--- title: Weak Quasistability in Linear Operators url: https://www.emergentmind.com/topics/weak-quasistability type: topic --- # Weak Quasistability in Linear Operators 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 [2311.14151]. ## 1. Definition and operator-theoretic setting The basic setting is a normed space \(X\) over \(\mathbb F\in\{\mathbb R,\mathbb C\}\), with dual \(X^*\), and the algebra \(B[X]\) of bounded linear operators. Weak convergence of a sequence \(\{x_n\}\subset X\) to \(x\in X\) means \[ \lim_{n\to\infty} f(x_n)=f(x)\qquad\text{for every }f\in X^*, \] and is denoted \(x_n\rightharpoonup x\). Weak stability of \(T\) is the stronger requirement \[ \lim_{n\to\infty}|f(T^n x)|=0\qquad\text{for every }x\in X,\ f\in X^*, \] while weak quasistability replaces the limit by a \(\liminf\). In Hilbert spaces this takes the familiar form \(\langle T^n x,y\rangle\to 0\) for weak stability, and \(\liminf_n |\langle T^n x,y\rangle|=0\) for weak quasistability [2508.17113]. This notion is intrinsically subsequential. Weak stability forces the entire orbit \(\{T^n x\}\) to converge weakly to \(0\) for every \(x\), whereas weak quasistability only asks that each orbit admit at least one subsequence converging weakly to \(0\). The subsequence may depend on the vector \(x\), and in general it may be very sparse. A stronger variant, introduced later, is homogeneous weak quasistability: there exists a single subsequence \(\{n_j\}\) such that for all \(x\in X\), \[ f(T^{n_j}x)\to 0 \qquad\text{for every }f\in X^*. \] Every weakly stable operator is homogeneously weakly quasistable, but the converse may fail [2508.17113]. Power boundedness remains central throughout this theory. An operator \(T\) is power bounded if \(\sup_{n\ge 0}\|T^n\|<\infty\). In Banach spaces this is equivalent, by Banach–Steinhaus, to \(\sup_n\|T^n x\|<\infty\) for every \(x\). 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 \(\liminf_{n\to\infty}\|T^n\|=0\), and Proposition 4.1 shows that this is equivalent to uniform stability: \[ \lim_{n\to\infty}\|T^n\|=0. \] Indeed, \(\inf_n\|T^n\|<1\) already forces \(\|T^n\|\to 0\). Thus uniform quasistability and uniform stability coincide for every operator on every normed space [2311.14151]. In the strong regime, strong quasistability means \(\liminf_{n\to\infty}\|T^n x\|=0\) for every \(x\in X\). Under power boundedness, Proposition 4.2 shows that strong quasistability is again equivalent to strong stability: \[ \|T^n x\|\to 0 \qquad \text{for every }x\in X. \] Hence the distinction between stability and quasistability disappears in the uniform and strong settings once the natural boundedness hypothesis is imposed [2311.14151]. 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 [2311.14151]. ## 3. Boundedly spaced subsequences as the bridging mechanism The mechanism that converts weak quasistability into weak stability is combinatorial. A subsequence \(\{n_k\}\) of the positive integers is boundedly spaced if \[ \sup_k (n_{k+1}-n_k)<\infty. \] Proposition 5.1 is a scalar bridging lemma: if \(\{a_{n_k}\}\) is a boundedly spaced subsequence of a scalar sequence \(\{a_n\}\) and \(a_{n_k+j}\to a\) for every fixed \(j\ge 1\), then \(a_n\to a\) [2311.14151]. Lemma 5.2 transfers this scalar fact to operator orbits. If \(\{T^{n_k}\}\) is a boundedly spaced subsequence of \(\{T^n\}\) and \(T^{n_k}x\rightharpoonup 0\), then \(T^n x\rightharpoonup 0\). The proof uses the scalar sequences \(g(T^n x)\), with \(g\in X^*\), and propagates convergence from boundedly spaced subsequences to the full sequence. Theorem 5.3 then globalizes the argument: if for each \(x\in X\) there exists a boundedly spaced subsequence \(\{n_k(x)\}\) such that \(T^{n_k(x)}x\rightharpoonup 0\), then \(T\) is weakly stable [2311.14151]. 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 \(\ell\)-sequential supercyclicity and dynamical consequences Weak quasistability is closely tied to weak \(\ell\)-sequential supercyclicity. For a nonzero vector \(y\), the projective orbit is \[ O_T([y])=\{\alpha T^n y:\alpha\in\mathbb F,\ n\ge 0\}. \] The vector \(y\) is weakly \(\ell\)-sequentially supercyclic if for every \(x\in X\) there exist scalars \(\{\alpha_k\}\) and indices \(\{n_k\}\) such that \[ \alpha_k T^{n_k}y \rightharpoonup x. \] An operator is weakly \(\ell\)-sequentially supercyclic if it has such a vector; the set of all such vectors is denoted \(Y_T\) [2306.08197]. A central theorem states that every power-bounded weakly \(\ell\)-sequentially supercyclic operator is weakly quasistable. Concretely, if \(T\) is power bounded and \(Y_T\neq\varnothing\), then for all \(y\in Y_T\) and all \(f\in X^*\), \[ \liminf_{n\to\infty}|f(T^n y)|=0, \] and by density of \(Y_T\) in \(X\), the same liminf statement holds for all \(x\in X\). Equivalently, \(0\) is a weak limit point of every orbit [2306.08197]. The later refinement is Theorem 6.2: if \(T\) is power bounded and weakly \(\ell\)-sequentially supercyclic, and if for each \(x\in X\setminus O_T([y])\) there exists a boundedly spaced subsequence of weak \(\ell\)-sequential supercyclicity, for every \(y\in Y_T\), then \(T\) is weakly stable. Thus weak \(\ell\)-sequential supercyclicity gives weak quasistability in general, and gives weak stability once the relevant subsequences are boundedly spaced [2311.14151]. The obstruction is also explicit. Theorem 6.3 states that if \(T\) is weakly \(\ell\)-sequentially supercyclic and weakly stable, then for every \(y\in Y_T\) and every \(x\in X\setminus O_T([y])\), all scalar sequences \(\{\alpha_k(x)\}\) arising in representations \(\alpha_k(x)T^{n_k(x)}y\rightharpoonup x\) must be unbounded. Corollary 6.4 packages this into a dichotomy: for a power-bounded weakly \(\ell\)-sequentially supercyclic operator, either boundedly spaced subsequences fail somewhere, or all such coefficient sequences are unbounded. For isometries on Hilbert spaces, weak \(\ell\)-sequential supercyclicity forces surjectivity and hence unitarity, and the resulting unitary is singular-continuous and weakly quasistable [2306.08197]. ## 5. Counterexamples, power boundedness, and measure-theoretic models The canonical example separating weak quasistability from weak stability is the Foguel operator. Let \(X\) be a separable Hilbert space with orthonormal basis \(\{e_k\}_{k\ge 0}\), let \(S\) be the unilateral shift, and let \(P\) be the orthogonal projection onto \(\operatorname{span}\{e_j:j\in J\}\), where \(J\subset\mathbb N\) is sparse and satisfies \(2i