scholarly journals A trace class operator inequality

2007 ◽  
Vol 328 (2) ◽  
pp. 1484-1486 ◽  
Author(s):  
Lei Liu
Keyword(s):  
Trace Class ◽  
Trace Class Operator ◽  
Operator Inequality ◽  
Class Operator

E-integrability of trace-class operator-valued functions

Positivity ◽  
2020 ◽  
Author(s):  
Minoo Khosheghbal-Ghorabayi ◽  
Ghorban-Ali Bagheri-Bardi
Keyword(s):  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator

scholarly journals Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

2008 ◽  
Vol 2008 ◽  
pp. 1-17
Author(s):  
Adrian P. C. Lim
Keyword(s):  
Integral Operator ◽  
Diffusion Process ◽  
Fredholm Determinant ◽  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator ◽  
Locally Lipschitz ◽  
Positive Matrices ◽  
The Law

This article aims to give a formula for differentiating, with respect to , an expression of the form , where and is a diffusion process starting from , taking values in a manifold, and the expectation is taken with respect to the law of this process. is a trace class operator defined by , where , are locally Lipschitz, positive matrices.

scholarly journals On zero-trace commutators

1986 ◽  
Vol 34 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Adjoint Operator ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Trace Class Operator ◽  
Class Operator ◽  
Closed Disc ◽  
Schmidt Operator

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

scholarly journals Gram matrix associated to controlled frames

2018 ◽  
Vol 16 (05) ◽  
pp. 1850035 ◽  
Author(s):  
E. Osgooei ◽  
A. Rahimi
Keyword(s):  
Hilbert Spaces ◽  
Diagnostic Tool ◽  
Riesz Basis ◽  
Trace Class ◽  
Trace Class Operator ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Frame Operator ◽  
Class Operator ◽  
Interactive Algorithms

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every [Formula: see text]-controlled Riesz basis [Formula: see text] is in the form [Formula: see text], where [Formula: see text] is a bijective operator on [Formula: see text]. Furthermore, we propose an equivalent accessible condition to the sequence [Formula: see text] being a [Formula: see text]-controlled Riesz basis.

On the Trace of a Trace Class Operator

1974 ◽  
Vol 6 (1) ◽  
pp. 47-50 ◽  
Author(s):  
J. A. Erdös
Keyword(s):  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator

Trace-Class Operators in CSL Algebras

1992 ◽  
Vol 35 (3) ◽  
pp. 416-422 ◽  
Author(s):  
Shlomo Rosenoer
Keyword(s):  
Distributive Lattice ◽  
Elementary Proof ◽  
Strong Operator Topology ◽  
Trace Class ◽  
Subspace Lattice ◽  
Completely Distributive ◽  
Class Operator ◽  
Rank One ◽  
Trace Class Operators ◽  
Completely Distributive Lattice

AbstractIn this note we show that if 𝓛 is a commutative subspace lattice, then every trace-class operator in Alg 𝓛 lies in the norm-closure of the span of rank-one operators in Alg 𝓛. We also give an elementary proof of a recent result of Davidson and Pitts that if 𝓛 is a CSL generated by completely distributive lattice and finitely many commuting chains, then 𝓛 is compact in the strong operator topology if and only if 𝓛 is completely distributive.

scholarly journals On the characters of unitary representations

1988 ◽  
Vol 45 (1) ◽  
pp. 62-65
Author(s):  
Michael Cowling
Keyword(s):  
Linear Functional ◽  
Trace Class ◽  
Locally Compact Group ◽  
Unitary Representations ◽  
Semisimple Lie Groups ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Convolution Algebra ◽  
Class Operator ◽  
Dense Subalgebra

AbstractLet G be a locally compact group, and let D(G) be a dense subalgebra of the convolution algebra L1(G). Suppose that π is a unitary representation of G and that, for each u in D(G), π(u)) is a trace-class operator. Then the linear functional u → tr(π(u)) (the trace of π(u)) is called the D-character of π. We give a simple proof that the D-character of such a representation determines the representation up to unitary equivalence. As an application, we give an easy proof of the result of Harish-Chandra that the K-finite characters of unitary representations of semisimple Lie groups determine the representations.

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.

scholarly journals An operator inequality and self-adjointness

2004 ◽  
Vol 377 ◽  
pp. 181-194 ◽  
Author(s):  
Bojan Magajna ◽  
Marko Petkovšek ◽  
Aleksej Turnšek
Keyword(s):  
Operator Inequality
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