On the Spectra of Unbounded Subnormal Operators

1986 ◽  
Vol 38 (5) ◽  
pp. 1135-1148 ◽  
Author(s):  
G. McDonald ◽  
C. Sundberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Line ◽  
Bounded Operator ◽  
Bounded Operators ◽  
The Real ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Image Position ◽  
Topological Boundary

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Fractional Part ◽  
Positive Definite ◽  
Its Sequence ◽  
Bounded Operators ◽  
Linear Bounded Operator

Abstract In 1993, Alcantara-Bode showed ([2]) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel function defined by the fractional part of (y/x), isinjective. Since then, the injectivity of the integral operator used inequivalent formulation of RH has not been addressed nor has beendissociated from RH.We provided in this paper methods for investigating the injectivityof linear bounded operators on separable Hilbert spaces using theirapproximations on dense families of subspaces.On the separable Hilbert space L2(0,1), an linear bounded operator(or its associated Hermitian), strict positive definite on a dense familyof including approximation subspaces in built on simple functions, isinjective if the rate of convergence of its sequence of injectivity pa-rameters on approximation subspaces is inferior bounded by a not nullconstant, that is the case with the Beurling - Alcantara-Bode integraloperator.We applied these methods to the integral operator used in RHequivalence proving its injectivity.

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

scholarly journals A characterization of subnormal operators

1984 ◽  
Vol 25 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Commutative Algebras ◽  
Binomial Expansion ◽  
Subnormal Operators

In this note a characterization of subnormality of operators on Hilbert space is given. The characterization is in terms of a sequence of polynomials in the operator and its adjoint reminiscent of the binomial expansion in commutative algebras. As such no external Hilbert spaces are needed, nor is it necessary to introduce forms dependent on arbitrary sequences of vectors from the Hilbert space.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

scholarly journals Factorization of hyponormal operators

1972 ◽  
Vol 13 (3) ◽  
pp. 323-326
Author(s):  
Mary R. Embry
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyponormal Operators ◽  
Image Position ◽  
Continuous Linear Operators

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

scholarly journals Zeros of Nonlinear Monotone Operators in Hilbert Space*

1978 ◽  
Vol 21 (2) ◽  
pp. 213-219 ◽  
Author(s):  
R. Schöneberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Russian Mathematician ◽  
Image Position

Around 1960, the Russian mathematician Kachurovski [1] introduced the notion of monotone operators in Hilbert spaces: Let E be a Hilbert space and X ⊂ E. An operator T:X→E is said to be monotone, iff.

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.

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