any vector in , then for every nonzero such that either

(a′) or(b′) for some subsequence of .

(Where stands for the dual of ) This is Theorem 6.2 from [7]. If a vector is such that , then (a′) is tautologically satisfied for every , and hence alternative (ii) can be rewritten as

(ii) if is any weakly l-sequentially supercyclic vector for , then for an arbitrary either (a′) or (b′) holds true.

If is a Hilbert space and if is a contraction, then is uniquely a direct sum of a completely nonunitary contraction and a unitary operator (where any of the these direct summands may be missing) by Nagy–Foiaş–Langer decomposition for Hilbert-space contractions (see, e.g., [14, p.8] or [15, p.76]). Since every completely nonunitary contraction is weakly stable (see, e.g., [16, p.55] or [15, p.106]), the above result when restricted to contractions is equivalent to the case of plain unitary operators, and this is stated as follows. If a unitary operator on a Hilbert space is weakly l-sequentially supercyclic, then either

(i) is weakly stable or (ii) if is a weakly l-sequentially supercyclic vector for , then for an arbitrary either

(a′′) or(b) for some subsequence of .

(Because the completely nonunitary part of any contraction is necessarily weakly stable and unitary operators are isometries).
Part 2. Suppose a unitary operator is weakly l-sequentially supercyclic and not weakly stable. Let be a weakly l-sequentially supercyclic vector for Thus either (a′′) or (b) holds. But property (a′′) holds if and only if (for an arbitrary )

(a) for every equivalently, for some boundedly spaced subsequence of .

Indeed, by Lemma 1 () for every if and only if for every for every (equivalently, for some) boundedly spaced subsequence of Hence () holds if and only if (a) holds. On the other hand, if () fails, then (b) holds. Thus suppose () fails; equivalently, suppose (a) fails. Fix an arbitrary weakly l-sequentially supercyclic vector for for which () fails so that (b) holds. Since is a weakly l-sequentially supercyclic operator, it has a noncollapsing vector and so has Let be an arbitrary noncollapsing vector for Now suppose (b) holds for some boundedly spaced subsequence. Then Lemma 1 () (with ) ensures

(since () fails). If

then converges to a positive number ,

Since is an l-sequentially weakly supercyclic vector for , for each there exists a scalar sequence such that for some subsequence of , which implies

for every Hence, since ,

for every , which is a contradiction because does not depend on and does not depend on (Indeed, since was taken to be an arbitrary noncollapsing vector for , the above limit holds for every such a and every and so take and a pair of noncollapsing vectors for such that is orthogonal to but not to .) Outcome is as follows

Therefore, if the subsequence in (b) is boundedly spaced, then

Remark 3. As we saw in the proof of Theorem 2, the Nagy–Foiaş–Langer decomposition for contractions on a Hilbert space says that admits an orthogonal decomposition , where a contraction is uniquely a direct sum of a completely nonunitary contraction and a unitary operator , where is the completely nonunitary part of and is the unitary part of (any of the these parcels may be missing). Moreover, a completely nonunitary contraction is weakly stable (see, e.g., [15, pp.76,106]). Thus a Hilbert-space contraction is not weakly stable if and only if it has a (nontrivial) unitary part which is not weakly stable (and so Therefore the question “does weak l-sequential supercyclicity for a Hilbert-space contraction implies weak stability?” is equivalent to the question “does weak l-sequential supercyclicity for a unitary operator implies weak stability?”

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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