scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set𝒮r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the𝒮r(ℬ)setting.

scholarly journals Schur Algebras overC*-Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong ◽  
Sheng-Wang Wang
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Trace Class ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Schur Algebras ◽  
Product Operation ◽  
The Matrix ◽  
Trace Class Operators

Let𝒜be aC*-algebra with identity1, and lets(𝒜)denote the set of all states on𝒜. Forp,q,r∈[1,∞), denote by𝒮r(𝒜)the set of all infinite matricesA=[ajk]j,k=1∞over𝒜such that the matrix(ϕ[A[2]])[r]:=[(ϕ(ajk*ajk))r]j,k=1∞defines a bounded linear operator fromℓptoℓqfor allϕ∈s(𝒜). Then𝒮r(𝒜)is a Banach algebra with the Schur product operation and norm‖A‖=sup{‖(ϕ[A[2]])r‖1/(2r):ϕ∈s(𝒜)}. Analogs of Schatten's theorems on dualities among the compact operators, the trace-class operators, and all the bounded operators on a Hilbert space are proved.

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

Range Inclusion for Multilinear Mappings: Applications

1985 ◽  
Vol 28 (3) ◽  
pp. 317-320
Author(s):  
C. K. Fong
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Trace Class ◽  
Inner Derivation ◽  
Finite Rank Operators ◽  
Multilinear Mappings ◽  
Trace Class Operators ◽  
The Ideal

AbstractThe result of S. Grabiner [5] on range inclusion is applied for establishing the following two theorems: 1. For A, B ∊ L(H), two operators on the Hilbert space H, we have DBC0(H) ⊆ DAL(H) if and only if DBC1(H) ⊆ DAL(H), where DA is the inner derivation which sends S ∊ L(H) to AS - SA, C1(H) is the ideal of trace class operators and C0(H) is the ideal of finite rank operators. 2. (Due to Fialkow [3]) For A, B ∊ L(H), we write T(A, B) for the map on L(H) sending S to AS - SB. Then the range of T(A, B)is the whole L(H) if it includes all finite rank operators L(H).

scholarly journals Decomposition algebras of Riesz operators

1980 ◽  
Vol 21 (1) ◽  
pp. 75-79 ◽  
Author(s):  
G. J. Murphy ◽  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Spectral Radius ◽  
Compact Operators ◽  
Linear Operators ◽  
Closed Ideal ◽  
Canonical Mapping ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators

Let H be a Hilbert space and let B denote the Banach algebra of all bounded linear operators on H with K denoting the closed ideal of compact operators in B. If T ∈ B, σ(T) and r(T) will denote the spectrum and spectral radius of T, respectively, and π the canonical mapping of B onto the Calkin algebra B/K.

Strong Extensions vs. Weak Extensions of C*-Algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 143-147
Author(s):  
S. J. Cho
Keyword(s):  
Hilbert Space ◽  
Partial Isometry ◽  
Compact Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Image Position ◽  
The Ideal

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such thatfor all a in A.

COVARIANCE-PARAMETER LÉVY PROCESSES IN THE SPACE OF TRACE-CLASS OPERATORS

2005 ◽  
Vol 08 (01) ◽  
pp. 33-54 ◽  
Author(s):  
VÍCTOR PÉREZ-ABREU ◽  
ALFONSO ROCHA-ARTEAGA
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Lévy Processes ◽  
Trace Class ◽  
Levy Processes ◽  
Trace Class Operators

The paper deals with Lévy processes with values in L1(H), the Banach space of trace-class operators in a Hilbert space H. Lévy processes with values and parameter in a cone K of L1(H) are introduced and several properties are established. A family of L1(H)-valued Lévy processes is obtained via the subordination of K-parameter, L1(H)-valued Lévy processes, identifying explicitly their generating triplets.

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),

PROBABILITY STRUCTURES FOR QUANTUM STATE SPACES

1995 ◽  
Vol 07 (07) ◽  
pp. 1105-1121 ◽  
Author(s):  
PAUL BUSCH ◽  
GIANNI CASSINELLI ◽  
PEKKA J. LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum State ◽  
Positive Operator ◽  
Trace Class ◽  
Measurable Space ◽  
Probability Measures ◽  
State Spaces ◽  
Trace Class Operators ◽  
Positive Operator Valued Measure

The theme of this paper is to represent the states of a quantum system by means of probability measures. We fix a positive operator valued measure E on a measurable space (Ω, ℬ(Ω)) acting in a Hilbert space ℋ, and we study the properties of the mapping that it induces from the set of trace class operators on ℋ to the set of measures on (Ω, ℬ(Ω)). In particular, the injectivity and the surjectivity of this map are characterised in terms of the properties of E.

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