scholarly journals Range-Kernel Orthogonality and Elementary Operators: The Nonsmoothness Case

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
A. Bachir ◽  
A. Segres ◽  
Nawal Sayyaf
Keyword(s):  
Trace Class ◽  
Smooth Case ◽  
Counter Example ◽  
Elementary Operators ◽  
Trace Class Operators

The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results.

scholarly journals Norm inequalities for elementary operators and other inner product type integral transformers with the spectra contained in the unit disc

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Danko Jocic ◽  
Stefan Milosevic ◽  
Vladimir Djuric
Keyword(s):  
Unit Disc ◽  
Trace Class ◽  
Inner Product ◽  
Bounded Operators ◽  
Shift Operators ◽  
Hilbert Space Operators ◽  
Type Integral ◽  
Normal Operators ◽  
Elementary Operators ◽  
Trace Class Operators

If {At}t?? and {Bt}t?? are weakly*-measurable families of bounded Hilbert space operators such that transformers X ??? At*XAtd?(t) and X ??? Bt*XBtd?(t) on B(H) have their spectra contained in the unit disc, then for all bounded operators X ||?AX?B|| ? ||X-?? At* XBtd?(t)||, (1) where ?A def= s-limr?1(I + ??,n=1 r2n ??...??|At1...Atn|2d?n(t1,...,tn)-1/2 and ?B by analogy. If additionally ??,n=1 ??n |A*t1...A*tn|2d?n(t1,...,tn) and ?,n=1 ??n|B*t1...B*tn|2d?n(t1,...,tn) both represent bounded operators, then for all p,q,s > 1 such that 1/q + 1/s = 2/p and for all Schatten p trace class operators X ||?1-1q,A X ?1-1s,B|| ? ||?-1/q,A*(X-?? At* XBtd?(t))?-1/s,B*||p.(2) If at least one of those families consists of bounded commuting normal operators, then (1) holds for all unitarily invariant Q-norms. Applications to the shift operators are also given.

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

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.

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.

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 Compactness properties for trace-class operators and applications to quantum mechanics

2008 ◽  
Vol 155 (1) ◽  
pp. 43-66 ◽  
Author(s):  
J. Dolbeault ◽  
P. Felmer ◽  
J. Mayorga-Zambrano
Keyword(s):  
Quantum Mechanics ◽  
Trace Class ◽  
Trace Class Operators
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