Hyponormality of Toeplitz operators on Newton spaces

In this paper we consider the following ‘Toeplitz completion’ problem: Complete the unspecified analytic Toeplitz entries of the partial block Toeplitz matrix $ \begin{linenomath} A:=\begin{bmatrix} T_{\overline\psi_1}& ?\\[4pt] \T_{\overline\psi_2} \end{bmatrix} \end{linenomath} $ to make A hyponormal, where ψi∈H∞ is a non-constant rational function for i=1,2.

Coburn's lemma and the finite section method for random Jacobi operators

A bounded Hilbert space operatorTis hyponormal ifT∗T−TT∗is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the formf+g¯wherefandgare bounded analytic functions on the unit disk. We then find sufficient conditions whenfis a monomial.

In this paper we consider the hyponormality of Toeplitz operators <TEX>$T_{\varphi}$</TEX> on the Bergman space <TEX>$L_a^2{(\mathbb{D})$</TEX> in the cases, where <TEX>${\varphi}\;:=f+\bar{g}$</TEX> (f and g are polynomials). We present some necessary or sufficient conditions for the hyponormality of <TEX>$T_{\varphi}$</TEX> under certain assumptions about the coefficients of <TEX>${\varphi}$</TEX>.

In this work we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form ψ + φ where ψ and φ are analytic on the annulus {z ∈ C,1/2 < |z| < 1} .

In this note we consider the hyponormality of Toeplitz operators <TEX>$T_{\varphi}$</TEX> on the Weighted Bergman space <TEX>$A^2_{\alpha}(\mathbb{D})$</TEX> with symbol in the class of functions <TEX>$f+\bar{g}$</TEX> with polynomials <TEX>$f$</TEX> and <TEX>$g$</TEX> of degree 2.

This paper concerns algebraic and spectral properties of Toeplitz operators T φ T_{\varphi } , on the Hardy space H 2 ( T ) H^{2}({\mathbb {T}}) , under certain assumptions concerning the symbols φ ∈ L ∞ ( T ) \varphi \in L^{\infty }({\mathbb {T}}) . Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at T φ T_{\varphi } , for each quasicontinuous φ \varphi . Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

Let w and \omega be two weight functions on R^{2d} and 1 \leq p,q \leq \infty. Also let M(p,q,\omega) (R^d) denote the subspace of tempered distributions S' (R^d) consisting of f \in S' (R^d) such that the Gabor transform V_g f of f is in the weighted Lorentz space L(p,q,\omega d\mu) (R^{2d}) . In the present paper we define a space Q_{g,w}^{M(p,q,\omega) (R^d) as counter image of M(p,q,\omega) (R^d) under Toeplitz operator with symbol w. We show that Q_{g,w}^{M(p,q,\omega)}(R^d) is a generalization of usual Sobolev-Shubin space Q_s (R^d). We also investigate the boundedness and Schatten-class properties of Toeplitz operators.

Toeplitz Operators and Hankel Operators on the Hardy Space of the Unit Sphere

We study the commuting problem for Toeplitz operators on the Hardy space of the polydisk. We consider two symbols where one is an arbitrarily bounded symbol and the other is a bounded symbol which has the form of a sum of an analytic function and a co-analytic function. We then give a necessary and sufficient condition on the symbols for which the corresponding Toeplitz operators commute each other.

The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces.Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.

Mellin Transform, Monomial Symbols, and Commuting Toeplitz Operators

We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar--Bernevig--Hughes' 2-D Hamiltonian and see that there still exists topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

Introduction.Let 1 <p^ao, L"=L"{0, 2n) and 77* the set of periodic functions x+{<f>)eLp with Fourier coefficients vanishing on the negative integers: x+{</>)-!?= o xne™.Let fi+) eL", l//> + l/a= 1.Then, for any x+{<f>) e H", f{<f>)x+{</>) e L1 and therefore formally f{<f>)x+{f>)~y+{<f>) + v+{t>)e~i<t'.We define the Toeplitz operator T=T{f) : 77p -> 77p associated with/ by D{T{f)) = {x+(¿) g 77" : f{<¡>)x+{</>) = y + {<t>)-rv + {^)e^ and y+ e 77"} y+it) = T{f)x+{<p).Below, we assume that a^2 (so that 1 </? = 2 and H"=>H9) to assure that D{T{f)) contains all polynomials in e10 and is dense in 77p.(We insist that/e Lq, l/p+ l/a= 1, so that fx e L1, for x e D{T(f)).)It should be noted that the results below remain valid if p is replaced by any index p', púp'Ú2, since f{f>) eL"' for 2^q'^q.In [4], for p = 2, Hartman and Wintner posed the problem of the determination of the location of the spectrum and point spectrum of T{f).A number of steps ([6], [9], [1], [12], [2], [3], [8]) toward solving this problem have been taken.In some investigations of the point spectrum, additional assumptions have been made, such as, for example, that T{f) is selfadjoint [6], that 2 |/n| <°° ([9], [3]), or that f{<f>) is bounded and satisfies (0.1) arg/0) -g{<f>) + K<f>)where g(<£) = 2"=i a/7(^ -0y), J{(f>) = <t>~ 2rr[<p\2rr\, [x] the integral part of x, and the conjugate of «(<£) = conj h{f>) eL™ [12].In [7], Hartman found necessary and sufficient conditions that 0 (and hence any complex A) belong to the point spectrum of T{f), p = 2. Hartman's general conditions are in terms of the conj arg/(<£).If N{T{f)) is the null space of Tif), we find necessary and sufficient conditions on f{<f>) that dim N{T(f)) = m (Theorem 1.1) under the assumption that (0.1) holds with h{j>) continuous.This result is in terms of conj h{<¡>).When it is assumed, for example, that conj h{f>) is bounded (Corollary 1.2), our results generalize those of [12].We make no use of the hypothesis ||/|| M < oo and, even in case/(<£) is bounded, our results are more general than those of [12].