Weak Quasistability in Linear Operators

• Weak quasistability is defined as the existence of a vector-dependent subsequence along which Tⁿx converges weakly to 0, serving as an intermediate property between weak stability and general weak convergence.
• The theory employs boundedly spaced subsequences to upgrade per-vector weak decay into full weak stability in power-bounded settings.
• Concrete examples like the Foguel operator and Rajchman measure models demonstrate that weak quasistability can occur with or without power boundedness, enriching the analysis of operator dynamics.

Weak quasistability is a weak-topological asymptotic property of bounded linear operators. For an operator $T\in B[X]$ on an infinite-dimensional normed space $X$, it requires

$$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$

or equivalently: for each $x\in X$ there exists a subsequence $\{n_k(x)\}$, depending on $x$ but not on $f$, such that $T^{n_k(x)}x\rightharpoonup 0$. In linear dynamics it occupies an intermediate position between weak stability and the mere existence of weakly convergent orbit subsequences, and recent work has clarified both its strict separation from weak stability and the precise mechanism by which it can collapse to weak stability in power-bounded settings (Kubrusly et al., 2023).

1. Definition and operator-theoretic setting

The basic setting is a normed space $X$ over $\mathbb F\in\{\mathbb R,\mathbb C\}$, with dual $X$0, and the algebra $X$1 of bounded linear operators. Weak convergence of a sequence $X$2 to $X$3 means

$X$4

and is denoted $X$5. Weak stability of $X$6 is the stronger requirement

$X$7

while weak quasistability replaces the limit by a $X$8. In Hilbert spaces this takes the familiar form $X$9 for weak stability, and $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$0 for weak quasistability (Kubrusly, 23 Aug 2025).

This notion is intrinsically subsequential. Weak stability forces the entire orbit $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$1 to converge weakly to $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$2 for every $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$3, whereas weak quasistability only asks that each orbit admit at least one subsequence converging weakly to $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$4. The subsequence may depend on the vector $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$5, and in general it may be very sparse. A stronger variant, introduced later, is homogeneous weak quasistability: there exists a single subsequence $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$6 such that for all $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$7,

$$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$8

Every weakly stable operator is homogeneously weakly quasistable, but the converse may fail (Kubrusly, 23 Aug 2025).

Power boundedness remains central throughout this theory. An operator $$\liminf_{n\to\infty}|f(T^n x)|=0 \qquad \text{for every }x\in X,\ f\in X^*,$$9 is power bounded if $x\in X$0. In Banach spaces this is equivalent, by Banach–Steinhaus, to $x\in X$1 for every $x\in X$2. Much of the recent structure theory asks which weak asymptotic properties become equivalent under power boundedness and which do not.

2. Stability, quasistability, and the weak-topology gap

A decisive structural point is that quasistability behaves differently in the uniform, strong, and weak regimes. For uniform asymptotics, uniform quasistability means $x\in X$3, and Proposition 4.1 shows that this is equivalent to uniform stability: $x\in X$4 Indeed, $x\in X$5 already forces $x\in X$6. Thus uniform quasistability and uniform stability coincide for every operator on every normed space (Kubrusly et al., 2023).

In the strong regime, strong quasistability means $x\in X$7 for every $x\in X$8. Under power boundedness, Proposition 4.2 shows that strong quasistability is again equivalent to strong stability: $x\in X$9 Hence the distinction between stability and quasistability disappears in the uniform and strong settings once the natural boundedness hypothesis is imposed (Kubrusly et al., 2023).

The weak regime is different. Weak stability always implies weak quasistability, but the converse fails even for power-bounded operators. This strict gap is the conceptual novelty of the weak theory: weak quasistability permits decay along tailored subsequences, possibly varying with the starting vector, while weak stability requires full-orbit decay in the weak topology. The operator-theoretic literature therefore treats weak quasistability as a genuinely intermediate phenomenon rather than a merely weakened reformulation of stability (Kubrusly et al., 2023).

3. Boundedly spaced subsequences as the bridging mechanism

The mechanism that converts weak quasistability into weak stability is combinatorial. A subsequence $\{n_k(x)\}$0 of the positive integers is boundedly spaced if

$\{n_k(x)\}$1

Proposition 5.1 is a scalar bridging lemma: if $\{n_k(x)\}$2 is a boundedly spaced subsequence of a scalar sequence $\{n_k(x)\}$3 and $\{n_k(x)\}$4 for every fixed $\{n_k(x)\}$5, then $\{n_k(x)\}$6 (Kubrusly et al., 2023).

