Equations in Hilbert Spaces Whose Sets of Solutions are Invariant Under a Group Isomorphic to a One-Parameter Group of Unitary Operators

2020 ◽  
Vol 72 (1) ◽  
pp. 98-113
Author(s):  
V. Yu. Slyusarchuk
Keyword(s):  
Hilbert Spaces ◽  
Unitary Operators ◽  
Parameter Group

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

scholarly journals Mutually unbiased unitary bases of operators on d-dimensional hilbert space

2020 ◽  
Vol 18 (01) ◽  
pp. 1941026 ◽  
Author(s):  
Rinie N. M. Nasir ◽  
Jesni Shamsul Shaari ◽  
Stefano Mancini
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Unitary Operator ◽  
Dimensional Subspace ◽  
Mutually Unbiased Bases ◽  
Unitary Operators ◽  
Dimension Formula ◽  
Dimensional Hilbert Space ◽  
Prime Dimension

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

SOME INEQUALITIES OF JENSEN TYPE FOR ARG-SQUARE CONVEX FUNCTIONS OF UNITARY OPERATORS IN HILBERT SPACES

2014 ◽  
Vol 90 (1) ◽  
pp. 65-73
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Unitary Operators

AbstractSome inequalities of Jensen type for Arg-square-convex functions of unitary operators in Hilbert spaces are given.

UNITARITY OF KUO'S FOURIER–MEHLER TRANSFORM

2004 ◽  
Vol 07 (01) ◽  
pp. 147-154 ◽  
Author(s):  
UN CIG JI ◽  
NOBUAKI OBATA
Keyword(s):  
Automorphism Group ◽  
White Noise ◽  
Gaussian Measure ◽  
Unitary Operators ◽  
Parameter Group

Kuo's Fourier–Mehler transforms {ℱΘ} form a one-parameter automorphism group of the space of white noise distributions [Formula: see text]. The adjoint transforms [Formula: see text] form a one-parameter automorphism group of the space of test white noise functions [Formula: see text] but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that [Formula: see text] admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

2007 ◽  
Vol 10 (02) ◽  
pp. 261-276 ◽  
Author(s):  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Separable Hilbert Space ◽  
Fock Spaces ◽  
Parameter Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

scholarly journals Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 174
Author(s):  
Koen Thas
Keyword(s):  
Quantum Theory ◽  
Finite Fields ◽  
Hilbert Spaces ◽  
Inner Product ◽  
Unitary Operators ◽  
Evolution Operators ◽  
Quantum Theories ◽  
Classical Quantum ◽  
Absolute Quantum ◽  
Made In

In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).

Trapezoid type inequalities for complex functions defined on the unit circle with applications for unitary operators in Hilbert spaces

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Sever S. Dragomir
Keyword(s):  
Unit Circle ◽  
Hilbert Spaces ◽  
Stieltjes Integral ◽  
Unitary Operators ◽  
Complex Functions ◽  
Continuous Complex ◽  
Complex Valued

AbstractSome trapezoid type inequalities for the Riemann–Stieltjes integral of continuous complex-valued integrands defined on the complex unit circle

scholarly journals Grüss Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces

2015 ◽  
Vol 49 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Unit Circle ◽  
Hilbert Spaces ◽  
Unitary Operators ◽  
Complex Functions ◽  
Grüss Type Inequalities

Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integrandos de valores continuos complejos definidos sobre el circulo unitario complejo C(0, 1) y varias subclases de integradores son dados. Aplicaciones naturales para funciones de operadores unitarios en espacios de Hilbert son proporcionadas.

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