Frame-cyclic operators and their properties

2021 ◽  
Vol 24 (01) ◽  
pp. 2150009
Author(s):  
Nastaran Alizadeh Moghaddam ◽  
Mohammad Janfada
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Hypercyclic Operators ◽  
Cyclic Operator ◽  
Infinite Dimensional ◽  
Cyclic Vectors ◽  
Cyclic Operators

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

2007 ◽  
Vol 10 (02) ◽  
pp. 261-276 ◽  
Author(s):  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Separable Hilbert Space ◽  
Fock Spaces ◽  
Parameter Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Dense Subalgebras of Left Hilbert Algebras

1982 ◽  
Vol 34 (6) ◽  
pp. 1245-1250 ◽  
Author(s):  
A. van Daele
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Quantum Field ◽  
Von Neumann ◽  
Hilbert Algebras

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

scholarly journals A functional Hodrick–Prescott filter

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Boualem Djehiche ◽  
Hiba Nassar
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Functional Data ◽  
Separable Hilbert Space ◽  
Underlying Distribution ◽  
Smoothing Operator ◽  
Infinite Dimensional ◽  
Optimal Smoothing ◽  
Functional Version

AbstractWe propose a functional version of the Hodrick–Prescott filter for functional data which take values in an infinite-dimensional separable Hilbert space. We further characterize the associated optimal smoothing operator when the associated linear operator is compact and the underlying distribution of the data is Gaussian.

Biholomorphic Mappings on Banach Spaces

2019 ◽  
Vol 62 (4) ◽  
pp. 913-924
Author(s):  
H. Carrión ◽  
P. Galindo ◽  
M. L. Lourenço
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Open Subset ◽  
Separable Hilbert Space ◽  
Derivative Operator ◽  
Infinite Dimensional ◽  
Dimensional Version ◽  
Dual Banach Space ◽  
Bounded Convex ◽  
Power Bounded

AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

scholarly journals Tomography in Abstract Hilbert Spaces

2006 ◽  
Vol 13 (03) ◽  
pp. 239-253 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
A. Simoni ◽  
F. Ventriglia
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Rank One

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

scholarly journals The Behavior of Basic Fields

2018 ◽  
Vol 1 (3) ◽  
Keyword(s):  
Mathematical Modeling ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Differential Calculus ◽  
Separable Hilbert Space ◽  
Floating Platform ◽  
Basic Field ◽  
Floating Platforms ◽  
The Continuum

A basic field is defined in the realm of a mathematical modeling platform that is based on a collection of floating platforms and an embedding platform. Each floating platform is represented by a quaternionic separable Hilbert space. The embedding platform is a non-separable Hilbert space. A basic field is a continuum eigenspace of an operator that resides in the non-separable embedding Hilbert space. The continuum can be described by a quaternionic function, and its behavior is described by quaternionic differential calculus. The separable Hilbert spaces contain the point-like artifacts that trigger the basic field. The floating platforms possess symmetry, which in combination with the background platform generates the sources of symmetry related fields.

Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

2014 ◽  
Vol 66 (5) ◽  
pp. 1143-1166 ◽  
Author(s):  
Lucijan Plevnik ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Idempotent Operators

AbstractLet and be infinite-dimensional separable Hilbert spaces and Lat the lattice of all closed subspaces oh . We describe the general form of pairs of bijective maps ϕ, Ψ : Lat → Lat having the property that for every pair U, V ∊ Lat we have . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

Factorization in the Invertible Group of a C*-Algebra

1997 ◽  
Vol 49 (6) ◽  
pp. 1188-1205 ◽  
Author(s):  
Michael J. Leen
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Upper Bounds ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Identity Component ◽  
C Algebras ◽  
Number Of Factors ◽  
Finite Products

AbstractIn this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

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