scholarly journals On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
M. De la Sen
Keyword(s):  
Fixed Points ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Finite Range ◽  
Worst Case ◽  
Orthonormal Bases ◽  
Finite Dimensional

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

scholarly journals Quantum ergodicity of random orthonormal bases of spaces of high dimension

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120511 ◽  
Author(s):  
Steve Zelditch
Keyword(s):  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Compact Riemannian Manifold ◽  
Random Sequence ◽  
Limit State ◽  
Flat Torus ◽  
Quantum Ergodicity ◽  
Orthonormal Bases ◽  
Finite Dimensional ◽  
Unique Limit

We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S 2 . In particular, it holds on the flat torus if d ≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

scholarly journals A new description of orthogonal bases

2012 ◽  
Vol 23 (3) ◽  
pp. 555-567 ◽  
Author(s):  
BOB COECKE ◽  
DUSKO PAVLOVIC ◽  
JAMIE VICARY
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Orthonormal Bases ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Categorical Structure ◽  
Categorical Quantum Mechanics ◽  
Composition Of Operations ◽  
Basis Vectors ◽  
Finite Dimensional Hilbert Space

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.

scholarly journals On Mann’s Method with Viscosity for Nonexpansive and Nonspreading Mappings in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Nawab Hussain ◽  
Giuseppe Marino ◽  
Afrah A. N. Abdou
Keyword(s):  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Common Fixed Points ◽  
Nonspreading Mapping ◽  
Type Mapping ◽  
Nonspreading Mappings

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

Best approximations and fixed points of nonexpansive maps in hilbert spaces sehie park

1997 ◽  
Vol 18 (5-6) ◽  
pp. 447-454 ◽  
Author(s):  
Sehie Park ◽  
Sehie Park ◽  
S. p. Singh ◽  
B. Watson ◽  
T. E. Williamson
Keyword(s):  
Fixed Points ◽  
Hilbert Spaces ◽  
Best Approximations ◽  
Nonexpansive Maps

ON A CLASS OF FOURTH ORDER NONLINEAR ELLIPTIC EQUATIONS UNDER NAVIER BOUNDARY CONDITIONS

2010 ◽  
Vol 08 (02) ◽  
pp. 185-197 ◽  
Author(s):  
F. J. S. A. CORRÊA ◽  
J. V. GONCALVES ◽  
ANGELO RONCALLI
Keyword(s):  
Boundary Conditions ◽  
Fixed Points ◽  
Fourth Order ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Compact Operators ◽  
Schmidt Method ◽  
Navier Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

We employ arguments involving continua of fixed points of suitable nonlinear compact operators and the Lyapunov–Schmidt method to prove existence and multiplicity of solutions in a class of fourth order non-homogeneous resonant elliptic problems. Our main result extends even similar ones known for the Laplacian.

GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS

2009 ◽  
Vol 20 (01) ◽  
pp. 45-76
Author(s):  
MAGNUS B. LANDSTAD ◽  
NADIA S. LARSEN
Keyword(s):  
Heisenberg Group ◽  
Hecke Algebra ◽  
Continuous Extension ◽  
Finite Range ◽  
Structure And Properties ◽  
Projection Formula ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
The Given

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra [Formula: see text], which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection [Formula: see text] such that [Formula: see text] is isomorphic to [Formula: see text]. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of [Formula: see text]. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

Sign in / Sign up

Export Citation Format

Share Document