Products of Positive Operators (original) (raw)

Factoring positive operators on reproducing kernel Hilbert spaces

Integral Equations and Operator Theory, 1996

We characterize when positive operators can be factored by "analytic Toeplitz" type operators. As a corollary, we give an operator theory characterization of those invariant subspaees of doubly commuting unilateral shifts, which are generated by a single inner function on the bidisk. The last result extends to shifts of arbitrary (countable) multiplicity.

Positive-Normal Operators in Semi-Hilbertian Spaces

International Mathematical Forum, Vol. 9, 2014, no. 11, 533 - 559, 2014

Given a bounded positive linear operator A on a Hilbert space H we consider the semi-Hilbertian space (H , | A), where ξ | η A := Aξ | η. In this paper we introduce a class of operators on a semi Hilbertian space H with inner product | A. We call the elements of this class A-positive-normal or A-posinormal. An operator T ∈ B(H) is said to be A-posinormal if there exists a A-positive operator P ∈ B(H) (i.e., AP ≥ 0) such that T AT * = T * AP T. We study some basic properties of these operators. Also we study the relationship between a special case of this class with the other kinds of classes of operators in semi-Hilbertian spaces.

Lebesgue-type decomposition of positive operators

Positivity, 2012

The present note is a revision of Ando's work with the same title. We give a new construction for the Lebesgue-decomposition of positive operators on Hilbert spaces with respect to each other. Our approach is similar to that of Kosaki: we use unbounded operator techniques and factorizations via two auxiliary Hilbert spaces associated to the positive operators in question.

On the Spectra of Left-Definite Operators

Complex Analysis and Operator Theory, 2013

If A is a self-adjoint operator that is bounded below in a Hilbert space H; Littlejohn and Wellman showed that, for each r > 0; there exists a unique Hilbert space Hr and a unique self-adjoint operator Ar in Hr satisfying certain conditions dependent on H and A: The space Hr and the operator Ar are called, respectively, the r th left-de…nite space and r th left-de…nite operator associated with (H; A): In this paper, we show that the operators A; Ar; and As (r; s > 0) are isometrically isomorphically equivalent and that the spaces H; Hr; and Hs (r; s > 0) are isometrically isomorphic. These results are then used to reproduce the left-de…nite spaces and left-de…nite operators: Furthermore, we will see that our new results imply that the spectra of A and Ar are equal, giving us another proof of this phenomenon that was …rst established in . words and phrases. right-de…nite operator, left-de…nite operator, left-de…nite space, Hilbert scale, Sobolev space, self-adjoint operator, isometric isomorphism, similarity transformation, point spectrum, continuous spectrum. 1 2 LANCE L. LITTLEJOHN AND RICHARD WELLMAN

A Spectral Characterization of Operators

Journal of the Australian Mathematical Society, 2016

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

Approximation by positive operators

Linear Algebra and its Applications, 1992

If A = B + iC is a normal operator, where B,C are Hermitian, then in each unitarily invariant norm, the positive part of B is a best approximation to A from the class of positive operators. This generalizes results proved earlier by P. R. Halmos, T. Ando, and R. Bouldin for special norms. Some related results are included.

On the composition and decomposition of positive linear operators (VII)

Applicable Analysis and Discrete Mathematics, 2021

In the present paper we study the compositions of the piecewise linear interpolation operator S∆ n and the Beta-type operator Bn, namely An := S∆ n •Bn and Gn := Bn • S∆ n. Voronovskaya type theorems for the operators An and Gn are proved, substantially improving some corresponding known results. The rate of convergence for the iterates of the operators Gn and An is considered. Some estimates of the differences between An, Gn, Bn and S∆ n , respectively, are given. Also, we study the behaviour of the operators An on the subspace of C[0, 1] consisting of all polygonal functions with nodes 0, 1 2 ,. .. , n−1 n , 1. Finally, we propose to the readers a conjecture concerning the eigenvalues of the operators An and Gn. If true, this conjecture would emphasize a new and strong relationship between Gn and the classical Bernstein operator Bn.

On the composition and decomposition of positive linear operators (II)

Studia Scientiarum Mathematicarum Hungarica, 2010

Following a 1939 article of Favard we consider the composition of classical Bernstein operators and piecewise linear interpolation at mutually distinct knots in [0, 1], not necessarily equidistant. We prove direct theorems in terms of the classical and the Ditzian-Totik modulus of second order. 2010 Mathematics Subject Classification: 41A10, 41A15, 41A17, 41A25, 41A36. Key words and phrases: Favard-Bernstein operator, Bernstein polynomials for arbitrary points of interpolation, piecewise linear interpolation, Bernstein operator, second order modulus of smoothness, Ditzian-Totik modulus, positive linear operator, degree of approximation.