Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

In this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

Multivariate version of slant Toeplitz operators on the Lebesgue space

The paper introduces the [Formula: see text]th-order slant Toeplitz operator on the Lebesgue space of [Formula: see text]-torus, where [Formula: see text] such that [Formula: see text] for all [Formula: see text]. It investigates certain properties of [Formula: see text]th-order slant Toeplitz operators on the Lebesgue space [Formula: see text]. The paper deals with a system of operator equations, characterizing the [Formula: see text]th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

Toeplitz operators between large Fock spaces in several complex variables

<abstract><p>Let $ \omega $ belong to the weight class $ \mathcal{W} $, the large Fock space $ \mathcal{F}_{\omega}^{p} $ consists of all holomorphic functions $ f $ on $ \mathbb{C}^{n} $ such that the function $ f(\cdot)\omega(\cdot)^{1/2} $ is in $ L^p(\mathbb{C}^{n}, dv) $. In this paper, given a positive Borel measure $ \mu $ on $ {\mathbb C}^n $, we characterize the boundedness and compactness of Toeplitz operator $ T_\mu $ between two large Fock spaces $ F^{p}_\omega $ and $ F^{q}_\omega $ for all possible $ 0 &lt; p, q &lt; \infty $.</p></abstract>

Spectral properties of Landau Hamiltonians with non-local potentials

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V, such that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ). First, we construct symbols V, possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ), and investigate its spectral asymptotics.

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.

Compressions of Multiplication Operators and Their Characterizations

Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

Normal Toeplitz Operators on the Fock Spaces

We characterize normal Toeplitz operator on the Fock spaces F2(C). First, we state basic properties for Toeplitz operator Tφ on F2(C). Next, we study the normal Toeplitz operator Tφ on F2(C) in terms of harmonic symbols φ. Finally, we characterize the normal Toeplitz operators Tφ with non-harmonic symbols acting on F2(C).