Radial Symbols Research Articles

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Toeplitz Operators on Harmonic Fock Spaces with Radial Symbols

The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established.

Dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space

<p>This paper aimed to give some partial answers to the zero-product problem and commutativity problem concerning dual Toeplitz operators with nonharmonic symbols on the orthogonal complement of the harmonic Bergman space. Using the symbol map, we described the necessary condition for $ S_{\varphi_1}S_{\varphi_2}\cdots S_{\varphi_N} = 0 $ with radial symbols. Furthermore, we established the sufficient and necessary conditions for $ S_{\varphi}S_{ \psi} = S_{\psi}S_{\varphi} $ with $ \varphi(z) = az^{p_1}\overline{z}^{q_1}+bz^{p_2}\overline{z}^{q_2} $ and $ \psi(z) = z^{s}\overline{z}^{t} $.</p>

Integral representation of radial operators on the Bergman space over the unit disc

Let D be the unit disc in C. In this article, we characterize all analytic functions g on D so that the integral operator Sg defined formally on A2(D) by(Sgf)(z)=∫Df(w)g(zw‾)dA(w),∀f∈A2(D),z∈D is bounded, where dA(w) is the normalized area measure on D. We show that the collection of all bounded operators Sg is exclusively the collection of all radial operators on A2(D). Further, we study several operator theoretic properties of these operators. Also, we give integral representation of the form Sg for all operators in the C⁎-algebra generated by Toeplitz operators Ta with radial symbols a∈L∞(D).

Global regularity of Weyl pseudo-differential operators with radial symbols in each phase-space variable

Global regularity of Weyl pseudo-differential operators with radial symbols in each phase-space variable

ΨDOs with radial symbols and spaces of type G

We investigate various G type spaces on Rd+ and their relations with the Gelfand-Shilov S type spaces on R2d through the mapping w : R2d ? Rd+, w(x, ?) = (2x21 + 2?21 ,..., 2x2d + 2?2d ). Sufficient conditions for the hypoellipticity of symbols originating from the coordinante radial symbols in G type spaces are also given. Two open problems explained in the introduction are posed.

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

$$L^p$$ regularity of Toeplitz operators on generalized Hartogs triangles

We obtain \(L^p\) estimates for Toeplitz operators on the generalized Hartogs triangles \(\mathbb {H}_\gamma = \{(z_1,z_2) \in \mathbb {C}^2\,{:}\, |z_1|^\gamma \!< |z_2|<1\}\) for two classes of positive radial symbols, one a power of the distance to the origin, and the other a power of the distance to the boundary.

Radial operators on polyanalytic weighted Bergman spaces

Let $$\mu _\alpha$$ be the Lebesgue plane measure on the unit disk with the radial weight $$\frac{\alpha +1}{\pi }(1-|z|^2)^\alpha$$ . Denote by $${\mathcal {A}}^{2}_{n}$$ the space of the n-analytic functions on the unit disk $${\mathbb {D}}$$ , square-integrable with respect to $$\mu _\alpha$$ . Extending results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wünsche in 2005) form an orthonormal basis of $${\mathcal {A}}^{2}_{n}$$ . Using this basis, we provide the Fourier decomposition of $${\mathcal {A}}^{2}_{n}$$ into the orthogonal sum of the subspaces associated with different frequencies. This leads to the decomposition of the von Neumann algebra of radial operators, acting in $${\mathcal {A}}^{2}_n$$ , into the direct sum of some matrix algebras. In other words, all radial operators are represented as matrix sequences. In particular, we represent in this form the Toeplitz operators with bounded radial symbols, acting in $${\mathcal {A}}^{2}_n$$ . Moreover, using ideas by Engliš (1996), we show that the set of the Toeplitz operators with bounded generating symbols is not weakly dense in $${\mathcal {B}}({\mathcal {A}}^{2}_n)$$ .

Weighted holomorphic mixed norm spaces in the unit disc defined in terms of Fourier coefficients

ABSTRACT Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the weighted case. We study the boundedness of the weighted Bergman projection and discuss boundedness and compactness of Toeplitz operators with radial symbols. We also provide a characterization of functions in these spaces.

