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 Existence of a bounded function of the maximal spectral type

1982 ◽  
Vol 2 (3-4) ◽  
pp. 259-261 ◽  
Author(s):  
V. M. Alexeyev
Keyword(s):  
Hilbert Space ◽  
Spectral Type ◽  
Measure Space ◽  
Bounded Function

Let us denote by the Hilbert space of all functions on the measure space Ω, square-integrable with respect to the measure μ.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

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 Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

On perturbed stochastic discrete systems

1974 ◽  
Vol 11 (3) ◽  
pp. 385-393 ◽  
Author(s):  
B.G. Pachpatte
Keyword(s):  
Asymptotic Behavior ◽  
Probability Measure ◽  
Measure Space ◽  
Discrete System ◽  
Discrete Systems ◽  
Image Position ◽  
Probability Measure Space

The object of this paper is to study a stochastic discrete system, including an operator T, of the formas a perturbation of the linear stochastic discrete systemwhere ω ∈ Ω, the supporting set of probability measure space (Ω, A, P) and n ∈ N, the set of nonnegative integers. We are concerned vith the existence, uniqueness, boundedness, and asymptotic behavior of random solutions of the above equation.

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 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 Notions of relative ubiquity for invariant sets of relational structures

1990 ◽  
Vol 55 (3) ◽  
pp. 948-986 ◽  
Author(s):  
Paul Bankston ◽  
Wim Ruitenburg
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
Measure Space ◽  
Invariant Sets ◽  
Natural Numbers ◽  
Compact Metric Space ◽  
Linear Orderings ◽  
Relational Structures ◽  
Probability Measure Space

AbstractGiven a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

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.

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