A Constructive Look at Orthonormal bases in Hilbert Space

1977 ◽  
Vol 84 (3) ◽  
pp. 189-191
Author(s):  
Douglas S. Bridges
Keyword(s):  
Hilbert Space ◽  
Orthonormal Bases

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.

Homeomorphic Images of Orthogonal Bases

2016 ◽  
Vol 31 ◽  
pp. 485-491
Author(s):  
M. Kebryaee ◽  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Finite Subset ◽  
Sufficient Conditions ◽  
Orthogonal Basis ◽  
Riesz Bases ◽  
Orthonormal Bases ◽  
Homeomorphic Image ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for a sequence $\{x_j:~j\in \mathbb J\}$ in a Hilbert space to be, up to the elimination of a finite subset of $\mathbb J$, the linear homeomorphic image of an orthogonal basis of some Hilbert space $K$. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques.

scholarly journals Hilbert modules revisited: orthonormal bases and Hilbert-Schmidt operators

1995 ◽  
Vol 37 (1) ◽  
pp. 45-54
Author(s):  
M. Cabrera ◽  
J. Martínez ◽  
A. Rodríguez
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Hilbert Module ◽  
Structure Theory ◽  
Complex Field ◽  
Hilbert Modules ◽  
Complete Structure ◽  
Orthonormal Bases

The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.

scholarly journals On a Class of Gaussian Distributions in Functional Space

2016 ◽  
Vol 34 (2) ◽  
Author(s):  
A.D. Egorov ◽  
A.V. Zherelo
Keyword(s):  
Hilbert Space ◽  
Functional Space ◽  
Gaussian Distributions ◽  
Orthonormal Bases ◽  
Special Cases ◽  
Reproducing Hilbert Space

An analog of reproducing Hilbert space of measure for a class of signed Gaussian distributions in the space of functionals is considered. Orthonormal bases in two special cases are constructed.

scholarly journals Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law

2021 ◽  
Vol 41 (6) ◽  
pp. 755-803
Author(s):  
Ilwoo Cho
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Orthonormal Bases ◽  
Shift Operators ◽  
Semicircular Law

In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\).

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