On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators Ta in weighted sup-normed Banach spaces Hv∞ of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol a such that Ta is not bounded in any space Hv∞ with an admissible weight v. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol a at the boundary of D guarantees the boundedness of Ta. The sufficient conditions for the boundedness and compactness of Ta, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition (B), which includes in addition to standard weight classes also many rapidly decreasing weights.

On boundedness and compactness of Toeplitz operators in weighted H∞-spaces

We characterize the boundedness and compactness of Toeplitz operators Ta with radial symbols a in weighted H∞-spaces Hv∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc.We also construct a bounded harmonic symbol f such that Tf is not bounded in Hv∞ for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on Hv∞.

Algebraic properties of Toeplitz operators on generalized Fock spaces on [formula omitted

We study two problems involving algebraic properties of Toeplitz operators on generalized Fock spaces on Cd with weights of the form |z|2se−|z|2m, m≥1,s≥0. We determine the commutant of a given Toeplitz operator with a radial symbol which satisfies certain growth conditions. We also discuss the equation TfTg=0, when f or g is radial.

Algebraic Properties of Toeplitz Operators on the Pluri-harmonic Fock Space

We study some algebraic properties of Toeplitz operators with radial and quasi homogeneous symbols on the pluriharmonic Fock space over $\mathbb{C}^{n}$. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator, the zero-product problem for the product of two Toeplitz operators. Next we characterize the commutativity of Toeplitz operators with quasi homogeneous symbols and finally we study finite rank of the product of Toeplitz operators with quasi homogeneous symbols.

G-type spaces of ultradistributions over $$\mathbb {R}^d_+$$ R + d and the Weyl pseudo-differential operators with radial symbols

The first part of the paper is devoted to the G-type spaces i.e. the spaces \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)1 and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justification of obtained results. The Laguerre type expansions of the elements in \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)1 and their duals characterise these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological description and by the nuclearity of \(G_\alpha ^\alpha (\mathbb {R}^d_+)\), \(\alpha \ge \)1 the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the G-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.

Separately radial and radial Toeplitz operators on the unit ball and representation theory

We study Toeplitz operators with radial and separately radial symbols on the weighted Bergman spaces on the unit ball. The unitary equivalence of such operators with multiplication operators on \(\ell ^2\) spaces was previously obtained by analytic methods in Grudsky et al. (J Oper Theory 49(2):325–346, 2003) and Quiroga-Barranco and Vasilevski (Integral Equ Oper Theory 59(1):67–98, 2007), respectively. We prove that the same constructions can be performed with a purely representation theoretic approach to obtain the same conclusions and formulas. However, our method is shorter, more elementary and more elucidating.

Which pseudodifferential operators with radial symbols are non-negative?

Operators the Weyl symbols of which are of the special kind \(f(x,\xi )=F(|x|^2+|\xi |^2)\) are in the spectral-theoretic sense functions \(\Psi \left(L^{(n)}\right)\) of the harmonic oscillator \(L^{(n)}\). We give the exact conditions on the real function F which ensure that the function \(\Psi \) (defined on the set \(\{\frac{n}{2}+k,\,k=0,1,\dots \}\)) will be non-negative, non-decreasing or convex. In the one-dimensional case, the first point can also be treated within a one-parameter relativistic deformation of the whole situation, involving in place of Hermite functions the no longer elementary (modified) Mathieu functions.

Toeplitz operators with quasi-separately radial symbols on the complex projective space

We study the Toeplitz operators with quasi-separately radial symbols over the complex projective space $${\mathbb {CP}}^n$$ . We describe such Toeplitz operators and we prove that each bounded operator is unitarily equivalent to a Toeplitz operator whose symbol is a finite sum of quasi-separately radial symbols.

Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball. Spherical coordinates previously used hid a possibility to detect an essentially wider class of symbols that can generate commutative Banach Toeplitz operator algebras. We characterize these new algebras describing their properties and, under a certain extra condition, construct the corresponding Gelfand theory.

Lower bounds for pseudodifferential operators with a radial symbol

In this paper we establish explicit lower bounds for pseudodifferential operators with a radial symbol. The proofs use classical Weyl calculus techniques and some useful, if not celebrated, properties of the Laguerre polynomials.