scholarly journals On Some Sampling-Related Frames inU-Invariant Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
H. R. Fernández-Morales ◽  
A. G. García ◽  
M. A. Hernández-Medina ◽  
M. J. Muñoz-Bouzo
Keyword(s):  
Unitary Operator ◽  
Separable Hilbert Space ◽  
General Setting ◽  
Continuous Group ◽  
Unitary Operators ◽  
Dual Frames ◽  
Generalized Sampling ◽  
Sampling Problem ◽  
Perturbation Problems ◽  
Invariant Spaces

This paper is concerned with the characterization as frames of some sequences inU-invariant spaces of a separable Hilbert spaceℋwhereUdenotes an unitary operator defined onℋ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces inL2ℝ, where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operatorUto belong to a continuous group of unitary operators.

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 Generalized sampling: From shift-invariant to U-invariant spaces

2015 ◽  
Vol 13 (03) ◽  
pp. 303-329 ◽  
Author(s):  
H. R. Fernández-Morales ◽  
A. G. García ◽  
M. A. Hernández-Medina ◽  
M. J. Muñoz-Bouzo
Keyword(s):  
Invariant Subspaces ◽  
Invertible Operator ◽  
Continuous Group ◽  
Sampling Frame ◽  
Generalize Convolution ◽  
Time Jitter ◽  
Generalized Sampling ◽  
Frame Expansions ◽  
Invariant Spaces ◽  
Convolution Systems

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.

scholarly journals Understanding mathematics of Grover’s algorithm

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Phase Change ◽  
Unitary Operator ◽  
Mathematical Structure ◽  
Complex Numbers ◽  
Unitary Operators ◽  
Grover’S Algorithm ◽  
Great Probability ◽  
Grover's Algorithm ◽  
One Step ◽  
The One

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

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].

scholarly journals Some notes on complex symmetric operators

2021 ◽  
Vol 2021 (1) ◽  
pp. 90-96
Author(s):  
Marcos S. Ferreira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Bounded Operator ◽  
Complex Symmetric ◽  
Symmetric Operators

Abstract In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * 𝒥T, where T is an unitary operator and 𝒥 f ( z ) = f ( z ¯ ) ¯ \mathcal{J}f\left( z \right) = \overline {f\left( {\bar z} \right)} with f ∈ H 2. Moreover we prove some relations of complex symmetry between the operators T and |T|, where T = U |T| is the polar decomposition of bounded operator T ∈ ℒ(ℋ) on the separable Hilbert space ℋ.

Genericity of Certain Classes of Unitary and Self-Adjoint Operators

1998 ◽  
Vol 41 (2) ◽  
pp. 137-139 ◽  
Author(s):  
J. R. Choksi ◽  
M. G. Nadkarni
Keyword(s):  
Hilbert Space ◽  
Conference Proceedings ◽  
Separable Hilbert Space ◽  
Strong Operator Topology ◽  
Slight Modification ◽  
Simple Spectrum ◽  
Cayley Transform ◽  
Unitary Operators ◽  
Full Support ◽  
Adjoint Operators

AbstractIn a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, forma dense Gδ. A similar theoremfor bounded self-adjoint operators with a given normbound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added.

scholarly journals REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES

2019 ◽  
Vol 36 (5) ◽  
pp. 773-802 ◽  
Author(s):  
Brendan K. Beare ◽  
Won-Ki Seo
Keyword(s):  
Separable Hilbert Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Operator Pencil ◽  
Second Order ◽  
Vector Autoregressive ◽  
Functional Time Series ◽  
Vector Autoregressive Processes ◽  
Necessary And Sufficient ◽  
Autoregressive Hilbertian Processes

We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

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