On the hyperreflexivity of power partial isometries

Piwowarczyk, Kamila; Ptak, Marek

doi:10.1016/j.laa.2012.02.008

Cited by 4 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Combining (19) and (20) we get the claim. Below we show that the adjoint of any operator belonging to Alg Lat M (λ) can be recovered from the data given by multiplication operators acting along the paths, whenever r P 2 (S λ ) > 0 for all P ∈ P. To this end, we need some notation: given a path P = (V P , E P ) ∈ P and a function ϕ P : N 0 → C we put λ P = {λ v } v∈V • P and define (using (1) and (2)) the multiplication operator M λ P ϕ P : ℓ 2 (V P ) → ℓ 2 (V P ) relative to P.…”

Section: Reflexivitymentioning

confidence: 76%

Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions

Budzyński¹,

Dymek²,

Płaneta³

et al. 2019

Studia Math.

View full text Add to dashboard Cite

We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent to the weighted shift being balanced.2010 Mathematics Subject Classification. Primary: 47B37; Secondary: 47L75. Key words and phrases. Weighted shift on directed tree, multiplication operator, reflexive algebras, Wold-type decomposition. 1 2 P. BUDZYŃSKI, P. DYMEK, A. PŁANETA, AND M. PTAKIt is worth noting that the analytic aspects of the theory of weighted shifts on directed trees were studied also in [9] and [10]. The approach used therein was different than ours and relied on Shimurin's work (see [22]) employing vector-valued analytic functions. PreliminariesLet N, R and C denote the set of all natural numbers, real numbers and complex numbers, respectively. Set N 0 = N ∪ {0} and R + = [0, ∞). Denote by T the unit circle {z ∈ C : |z| = 1} and by D the open unit disc {z ∈stands for the set of all complex polynomials in one variable, whereas T denotes the set of trigonometric polynomials on T. In all what follows we use the convention that i∈∅ x i = 0.Let V be a nonempty set. Then ℓ 2 (V ) denotes the Hilbert space of all functions f :The norm induced by ·, − is denoted by · . For u ∈ V , we define e u ∈ ℓ 2 (V ) to be the characteristic function of the one-point set {u}; clearly, {e u } u∈V is an orthogonal basis of ℓ 2 (V ). We will denote by E the linear span of {e u } u∈V . Given a subset W of V , ℓ 2 (W ) stands for the subspace of ℓ 2 (V ) composed of all functions f such that f (v) = 0 for all W c , and E W denotes the set of all f ∈ ℓ 2 (W ) such that {v ∈ V : f (v) = 0} is finite. By P K we denote the orthogonal projection from ℓ 2 (V ) onto its closed subspace K.Let H be a Hilbert space, J be a nonempty set, and {X j } j∈J ⊆ H be a family of sets. Then j∈J X j stands for the smallest closed linear subspace of H such that X i ⊂ j∈J X j for every i ∈ J. Throughout the paper, unless otherwise stated, every subspace of a Hilbert space is assumed to be closed.Let H be a (complex) Hilbert space. If A is a (linear) operator in H, then D(A), N (A), R(A), and A * denote the domain, the kernel, the range, and the adjoint of A, respectively (in case it exists). We write B(H) for the algebra of all bounded operators on H equipped with the standard operator norm. By F 1 (H) and T(H) we denote the sets of rank one and trace class, respectively, operators on H. Let W be a subalgebra of B(H). Then the preannihilator W ⊥ of W is given by {T ∈ T(H) : tr(AT ) = 0 for all A ∈ W }. The set of all invariant subspaces of all operators A ...

show abstract

Section: Reflexivitymentioning

confidence: 76%

Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions

Budzyński¹,

Dymek²,

Płaneta³

et al. 2019

Studia Math.

View full text Add to dashboard Cite

show abstract

“…Recall that W(a s ) has property A 1 (1) (see [9,Proposition 60.5]), thus using Theorem 4.3 we get hyporeflexivity of W(V s ). It also has property A 1 (1) by [21,Proposition 2.3]. The backward shift a c ∈ B(l 2 + ) is reflexive and has property A 1 (1) (since both properties are preserved by taking the adjoint of operator), thus in the same way as above we get hyporeflexivity and property A 1 (1) of W(V c ).…”

Section: Power Partial Isometries Are Hyporeflexivementioning

confidence: 78%

“…In [5] full characterization of reflexivity of an algebra generated by completely non-unitary power partial isometries was given. In [21] it was shown that the same conditions given in [5] characterize hyperreflexive algebras generated by power partial isometries. In the present paper we will show that algebras generated by power partial isometries are hyporeflexive, 2-reflexive and even 2-hyperreflexive.…”

Section: Introductionmentioning

confidence: 85%

“…Similarly, since W(V s ) is hyperreflexive, κ(W(a s )) < 11, 4 (see [10], [15]) and has property A 1 (1) ([9, Proposition 60.5]), thus W(V (2) s ) is hyperreflexive and has property A 1 (1) ([21, Proposition 2.3]). Recall also that both hyperreflexivity (with the same hyperreflexivity constant) and property A 1 (1) are kept, when we take the adjoint, hence W(V (2) c ) is hyperreflexive and has property A 1 (1) (see also [21,Proposition 3.1]). By [6, Proposition III.1.21] we get that W(V…”

Section: Power Partial Isometries Are 2-hyperreflexivementioning

confidence: 99%

See 1 more Smart Citation

2-hyperreflexivity and hyporeflexivity of power partial isometries

Piwowarczyk¹,

Ptak²

2016

OpMath

Self Cite

View full text Add to dashboard Cite

show abstract

“…Subsequently, in [Ber98], Bercovici established the hyperreflexivity of much larger class of algebras, and substantially improved the distance constant from [DP99]. See also the papers [LS75], [Ros82], [KL85], [KL86], [MP05], [DL06], [KP06], [JP06] and [PP12]. Despite these results, hyperreflexivity is still not well understood.…”

Section: Introductionmentioning

confidence: 99%