Scalability of Frames Generated by Dynamical Operators
نویسندگان
چکیده
Let H be a separable Hilbert space, let G ⊂ H, and let A be an operator on H. Under appropriate conditions on A andG, it is known that the set of iterations FG(A) = {Ag | g ∈ G, 0 ≤ j ≤ L(g)} is a frame for H. We call FG(A) a dynamical frame for H, and explore further its properties; in particular, we show that the canonical dual frame of FG(A) also has an iterative set structure. We explore the relations between the operator A, the set G and the number of iterations L which ensure that the system FG(A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when FG(A) is a scalable frame in several special cases involving block-diagonal and companion operators.
منابع مشابه
Coherent Frames
Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of ...
متن کاملDynamical System and Semi-Hereditarily Hypercyclic Property
In this paper we give conditions for a tuple of commutative bounded linear operators which holds in the property of the Hypercyclicity Criterion. We characterize topological transitivity and semi-hereiditarily of a dynamical system given by an n-tuple of operators acting on a separable infinite dimensional Banach space .
متن کامل$G$-Frames for operators in Hilbert spaces
$K$-frames as a generalization of frames were introduced by L. Gu{a}vruc{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new ge...
متن کاملOn Two-parameter Dynamical Systems and Applications
In this note some useful properties of strongly continuous two-parameter semigroups of operators are studied, an exponential formula for two-parameter semigroups of operators on Banach spaces is obtained and some applied examples of two-parameter dynamical systems are discussed
متن کاملSome Properties of $ ast $-frames in Hilbert Modules Over Pro-C*-algebras
In this paper, by using the sequence of adjointable operators from pro-C*-algebra $ mathcal{A} $ into a Hilbert $ mathcal{A} $-module $ E $. We introduce frames with bounds in pro-C*-algebra $ mathcal{A} $. New frames in Hilbert modules over pro-C*-algebras are called standard $ ast $-frames of multipliers. Meanwhile, we study several useful properties of standard $ ast $-frames in Hilbert modu...
متن کامل