Generalized Toeplitz Operators Research Articles

The Generalized Volterra Integral Operator and Toeplitz Operator on Weighted Bergman Spaces

We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disc. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class membership of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $$H^2$$ .

English

Let μ be a finite positive measure on the unit disk and let j ⩾ 1 be an integer. D. Suarez (2015) gave some conditions for a generalized Toeplitz operator $$T_\mu ^{(j)}$$ to be bounded or compact. We first give a necessary and sufficient condition for $$T_\mu ^{(j)}$$ to be in the Schatten p-class for 1 ⩽ p < ∞ on the Bergman space A2, and then give a sufficient condition for $$T_\mu ^{(j)}$$ to be in the Schatten p-class (0 < p < 1) on A2. We also discuss the generalized Toeplitz operators with general bounded symbols. If ϕ ∈ L∞ (D, dA) and 1 < p < ∞, we define the generalized Toeplitz operator $$T_\varphi ^{(j)}$$ on the Bergman space Ap and characterize the compactness of the finite sum of operators of the form $$T_{{\varphi _1}}^{(j)} \ldots T_{{\varphi _n}}^{(j)}$$ .

Algebraic Properties of Toeplitz Operators on the Symmetrized Polydisk

This paper is an effort to continue the legacy of the classically successful theory of Toeplitz operators on the Hardy space over the unit disk to a new domain in \({\mathbb {C}}^d\)—the symmetrized polydisk. We obtain algebraic characterizations of Toeplitz operators, analytic Toeplitz operators, compact perturbation of Toeplitz operators and dual Toeplitz operators. We then revisit the operator theory of this domain considered first in Biswas and Shyam Roy (J Funct Anal 266:6224–6255, 2014), to study the generalized Toeplitz operators and find a commutant lifting type result.

Toeplitz operators on Bergman spaces of polygonal domains

AbstractWe study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 &lt; p &lt; ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

On generalized Toeplitz and little Hankel operators on Bergman spaces

We find a concrete integral formula for the class of generalized Toeplitz operators $$T_a$$ in Bergman spaces $$A^p$$ , $$1<p<\infty $$ , studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $$L^2$$ -symbol a such that $$T_{|a|} $$ fails to be bounded in $$A^2$$ , although $$T_a : A^2 \rightarrow A^2$$ is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.

Properties of two variables Toeplitz type operators

AbstractThe investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator.

Toeplitz operators with locally integrable symbols on Bergman spaces of bounded simply connected domains

We study the boundedness and compactness of generalized Toeplitz operators with locally integrable symbols on Bergman spaces where is a bounded simply connected domain with smooth boundary. We give sufficient conditions for boundedness and compactness of in terms of “averages” of symbol a over certain Cartesian squares. The main tool in the proof is the Whitney covering: is decomposed into union of countably many squares whose side lengths are comparable to the boundary distance. If a is nonnegative, we show that the given conditions are also necessary.

Spectral triples and Toeplitz operators

We give examples of spectral triples, in the sense of A. Connes, constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in \mathbb C^n , or the star product for the Berezin–Toeplitz quantization. Our main tool is the theory of generalized Toeplitz operators on the boundary of such domains, due to Boutet de Monvel and Guillemin.

Geometric Arveson–Douglas conjecture

We prove the Arveson–Douglas essential normality conjecture for graded Hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball, smooth away from the origin. Our main tool is the theory of generalized Toeplitz operators of Boutet de Monvel and Guillemin.

Ergodic properties of operators in some semi-Hilbertian spaces

This article deals with linear operators T on a complex Hilbert space ℋ, which are bounded with respect to the seminorm induced by a positive operator A on ℋ. The A-adjoint and A 1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T * is not a quasiaffine transform of an orthogonally mean ergodic operator.

Hankel operators and the Dixmier trace on strictly pseudoconvex domains

Generalizing earlier results for the disc and the ball, we give a formula for the Dixmier trace of the product of 2n Hankel operators on Bergman spaces of strictly pseudoconvex domains in \bold C^n . The answer turns out to involve the dual Levi form evaluated on boundary derivatives of the symbols. Our main tool is the theory of generalized Toeplitz operators due to Boutet de Monvel and Guillemin.

An index theorem for Toeplitz operators on the quarter-plane

We prove an index theorem for Toeplitz operators on the quarter-plane using the index theory for generalized Toeplitz operators introduced by G. J. Murphy. To prove this index theorem we construct an indicial triple on the tensor product of two commutative symbol $\mathrm {C}^\ast$–algebras. We extend our results to matrices of Toeplitz operators on the quarter-plane.

Asymptotic behaviours and generalized Toeplitz operators

We study some generalized Toeplitz operators associated to operators T on a Hilbert space H , for which there exists the limit of { ‖ T n h ‖ } for every h ∈ H . We refer to the asymptotic limit S T of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571–605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373–400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101–140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation S ∗ X T = X and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19–32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions.

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b e \(\dot \Lambda _{\beta _0 } \) (ℝn) is discussed from Lp(ℝn) to Lq(ℝn), \(\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\beta _0 }}{n}\), and from Lp(ℝn) to Triebel-Lizorkin space \(\dot F_p^{\beta _0 ,\infty } \). We also obtain the boundedness of generalized Toeplitz operator Θα0b from Lp(ℝn) to Lq(ℝn), \(\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\alpha _0 + \beta _0 }}{n}\). All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on Lp(ℝn), 1 < p < ∞.

Topological and analytical indices in [formula omitted]-algebras

A notion of topological index for the continuous symbol functions of generalized Toeplitz operators is introduced. This generalizes the winding number of functions on the circle and the average winding number of almost periodic functions on the real line and makes fundamental use of the quantized differential calculus of Alain Connes. The analytic index of a generalized Toeplitz operator—defined in terms of a trace—is shown to be equal to minus the topological index of the symbol function of the operator, a result that extends some well-known index theorems.

CUMULANTS IN NONCOMMUTATIVE PROBABILITY THEORY III: CREATION AND ANNIHILATION OPERATORS ON FOCK SPACES

Exchangeability systems arising from Fock space constructions are considered and the corresponding cumulants are computed for generalized Toeplitz operators and similar noncommutative random variables. In particular, simplified calculations are given for the two known examples of q-cumulants.In the second half of the paper we consider in detail the Fock states associated to characters of the infinite symmetric group recently constructed by Bożejko and Guta. We express moments of multidimensional Dyck words in terms of the so-called cycle indicator polynomials of certain digraphs.

Projections onto the spaces of Toeplitz operators

Projections onto the spaces of all Toeplitz operators on the $N$-torus and the unit sphere are constructed. The constructions are also extended to generalized Toeplitz operators and applied to show hyperreflexivity results.

On Ptak's Generalization of Hankel Operators

In 1997 Ptak defined generalized Hankel operators as follows: Given two contractions \(T_1 \in \mathcal{B}(\mathcal{H}_1 )\) and \(T_2 \in \mathcal{B}(\mathcal{H}_2 )\), an operator \(X:\mathcal{H}_1 \to \mathcal{H}_2 \) is said to be a generalized Hankel operator if \(T_2 X = XT_1^* \) and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T 1 and T 2. This approach, call it (P), contrasts with a previous one developed by Ptak and Vrbova in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some T 1 and T 2, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Ptak.

Properties of generalized Toeplitz operators

An operatorX:\(X:\mathcal{H}_1 \to \mathcal{H}_2 \) is said to be a generalized Toeplitz operator with respect to given contractionsT1 andT2 ifX=T2XT1*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foias, and Ptak and Vrbova, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.