scholarly journals Besselian g-frames and near g-Riesz bases

2011 ◽  
Vol 5 (2) ◽  
pp. 259-270 ◽  
Author(s):  
M.R. Abdollahpour ◽  
A. Najati
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Orthonormal Basis ◽  
Inner Product ◽  
Riesz Bases ◽  
Finite Dimensional

In this paper we introduce and study near g-Riesz basis, Besselian g-frames and unconditional g-frames. We show that a near g-Riesz basis is a Besselian g-frame and we conclude that under some conditions the kernel of associated synthesis operator for a near g-Riesz basis is finite dimensional. Finally, we show that a g-frame is a g-Riesz basis for a Hilbert space H if and only if there is an equivalent inner product on H, with respect to which it becomes an g-orthonormal basis for H.

Finite-Dimensional Extensions of Certain Symmetric Operators(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 455-456
Author(s):  
I. M. Michael
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Inner Product ◽  
Selfadjoint Extension ◽  
Von Neumann ◽  
Deficiency Indices ◽  
Finite Dimensional ◽  
Symmetric Operators

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals On normal derivations of Hilbert–Schmidt type

1987 ◽  
Vol 29 (2) ◽  
pp. 245-248 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Trace Class ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Schmidt Norm ◽  
Image Position

Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner productwhere {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of X ∈ C2 is given by ⅡXⅡ2=(X, X)½.

scholarly journals Hamiltonians Generated by Parseval Frames

2020 ◽  
Vol 171 (1) ◽  
Author(s):  
F. Bagarello ◽  
S. Kużel
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Physical System ◽  
Orthonormal Basis ◽  
Orthonormal Bases ◽  
Parseval Frames ◽  
Orthogonal Projectors ◽  
Physical Reasons ◽  
Math Phys ◽  
Infinite Linear

AbstractIt is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.

scholarly journals SISTEM ORTONORMAL DALAM RUANG HILBERT

2014 ◽  
Vol 8 (2) ◽  
pp. 19-26
Author(s):  
Zeth A. Leleury
Keyword(s):  
Hilbert Space ◽  
Quadratic Form ◽  
Vector Space ◽  
Integral Equations ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Finite Dimensional

Hilbert space is one of the important inventions in mathematics. Historically, the theory of Hilbert space originated from David Hilbert’s work on quadratic form in infinitely many variables with their applications to integral equations. This paper contains some definitions such as vector space, normed space and inner product space (also called pre-Hilbert space), and which is important to construct the Hilbert space. The fundamental ideas and results are discussed with special attention given to finite dimensional pre-Hilbert space and some basic propositions of orthonormal systems in Hilbert space. This research found that each finite dimensional pre- Hilbert space is a Hilbert space. We have provided that a finite orthonormal systems in a Hilbert space X is complete if and only if this orthonormal systems is a basis of X.

scholarly journals Wavelet bases for a unitary operator

1995 ◽  
Vol 38 (2) ◽  
pp. 233-260 ◽  
Author(s):  
S. L. Lee ◽  
H. H. Tan ◽  
W. S. Tang
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Unitary Operator ◽  
Linear Span ◽  
General Setting ◽  
Complex Hilbert Space ◽  
Riesz Bases ◽  
Duality Principle ◽  
Spline Wavelets ◽  
Necessary And Sufficient

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

scholarly journals Operator Characterizations and Some Properties ofg-Frames on Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Riesz Bases ◽  
Dual Frames

Given theg-orthonormal basis for Hilbert spaceH, we characterize theg-frames, normalized tightg-frames, andg-Riesz bases in terms of theg-preframe operators. Then we consider the transformations ofg-frames, normalized tightg-frames, andg-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums andg-dual frames ofg-frames by applying the results of characterizations.

scholarly journals Computing the numerical range of Krein space operators

2014 ◽  
Vol 13 (1) ◽  
Author(s):  
Natalia Bebiano ◽  
J. da Providência ◽  
A. Nata ◽  
J.P. da Providência
Keyword(s):  
Hilbert Space ◽  
Quadratic Form ◽  
Numerical Range ◽  
Krein Space ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Finite Dimensional ◽  
Alternative Algorithm ◽  
Kreĭn Space ◽  
Finite Dimensional Case

Abstract Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined elliptical and hyperbolical numerical ranges. The numerical results reported here indicate that this method is very efficient, since it is faster and more accurate than either of the existing algorithms. Further, it may yield easy solutions for the inverse indefinite numerical range problem. Our algorithm uses an idea of Marcus and Pesce from 1987 for generating Hilbert space numerical ranges of matrices of size n.

Perturbation Theorems for Frames and Riesz Bases

2013 ◽  
Vol 433-435 ◽  
pp. 44-47
Author(s):  
Xue Mei Xiao
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Linear Operator ◽  
Riesz Basis ◽  
Operator Theory ◽  
Riesz Bases ◽  
Perturbation Theorem

This paper gives a perturbation theorem for frames in a Hilbert space which is a generalization of a result by Ping Zhao. It is proved that the condition a linear operator is invertible can be weakened to be surjective, and a similar result also be obtained for a Riesz basis. The perturbation theorems for frames and Riesz bases in a Hilbert space were studied by operator theory in functional analysis.

N-tangle, Entangled Orthonormal Basis, and a Hierarchy of Spin Hamilton Operators

2011 ◽  
Vol 66 (10-11) ◽  
pp. 615-619 ◽  
Author(s):  
Willi-Hans Steeb ◽  
Yorick Hardy
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Entangled States ◽  
Hamilton Operator ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

An N-tangle can be defined for the finite dimensional Hilbert space H = C2N , with N = 3 or N even.We give an orthonormal basis which is fully entangled with respect to this measure.We provide a spin Hamilton operator which has this entangled basis as normalized eigenvectors if N is even. From these normalized entangled states a Bell matrix is constructed and the cosine-sine decomposition is calculated. If N is odd the normalized eigenvectors can be entangled or unentangled depending on the parameters.

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