Factoring positive operators on reproducing kernel Hilbert spaces (original) (raw)
Related papers
Products of Positive Operators
Complex Analysis and Operator Theory
On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class {\mathcal {L}^{+\,2}}L+2ofboundedoperatorsonseparableinfinitedimensionalHilbertspaceswhichcanbewrittenastheproductoftwoboundedpositiveoperatorsisstudied.Thestructureismuchricher,andconnects(butisnotequivalentto)quasi−similarityandquasi−affinitytoapositiveoperator.ThespectralpropertiesofoperatorsinL + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators inL+2ofboundedoperatorsonseparableinfinitedimensionalHilbertspaceswhichcanbewrittenastheproductoftwoboundedpositiveoperatorsisstudied.Thestructureismuchricher,andconnects(butisnotequivalentto)quasi−similarityandquasi−affinitytoapositiveoperator.Thespectralpropertiesofoperatorsin{\mathcal {L}^{+\,2}}L+2aredeveloped,andmembershipinL + 2 are developed, and membership inL+2aredeveloped,andmembershipin{\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.
Post-composition transforms of totally positive kernels
2020
Author(s): Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai | Abstract: The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitney's density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-nega...
Positive Toeplitz operators on the Bergman space
Annals of Functional Analysis, 2013
In this paper we find conditions on the existence of bounded linear operators A on the Bergman space L 2 a (D) such that A * T φ A ≥ S ψ and A * T φ A ≥ T φ where T φ is a positive Toeplitz operator on L 2 a (D) and S ψ is a self-adjoint little Hankel operator on L 2 a (D) with symbols φ, ψ ∈ L ∞ (D) respectively. Also we show that if T φ is a non-negative Toeplitz operator then there exists a rank one operator R 1 on L 2 a (D) such that φ(z) ≥ α 2 R 1 (z) for some constant α ≥ 0 and for all z ∈ D where φ is the Berezin transform of T φ and R 1 (z) is the Berezin transform of R 1 .
Proceedings of The Edinburgh Mathematical Society, 2008
We construct and examine an operator space X, isometric to 2 , such that every completely bounded map from its subspace Y into X is a compact perturbation of a linear combination of multiples of a shift of given multiplicity and their adjoints. Moreover, every completely bounded map on X is a Hilbert-Schmidt perturbation of such a linear combination.
Operators That Commute with Slant Toeplitz Operators
Applied Mathematics Research eXpress, 2010
Let H be a separable Hilbert space and {e n : n ∈ Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if T e j , e i = c 2i− j , where c n is the nth Fourier coefficient of a bounded Lebesgue measurable function ϕ on the unit circle T = {z ∈ C : |z| = 1}. It has been shown [9], with some assumption on the smoothness and the zeros of ϕ, that T * is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. These results, together with the theory of shifts (e.g., in [11]), allows us to identify all bounded operators on H commuting with such T. 1 Introduction Let H be a separable Hilbert space and E = {e n : n ∈ Z} be an orthonormal basis in H. Consider a problem of the following simple form: Characterize all bounded operators on H that commute with the bounded operator whose matrix is
Invariant subspaces for positive operators on Banach spaces with unconditional basis
2022
We prove that every lattice homomorphism acting on a Banach space X with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal, which is no longer true for every positive operator on such a space. Motivated by these later examples, we characterize tridiagonal positive operators without nontrivial closed invariant ideals on X extending to this context a result of Grivaux on the existence of non-trivial closed invariant subspaces for tridiagonal operators.
Berezin Transform of Invertible Positive Operators
Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application
In this paper we introduce a class A ⊂ L∞(D) such that if φ ∈ Aand satisfies certain positive-definite condition, then there exists aψ ∈ A such that φ(z) ≤ αeψ(z), for some constant α > 0. Further,if φ(z) = hAkz, kzi, for some bounded positive, invertible operator Afrom the Bergman space L2a(D) into itself then ψ(z) = h(log A)kz, kzi.Here kz, z ∈ D are the normalized reproducing kernel of L2a(D). Ap-plications of these results are also discusseonly a non-standard growth condition. We show that our problem admits at least one weak solution. In order to do this, the main tool is the Berkovits degree theory for abstract Hammerstein type mappings.