ON THE DOMAIN OF SINGULAR TRACES

2002 ◽  
Vol 13 (06) ◽  
pp. 667-674 ◽  
Author(s):  
DANIELE GUIDO ◽  
TOMMASO ISOLA
Keyword(s):  
Positive Operator ◽  
Trace Class ◽  
Regular Operator ◽  
Infinite Rank

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are genrated by a "regular" operator, the answers change, namely such traces alway vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor and make use of a new characterization of singular traceability (cf. [7]).

scholarly journals Dunford-Pettis and strongly Dunford-Pettis operators on L1(μ)

1989 ◽  
Vol 31 (1) ◽  
pp. 49-57 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Positive Operator ◽  
Finite Measure ◽  
Regular Operator ◽  
Mathematical Economics ◽  
Canonical Injection ◽  
Do So ◽  
Convergent Sequences

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.

On Extending the Trace as a Linear Functional, II

1980 ◽  
Vol 32 (6) ◽  
pp. 1288-1298
Author(s):  
George A. Elliott
Keyword(s):  
Positive Operator ◽  
Selfadjoint Operator ◽  
Linear Functional ◽  
Orthonormal Basis ◽  
Unitary Operator ◽  
Trace Class ◽  
Diagonal Matrix ◽  
Matrix Elements ◽  
Von Neumann ◽  
The Difference

A positive bounded selfadjoint operator is in the trace class of von Neumann and Schatten ([4]) if the sum of its diagonal matrix elements with respect to some orthonormal basis is finite, and the trace is then defined to be this sum, which is independent of the basis. A bounded selfadjoint but not necessarily positive operator x is in the trace class if in the decomposition x = x+ – x−, with x+ and x− positive and x+x− = 0, both x+ and x−are in the trace class; the trace of x is then defined to be the difference of the finite traces of x+ and x−. The trace defined in this way is a linear functional on the trace class, and is unitarily invariant; if u is a unitary operator, the trace of uxu−1 is the same as the trace of x.

scholarly journals An inverse spectral problem for a star graph of Krein strings

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Jonathan Eckhardt
Keyword(s):  
Spectral Problem ◽  
Spectral Data ◽  
Inverse Spectral Problem ◽  
Trace Class ◽  
Star Graph ◽  
Borel Measures ◽  
Inverse Spectral ◽  
The Individual

AbstractWe solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.

scholarly journals Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements

2011 ◽  
Vol 83 (5) ◽  
Author(s):  
Z. E. D. Medendorp ◽  
F. A. Torres-Ruiz ◽  
L. K. Shalm ◽  
G. N. M. Tabia ◽  
C. A. Fuchs ◽  
...  
Keyword(s):  
Positive Operator ◽  
Experimental Characterization

scholarly journals A note on Dunford-Pettis operators

1987 ◽  
Vol 29 (2) ◽  
pp. 271-273 ◽  
Author(s):  
J. R. Holub
Keyword(s):  
Linear Operator ◽  
Positive Operator ◽  
Continuous Linear Operator ◽  
Positive Operators ◽  
Regular Operator ◽  
Continuous Linear

Talagrand has shown [4, p. 76] that there exists a continuous linear operator from L1[0, 1] to c0 which is not a Dunford-Pettis operator. In contrast to this result, Gretsky and Ostroy [2] have recently proved that every positive operator from L[0, 1] to c0 is a Dunford-Pettis operator, hence that every regular operator between these spaces (i.e. a difference of positive operators) is Dunford-Pettis.

Experimental characterization of qutrits using symmetric, informationally complete positive operator-valued measures

2011 ◽  
Author(s):  
Z. E. D. Medendorp ◽  
F. A. Torres-Ruiz ◽  
L. K. Shalm ◽  
G. N. M. Tabia ◽  
C. A. Fuchs ◽  
...  
Keyword(s):  
Positive Operator ◽  
Experimental Characterization ◽  
Positive Operator Valued Measures

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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