Trace class integral kernels

1990 ◽  
pp. 61-64 ◽  
Author(s):  
Chris Brislawn
Keyword(s):  
Trace Class

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

Trace-Class for an Arbitrary H ∗ -Algebra

1970 ◽  
Vol 26 (1) ◽  
pp. 95 ◽  
Author(s):  
Parfeny P. Saworotnow ◽  
John C. Friedell
Keyword(s):  
Trace Class

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.

scholarly journals Fisher information for inverse problems and trace class operators

2012 ◽  
Vol 53 (12) ◽  
pp. 123503 ◽  
Author(s):  
S. Nordebo ◽  
M. Gustafsson ◽  
A. Khrennikov ◽  
B. Nilsson ◽  
J. Toft
Keyword(s):  
Inverse Problems ◽  
Fisher Information ◽  
Trace Class ◽  
Trace Class Operators

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

Trace Class Elements and Cross-Sections in Kac-Moody Groups

1998 ◽  
Vol 50 (5) ◽  
pp. 972-1006 ◽  
Author(s):  
Gerd Brüchert
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Trace Class ◽  
Irreducible Representations ◽  
Functional Identity ◽  
Trace Class Operators ◽  
Regular Classes

AbstractLet G be an affine Kac-Moody group, π0, … ,πr, πδ its fundamental irreducible representations and χ0, … , χr, χδ their characters. We determine the set of all group elements x such that all πi(x) act as trace class operators, i.e., such that χi(x) exists, then prove that the χ i are class functions. Thus, χ := (χ0, … , χr, χδ) factors to an adjoint quotient χ for G. In a second part, following Steinberg, we define a cross-section C for the potential regular classes in G. We prove that the restriction χ|C behaves well algebraically. Moreover, we obtain an action of C ℂ✗ on C, which leads to a functional identity for χ|C which shows that χ|C is quasi-homogeneous.

On the Uniqueness of Wave Operators Associated with Non-Trace Class Perturbations

2004 ◽  
Vol 47 (1) ◽  
pp. 144-151
Author(s):  
Jingbo Xia
Keyword(s):  
Wave Operator ◽  
Trace Class ◽  
Uniqueness Result ◽  
Orthogonal Projections ◽  
Wave Operators ◽  
Adjoint Operators ◽  
Norm Ideal

AbstractVoiculescu has previously established the uniqueness of the wave operator for the problem of -perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal has the property , where {Pn} is any sequence of orthogonal projections with rank(Pn) = n. We prove that the same uniqueness result holds true so long as is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

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