Generalised Trapezoid-Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces

2014 ◽  
Vol 12 (3) ◽  
pp. 573-591 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Unit Circle ◽  
Hilbert Spaces ◽  
Unitary Operators ◽  
Complex Functions

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.

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.

scholarly journals The spectra of compact operators in Hilbert spaces

1965 ◽  
Vol 7 (1) ◽  
pp. 34-38
Author(s):  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Real Axis ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Conformal Mappings ◽  
The Real ◽  
Finite Set

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

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).

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