scholarly journals Lower Semi-frames, Frames, and Metric Operators

2020 ◽  
Vol 18 (1) ◽  
Author(s):  
J.-P. Antoine ◽  
R. Corso ◽  
C. Trapani
Keyword(s):  
Hilbert Space ◽  
Measurable Function ◽  
Adjoint Operator ◽  
Necessary Condition ◽  
Frame Operator ◽  
Parseval Frame ◽  
Lower Semi ◽  
Metric Operator ◽  
Generalized Frame ◽  
Continuous Frame

AbstractThis paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

scholarly journals The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180264 ◽  
Author(s):  
David Krejčiřík ◽  
Vladimir Lotoreichik ◽  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Adjoint Operator ◽  
Inner Product ◽  
Sufficient Condition ◽  
Lagrange Equations ◽  
Entire Class ◽  
Inner Products ◽  
Metric Operator ◽  
A Charge

We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all. In the former case, we derive a system of Euler–Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

scholarly journals Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 471 ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Quantum Systems ◽  
Time Dependent ◽  
Inner Product ◽  
Geometric Framework ◽  
Heisenberg Picture ◽  
Metric Operator ◽  
Recent Developments

A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

scholarly journals Self-Adjoint Extensions with Friedrichs Lower Bound

2020 ◽  
Vol 14 (7) ◽  
Author(s):  
Matteo Gallone ◽  
Alessandro Michelangeli
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Symmetric Operator ◽  
Simple Criterion ◽  
Abstract Result ◽  
Friedrichs Extension ◽  
Lower Semi

Abstract We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

scholarly journals Pseudo-Hermitian quantum mechanics with unbounded metric operators

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120050 ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Discrete Spectrum ◽  
Metric Operator ◽  
Hamiltonian Operators

I extend the formulation of pseudo-Hermitian quantum mechanics to η + -pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η + . In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η + and consequently .

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.

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