Positive Toeplitz Research Articles

Positive Toeplitz Operators from aWeighted Harmonic Bloch Space into Another

Positive Toeplitz Operators from aWeighted Harmonic Bloch Space into Another

Invertibility of Positive Toeplitz Operators and Associated Uncertainty Principle

Invertibility of Positive Toeplitz Operators and Associated Uncertainty Principle

Toeplitz separability, entanglement, and complete positivity using operator system duality

A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) C(S^1)^{(n)} of n × n n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) , where H \mathcal {H} is an arbitrary Hilbert space and C ( S 1 ) ( n ) C(S^1)_{(n)} is the operator system dual of C ( S 1 ) ( n ) C(S^1)^{(n)} . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from B ( H ) \mathcal {B}(\mathcal {H}) when H \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps ψ \psi on B ( H ) \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that ψ ( n ) ( x ) \psi ^{(n)}(x) is positive whenever x x is a positive n × n n\times n Toeplitz matrix with entries from B ( H ) \mathcal {B}(\mathcal {H}) . We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality.

Positive Toeplitz operators from a harmonic Bergman–Besov space into another

We define positive Toeplitz operators between harmonic Bergman–Besov spaces \(b^p_\alpha \) on the unit ball of \({\mathbb {R}}^n\) for the full ranges of parameters \(0<p<\infty \), \(\alpha \in {\mathbb {R}}\). We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman–Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on \(b^{2}_{\alpha }\) to be a Schatten class operator \(S_{p}\) in terms of averaging functions and Berezin transforms for \(1\le p<\infty \), \(\alpha \in {\mathbb {R}}\). Our results extend those known for harmonic weighted Bergman spaces.

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

Schatten class positive Toeplitz operators on Bergman spaces of the Siegel upper half-space

We characterize Schatten class membership of positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space in terms of averaging functions and Berezin transforms in the range of \(0<p<\infty \).

Total positivity of Toeplitz matrices of recursive hypersequences

We present a new class of totally positive Toeplitz matrices composed of recently introduced hyperfibonacci numbers of the r -th generation. As a consequence, we obtain that all sequences F n ( r ) of hyperfibonacci numbers of r -th generation are log-concave for r ≥ 1 and large enough n .

Positive Toeplitz Operators on the Bergman Spaces of the Siegel Upper Half-Space

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.

Carleson Measures and Toeplitz Operators on Doubling Fock Spaces

Given ϕ a subharmonic function on the complex plane ℂ, with ΔϕdA being a doubling measure, the author studies Fock Carleson measures and some characterizations on μ such that the induced positive Toeplitz operator Tμ is bounded or compact between the doubling Fock space $$F_\phi ^p$$ and $$F_\phi ^\infty $$ with 0 < p ≤ ∞, where μ is a positive Borel measure on ℂ.

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.

Generating Totally Positive Toeplitz Matrix from an Upper Bidiagonal Matrix

In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factorization are presented.

Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z| on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.

Positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain

We give criteria for the boundedness of positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain in terms of the Berezin symbol or the averaging function of the symbol. Moreover, we estimate the essential norm of positive Toeplitz operators assuming that they are bounded. As an application of these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.

Positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain

Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property

The q-Gelfand–Tsetlin graph, Gibbs measures and q-Toeplitz matrices

The problem of the description of finite factor representations of the infinite-dimensional unitary group investigated by Voiculescu in 1976, is equivalent to the description of all totally positive Toeplitz matrices. Vershik–Kerov showed that this problem is also equivalent to the description of the simplex of central (i.e. possessing a certain Gibbs property) measures on paths in the Gelfand–Tsetlin graph. We study a quantum version of the latter problem. We introduce a notion of a q-centrality and describe the simplex of all q-central measures on paths in the Gelfand–Tsetlin graph. Conjecturally, q-central measures are related to representations of the quantized universal enveloping algebra U ϵ ( gl ∞ ) . We also define a class of q-Toeplitz matrices and show that extreme q-central measures correspond to q-Toeplitz matrices with non-negative minors. Finally, our results can be viewed as a classification theorem for certain Gibbs measures on rhombus tilings of the halfplane. We use a class of q-interpolation polynomials related to Schur functions. One of the key ingredients of our proofs is the binomial formula for these polynomials proved by Okounkov.

Toeplitz matrices, ergodicity and Fréchet spaces

Abstract Yosida´s version of the mean ergodic theorem is extended to positive Toeplitz matrices and locally convex spaces. This is used to provide extensions of the results of Albanese, Bonet and Ricker on the structure of Fréchet spaces and also Köthe echelon spaces.

Gibbs Ensembles of Nonintersecting Paths

We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices.

Positive Schatten(-Herz) Class Toeplitz Operators on Pluriharmonic Bergman Spaces

We study characterizations of arbitrary positive Toeplitz operators of Schatten (or Schatten-Herz) type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman space.

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.

Positive Toeplitz operators between the pluriharmonic Bergman spaces

We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space bp into another bq for 1 < p, q < ∞ in terms of certain Carleson and vanishing Carleson measures.