scholarly journals Lorentz-Marcinkiewicz property of direct sum of operators

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 391-398
Author(s):  
Ala Ipek
Keyword(s):  
Hilbert Spaces ◽  
Direct Sum

In this paper, the relations between Lorentz-Marcinkiewicz property of the direct sum of operators in the direct sum of Hilbert spaces and its coordinate operators are studied. Then, the obtained results are supported by applications.

scholarly journals Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hadamard Product ◽  
Operator Matrices ◽  
Linear Map ◽  
Direct Sum ◽  
Positive Linear Map ◽  
Algebraic Properties ◽  
Product Of Matrices

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

scholarly journals TWO-INTERVAL EVEN-ORDER DIFFERENTIAL OPERATORS IN MODIFIED HILBERT SPACES

2011 ◽  
Vol 85 (2) ◽  
pp. 241-260
Author(s):  
JIANQING SUO ◽  
WANYI WANG
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Order Equation ◽  
Inner Product ◽  
Coupling Matrix ◽  
Direct Sum ◽  
Even Order

AbstractBy modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the ‘unmodified’ theory.

scholarly journals Inverse resonance scattering on rotationally symmetric manifolds

2020 ◽  
pp. 1-17
Author(s):  
Hiroshi Isozaki ◽  
Evgeny Korotyaev
Keyword(s):  
Hilbert Spaces ◽  
Counting Function ◽  
Resonance Scattering ◽  
Compact Interval ◽  
Large Radius ◽  
One Dimensional ◽  
Symmetric Manifold ◽  
Direct Sum ◽  
Rotationally Symmetric ◽  
Real Analytic

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = ( 0 , ∞ ) × Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove Asymptotics of counting function of resonances at large radius. The rotation radius is uniquely determined by its eigenvalues and resonances. There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.

BRAID STATISTICS FOR THE SOURCES OF THE NONABELIAN CHERN-SIMONS TERM

1990 ◽  
Vol 05 (12) ◽  
pp. 2423-2470 ◽  
Author(s):  
A. P. BALACHANDRAN ◽  
M. BOURDEAU ◽  
S. JO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Point Particle ◽  
Yang Mills ◽  
Direct Sum ◽  
Identical Point ◽  
Effective Lagrangian ◽  
All Solutions ◽  
Chern Simons

A Lagrangian consisting of an Abelian Chern-Simons term and N identical point particle sources is known to lead to fractional statistics for the sources. In this paper, we investigate the non-Abelian generalization of this system with special emphasis on source statistics. All solutions for the Yang-Mills potential in the presence of identical or nonidentical sources are found. For two or more sources, they fall in many gauge inequivalent classes whereas in the Abelian problem, there is only one such class. An effective Lagrangian for N sources is found for each of these solutions. The quantum mechanics and statistics of the sources are sensitive to the potential leading to the effective Lagrangian. There is for instance, a class of solutions for identical sources which are not invariant under exchange of sources. For these solutions, the identity of the sources obliges us to consider such a potential and all its exchange transforms at the same time, and to introduce a Hilbert space of states which is the direct sum of the Hilbert spaces associated with each of these potentials. There are also exchange invariant potentials for identical sources. For SU(3) and N = 3, all exchange invariant potentials are shown to lead to statistics defined by S3 representations. The nature of statistics for SU (M) for higher M as also the creation of intrinsic spin by self interaction are briefly considered.

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