Lemma 5.2 transfers this scalar fact to operator orbits. If $\{n_k(x)\}$7 is a boundedly spaced subsequence of $\{n_k(x)\}$8 and $\{n_k(x)\}$9, then $x$0. The proof uses the scalar sequences $x$1, with $x$2, and propagates convergence from boundedly spaced subsequences to the full sequence. Theorem 5.3 then globalizes the argument: if for each $x$3 there exists a boundedly spaced subsequence $x$4 such that $x$5, then $x$6 is weakly stable (Kubrusly et al., 2023).

This result identifies bounded spacing as the exact device that removes subsequence-dependence. Weak quasistability by itself only guarantees per-vector subsequential vanishing. Once the vanishing subsequences have uniformly bounded gaps, the distinction between subsequential and full-sequence weak decay disappears. The theory therefore isolates boundedly spaced subsequences not as a technical convenience but as the precise combinatorial condition under which weak quasistability upgrades to weak stability.

4. Weak $x$7-sequential supercyclicity and dynamical consequences

Weak quasistability is closely tied to weak $x$8-sequential supercyclicity. For a nonzero vector $x$9, the projective orbit is

$f$0

The vector $f$1 is weakly $f$2-sequentially supercyclic if for every $f$3 there exist scalars $f$4 and indices $f$5 such that

$f$6

An operator is weakly $f$7-sequentially supercyclic if it has such a vector; the set of all such vectors is denoted $f$8 (Kubrusly et al., 2023).

A central theorem states that every power-bounded weakly $f$9-sequentially supercyclic operator is weakly quasistable. Concretely, if $T^{n_k(x)}x\rightharpoonup 0$0 is power bounded and $T^{n_k(x)}x\rightharpoonup 0$1, then for all $T^{n_k(x)}x\rightharpoonup 0$2 and all $T^{n_k(x)}x\rightharpoonup 0$3,

$T^{n_k(x)}x\rightharpoonup 0$4

and by density of $T^{n_k(x)}x\rightharpoonup 0$5 in $T^{n_k(x)}x\rightharpoonup 0$6, the same liminf statement holds for all $T^{n_k(x)}x\rightharpoonup 0$7. Equivalently, $T^{n_k(x)}x\rightharpoonup 0$8 is a weak limit point of every orbit (Kubrusly et al., 2023).

The later refinement is Theorem 6.2: if $T^{n_k(x)}x\rightharpoonup 0$9 is power bounded and weakly $X$0-sequentially supercyclic, and if for each $X$1 there exists a boundedly spaced subsequence of weak $X$2-sequential supercyclicity, for every $X$3, then $X$4 is weakly stable. Thus weak $X$5-sequential supercyclicity gives weak quasistability in general, and gives weak stability once the relevant subsequences are boundedly spaced (Kubrusly et al., 2023).

The obstruction is also explicit. Theorem 6.3 states that if $X$6 is weakly $X$7-sequentially supercyclic and weakly stable, then for every $X$8 and every $X$9, all scalar sequences $\mathbb F\in\{\mathbb R,\mathbb C\}$0 arising in representations $\mathbb F\in\{\mathbb R,\mathbb C\}$1 must be unbounded. Corollary 6.4 packages this into a dichotomy: for a power-bounded weakly $\mathbb F\in\{\mathbb R,\mathbb C\}$2-sequentially supercyclic operator, either boundedly spaced subsequences fail somewhere, or all such coefficient sequences are unbounded. For isometries on Hilbert spaces, weak $\mathbb F\in\{\mathbb R,\mathbb C\}$3-sequential supercyclicity forces surjectivity and hence unitarity, and the resulting unitary is singular-continuous and weakly quasistable (Kubrusly et al., 2023).

