scholarly journals Krylov Solvability of Unbounded Inverse Linear Problems

2020 ◽  
Vol 93 (1) ◽  
Author(s):  
Noè Angelo Caruso ◽  
Alessandro Michelangeli
Keyword(s):  
The Self ◽  
Linear Combinations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Unbounded Linear Operator ◽  
Gradient Based ◽  
Hilbert Norm ◽  
Linear Problems ◽  
Finite Linear ◽  
Densely Defined

AbstractThe abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem $$Af=g$$ A f = g where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of $$g,Ag,A^2g,\dots $$ g , A g , A 2 g , ⋯ , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.

scholarly journals Jensen's Trace Inequality in Several Variables

2003 ◽  
Vol 14 (06) ◽  
pp. 667-681 ◽  
Author(s):  
Frank Hansen ◽  
Gert K. Pedersen
Keyword(s):  
Symmetric Matrices ◽  
Trace Inequality ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Several Variables ◽  
Lower Semi ◽  
Continuous Trace ◽  
Abelian Fields ◽  
Densely Defined

For a convex, real function f, we present a simple proof of the formula [Formula: see text] valid for each tuple (x1,…, xm) of symmetric matrices in [Formula: see text] and every unital column (a1,…, am) of matrices, i.e. [Formula: see text]. This is the standard Jensen trace ine-quality. If f ≥ 0 it holds also for the unbounded trace on [Formula: see text], where [Formula: see text] is an infinite-dimensional Hilbert space. We then investigate the more general case where τ is a densely defined, lower semi-continuous trace on a C*-algebra [Formula: see text] and f is a convex, continuous function of n variables, and show that we have the inequality [Formula: see text] for every family of abeliann-tuples [Formula: see text], i.e. tuples of self-adjoint elements in [Formula: see text] such that [xik, xjk] = 0 for all i, j and k, where 1 ≤ k ≤ m, and every unital m-column (a1,…, am) in [Formula: see text], provided that the elements [Formula: see text] also form an abelian n-tuple. We even establish this result for weak* measurable, self-adjoint, abelian fields (xit)t∈T, 1 ≤ i ≤ n, i.e. [xit, xjt] = 0 for all i, j and t, and weak* measurable, unital column field (at)t∈T in [Formula: see text] paired with any trace or trace-like functional φ, i.e. one that contains the n-tuple (presumed abelian) with elements [Formula: see text] in its centralizer. This takes the form of the inequality [Formula: see text] We also study functions of n variables that are monotone increasing in each variable, and show in two important cases that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are abelian n-tuples with xi ≤ yi for each i and φ is a trace or a trace-like functional.

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.

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.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

Multi-particle Spaces

2020 ◽  
pp. 187-200
Author(s):  
Daniel Canarutto
Keyword(s):  
Operator Algebra ◽  
Particle Physics ◽  
Arbitrary Number ◽  
Exterior Algebra ◽  
Linear Combinations ◽  
Absorption And Emission ◽  
Dirac Type ◽  
Natural Generalisation ◽  
Finite Linear ◽  
Number Of Particles

The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.

Classifying Algebras for the K-Theory of σ-C*-Algebras

1989 ◽  
Vol 41 (6) ◽  
pp. 1021-1089 ◽  
Author(s):  
N. Christopher Phillips
Keyword(s):  
Hilbert Space ◽  
Topological Space ◽  
Fredholm Operators ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Continuous Maps ◽  
Group Structures

In topology, the representable K-theory of a topological space X is defined by the formulas RK0(X) = [X,Z x BU] and RKl(X) = [X, U], where square brackets denote sets of homotopy classes of continuous maps, is the infinite unitary group, and BU is a classifying space for U. (Note that ZxBU is homotopy equivalent to the space of Fredholm operators on a separable infinite-dimensional Hilbert space.) These sets of homotopy classes are made into abelian groups by using the H-group structures on Z x BU and U. In this paper, we give analogous formulas for the representable K-theory for α-C*-algebras defined in [20].

On Products of Symmetries

1966 ◽  
Vol 18 ◽  
pp. 897-900 ◽  
Author(s):  
Peter A. Fillmore
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Type Ii ◽  
Central Projections ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Principal Tool ◽  
The Subject

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

Sign in / Sign up

Export Citation Format

Share Document