Hilbert Spaces and Bounded Operators

2017 ◽  
pp. 107-196
Author(s):  
Valter Moretti
Keyword(s):  
Hilbert Spaces ◽  
Bounded Operators

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

(F,G)-operator frames for ℒ(ℋ,𝒦)

2020 ◽  
Vol 18 (05) ◽  
pp. 2050031
Author(s):  
J. Sedghi Moghaddam ◽  
A. Najati ◽  
F. Ghobadzadeh
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Atomic Systems ◽  
Pseudo Inverse

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

scholarly journals Decompositions ofg-Frames and Duals and Pseudoduals ofg-Frames in Hilbert Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Riesz Bases ◽  
Bounded Operators ◽  
Linear Combinations ◽  
Orthonormal Bases ◽  
Critical Components

Firstly, we study the representation ofg-frames in terms of linear combinations of simpler ones such asg-orthonormal bases,g-Riesz bases, and normalized tightg-frames. Then, we study the dual and pseudodual ofg-frames, which are critical components in reconstructions. In particular, we characterize the dualg-frames in a constructive way; that is, the formulae for dualg-frames are given. We also give someg-frame like representations for pseudodualg-frame pairs. The operator parameterizations ofg-frames and decompositions of bounded operators are the key tools to prove our main results.

scholarly journals On the uniqueness of the polar decomposition of bounded operators in Hilbert spaces

2103 ◽  
Vol 70 (1) ◽  
pp. 175-180 ◽  
Author(s):  
Wataru Ichinose ◽  
Kanako Iwashita
Keyword(s):  
Hilbert Spaces ◽  
Polar Decomposition ◽  
Bounded Operators

scholarly journals On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

On the Spectra of Unbounded Subnormal Operators

1986 ◽  
Vol 38 (5) ◽  
pp. 1135-1148 ◽  
Author(s):  
G. McDonald ◽  
C. Sundberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Line ◽  
Bounded Operator ◽  
Bounded Operators ◽  
The Real ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Image Position ◽  
Topological Boundary

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

scholarly journals Infinite-dimensional numerical linear algebra: theory and applications

2010 ◽  
Vol 466 (2124) ◽  
pp. 3539-3559 ◽  
Author(s):  
Anders C. Hansen
Keyword(s):  
Linear Algebra ◽  
Hilbert Spaces ◽  
Numerical Linear Algebra ◽  
New Method ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
The Core ◽  
Algebra Theory ◽  
Spectra And Pseudospectra

We present a new method for computing spectra and pseudospectra of bounded operators on separable Hilbert spaces. The core in this theory is a generalization of the pseudospectrum called the n -pseudospectrum.

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