Narrow Results

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian $A_{\alpha }={(i\nabla +A)}^2+\alpha {\delta }_{\mathrm{\Sigma }}$ in $L^2({ℝ}^2)$ with a $\delta$-potential supported on a finite $C^{1,1}$-smooth curve $\mathrm{\Sigma }$ are studied. Here $A=\frac{1}{2}B{(-x_2,x_1)}^{\mathrm{\text{T}}}$ is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $\alpha \in L^{\infty }(\mathrm{\Sigma })$ is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $\mathrm{\Sigma }$. After a general discussion of the qualitative spectral properties of $A_{\alpha }$ and its resolvent, one of the main objectives in the present paper is a local spectral analysis of $A_{\alpha }$ near the Landau levels $B(2q+1)$, $q\in {\mathbb{N}}_0$. Under various conditions on $\alpha$, it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of $\alpha$. Furthermore, the use of Landau Hamiltonians with $\delta$-perturbations as model operators for more realistic quantum systems is justified by showing that $A_{\alpha }$ can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.

We characterize binormal Toeplitz operators with analytic, or, coanalytic symbol functions. Furthermore, for a large class of nonanalytic, noncoanalytic Toeplitz operators which include Toeplitz operators with trigonometric or rational symbols, we prove that those Toeplitz operators are binormal if and only if they are normal. Some of the historically important examples of Toeplitz operators in the paper show that our problem is subtle and the above result is sharp.

When is the collection of $S$-Toeplitz operators with respect to a tuple of commuting bounded operators $S=(S_1,S_2,\dots ,S_{d-1},P)$, which has the symmetrized polydisc as a spectral set, nontrivial? The answer is in terms of powers of $P$ as well as in terms of a unitary extension. En route, the Brown–Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the $C^{*}$-algebra generated by the commutant of $S$ and the commutant of its unitary extension $R$.

In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators $W_{\psi ,\phi }J$ are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators $T_{z^{p\overline{z}q}}$ with $p=Vq$ for some symmetric permutation matrix $V$. Surprisingly, these operators $T_{z^{Vq\overline{z}q}}$ are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk.

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

The automorphism group of the Toeplitz C*- algebra, , generated by Toeplitz operators with C¹-symbols on Dirichlet space is discussed; the K₀, K₁-groups and the first cohomology group of are computed. In addition, the author proves that the spectra of Toeplitz operators with C¹-symbols are always connected, and discusses the algebraic properties of Toeplitz operators. In particular, it is proved that there is no nontrivial selfadjoint Toeplitz operator on and if and only if T_φ is a scalar operator.

On the setting of the unit ball U the author considers Toeplitz operators on Bergman space. The Bergman space B^p(U)(1 ≤ p < ∞) is the closed subspace of the usual Lebesgue space L^p(U) consisting of holomorphic functions. For a function β ∈ L²(U), the Toeplitz operator T_β with symbol β is defined by T_β f = ℘(β f) for function f ∈ B²(U). Here ℘ is the Bergman projection from L²(U) onto B²(U). Two bounded linear operators S, T on the Hilbert H are said to be essentially commuting on H if the commutator ST-TS is compact on H. In this paper, a criterion of essentially Toeplitz operators with the vanishing property is obtained.

In this article we highlight the interplay of multi-parameter BMO spaces and boundedness of corresponding commutators. In a variety of settings, we discuss two-sided norm estimates for commutators of classical singular operators with a symbol function. In its classical form, this concerns a theorem by Nehari, factorisation of Hardy space, Hankel and Toeplitz forms. We highlight recent results in which a characterization of L^p boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms is obtained, completing a theory on characterisation of BMO spaces begun by Cotlar, Ferguson and Sadosky. In the light of real analysis, we discuss results in a more intricate situation; commutators of multiplication by a symbol function and Calderón-Zygmund or Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the Hilbert or Riesz transforms or their tensor products are a representative testing class for Calderón-Zygmund or Journé operators.