Covariant loops and strings in a positive definite Hilbert space

1977 ◽  
Vol 37 (3) ◽  
pp. 242-265 ◽  
Author(s):  
F. Rohrlich
Keyword(s):  
Hilbert Space ◽  
Positive Definite

Positive Definite Sequence of Operators and a Fixed Point Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 295-295
Author(s):  
A. T. Dash
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Positive Definite ◽  
Definite Sequence ◽  
Positive Definite Sequence ◽  
Image Position

The purpose of this note is to prove the following:Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite iffor every finitely nonzero sequence {fn} of vectors in H [1].

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

scholarly journals Correct selfadjoint and positive extensions of nondensely defined minimal symmetric operators

2005 ◽  
Vol 2005 (7) ◽  
pp. 767-790 ◽  
Author(s):  
I. Parassidis ◽  
P. Tsekrekos
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Selfadjoint Extension ◽  
Symmetric Operators ◽  
Selfadjoint Extensions ◽  
Positive Extensions

LetA0be a closed, minimal symmetric operator from a Hilbert spaceℍintoℍwith domain not dense inℍ. LetA^also be a correct selfadjoint extension ofA0. The purpose of this paper is (1) to characterize, with the help ofA^, all the correct selfadjoint extensionsBofA0with domain equal toD(A^), (2) to give the solution of their corresponding problems, (3) to find sufficient conditions forBto be positive (definite) whenA^is positive (definite).

SUPERSYMMETRY IN 2+2 DIMENSIONS

2000 ◽  
Vol 15 (21) ◽  
pp. 1349-1355 ◽  
Author(s):  
F. T. BRANDT ◽  
D. G. C. MCKEON ◽  
T. N. SHERRY
Keyword(s):  
Hilbert Space ◽  
Positive Definite ◽  
The Other ◽  
Dirac Spinors

The analysis of spinors in 2+2 dimensions is reviewed. The two most basic supersymmetry algebras in this space are considered. In the simpler case, the supersymmetry generators are Majorana spinors. In the other algebra we consider, the supersymmetry generators are Dirac spinors. Consistency with having a positive definite Hilbert space is shown to be impossible in both of these particular cases.

scholarly journals Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

2020 ◽  
Vol 12 (2) ◽  
pp. 289-296
Author(s):  
O.G. Storozh
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Differential Operators ◽  
Boundary Value ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Multivalued Operator ◽  
Linear Transformations ◽  
Accretive Extension

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

scholarly journals On first order wave equations for elementary particles without subsidiary conditions

1947 ◽  
Vol 191 (1025) ◽  
pp. 253-268 ◽  
Keyword(s):  
Hilbert Space ◽  
Total Energy ◽  
Wave Equations ◽  
Positive Definite ◽  
Elementary Particles ◽  
Indefinite Metric ◽  
First Order ◽  
Fermi Statistics

The possibility of extending the application of the formalism used for the electron and meson fields, as developed from the equation ∂ k β k ψ ═ ik ψ to other types of field, is investigated. It is shown that, subject to the conditions that (1) the equations must contain no subsidiary conditions, (2) either the total energy must be positive definite, or it must be possible to quantize the equations according to Fermi statistics without using an indefinite metric in Hilbert space, no such extension is possible.

Positive definite functions on spheres

1973 ◽  
Vol 73 (1) ◽  
pp. 145-156 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Hilbert Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Distance ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Necessary Condition ◽  
Unit Hypersphere

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

scholarly journals The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space

2010 ◽  
Vol 2010 ◽  
pp. 1-7
Author(s):  
S. J. Aneke
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Method Of Moments ◽  
Iterative Scheme ◽  
Inverse Function ◽  
Positive Definite ◽  
Inverse Function Theorem ◽  
Definite Operator ◽  
Fréchet Differentiable ◽  
Positive Definite Operator

The equation , where , with being a K-positive definite operator and being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point , where is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.

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