5. Counterexamples, power boundedness, and measure-theoretic models

The canonical example separating weak quasistability from weak stability is the Foguel operator. Let $\mathbb F\in\{\mathbb R,\mathbb C\}$4 be a separable Hilbert space with orthonormal basis $\mathbb F\in\{\mathbb R,\mathbb C\}$5, let $\mathbb F\in\{\mathbb R,\mathbb C\}$6 be the unilateral shift, and let $\mathbb F\in\{\mathbb R,\mathbb C\}$7 be the orthogonal projection onto $\mathbb F\in\{\mathbb R,\mathbb C\}$8, where $\mathbb F\in\{\mathbb R,\mathbb C\}$9 is sparse and satisfies $X$00 whenever $X$01 in $X$02. On $X$03, define

$X$04

This operator is power bounded, weakly quasistable, and not weakly stable. The proof of quasistability uses the subsequence $X$05, $X$06, along which the middle block vanishes; crucially, $X$07, so the subsequence is not boundedly spaced (Kubrusly et al., 2023).

Weak quasistability also does not imply power boundedness. An explicit diagonal operator on $X$08 satisfies

$X$09

It is therefore power unbounded and weakly unstable, but still weakly quasistable, indeed strongly quasistable. This shows that weak quasistability neither implies weak stability nor power boundedness (Kubrusly, 23 Aug 2025).

There is, however, a sharp restriction on power-unbounded examples. If $X$10 is weakly quasistable, then either $X$11 is power bounded, or $X$12 is power unbounded but noncoercive in the sense that no single orbit satisfies $X$13. In particular, coercive power-unbounded operators cannot be weakly quasistable (Kubrusly, 23 Aug 2025).

A complementary model comes from harmonic analysis. For a finite positive measure $X$14 on the unit circle $X$15, the position operator $X$16 on $X$17 is given by

$X$18

The measure $X$19 is Rajchman if $X$20, and Proposition 5.3 shows that

$X$21

By contrast, if $X$22 is finite and continuous, then $X$23 is weakly quasistable; indeed the construction yields homogeneous weak quasistability. Thus continuous non-Rajchman measures provide natural unitary models that are weakly unstable but homogeneously weakly quasistable (Kubrusly, 23 Aug 2025).

6. Spectral structure, open problems, and terminological variation

Weak quasistability has spectral consequences. In Hilbert spaces it is preserved under adjoints, and if $X$24 is weakly quasistable then

$X$25

so the part of the spectrum on the unit circle can only be continuous spectrum. If $X$26 is also power bounded, then $X$27 and therefore

$X$28

Survey work places these facts into a broader hierarchy: for power-bounded weakly $X$29-sequentially supercyclic operators, weak quasistability is always present and $X$30 is a weak limit point of every orbit, but whether such operators must be weakly stable remains open. The same open status holds for the stronger question of whether weak supercyclicity implies weak stability (Kubrusly, 2024).

A further open problem, raised in the operator-theoretic literature, concerns the Foguel operator itself: it is known to be power bounded, weakly quasistable, and not weakly stable, but it remains asked whether it is weakly $X$31-sequentially supercyclic (Kubrusly et al., 2023). This question is representative of the current frontier. Weak quasistability is now well understood as an intermediate asymptotic property, but the full weak-supercyclicity-to-weak-stability program remains unresolved.

The phrase is not uniform across disciplines. In distributed systems, the corresponding notion is weak-stabilization: closure of the legitimate set together with possible convergence from every initial configuration (0711.3672). In stochastic ecology, weak quasistability refers to shallow basins in a quasi-potential landscape, quantified by small barrier height $X$32 and associated with frequent noise-induced escapes (Nolting et al., 2015). In partially hyperbolic dynamics, topological quasi-stability means quasi-conjugacy modulo motions along center directions (Hu et al., 2012). In the planar circular restricted three-body problem, generalized weak stability boundaries are identified with cuts of the stable manifold of a Lyapunov orbit and organize weakly $X$33-stable motion near a primary (Belbruno et al., 2012). Power-systems and long-range-interaction literatures use yet other quasi-stability notions, concerning respectively the failure of QSS reduction and the conversion of quasistationarity into a crossover phenomenon under stochastic dynamics (Wang et al., 2014, Gupta et al., 2010).

Within operator theory, however, weak quasistability now has a precise and stable meaning: per-vector subsequential weak vanishing of the orbit, strictly weaker than weak stability, compatible with both power-bounded and certain power-unbounded dynamics, and governed in its passage to full weak stability by the bounded-spacing structure of the relevant subsequences.