arXiv subjects
Publications and source records attributed to Yuanqi Sang.
Let $\mathbf T$ denote the $C^*$-algebra generated by all bounded Toeplitz operators on the Hardy space of the unit disk. Barr\'ıa and Halmos [\emph{Trans. Amer. Math. Soc.} \textbf{273} (1982), no.~2, 621--630] asked whether the Cesàro operator belongs to $\mathbf T$. We answer this question affirmatively.
We characterize all bounded complex symmetric weighted composition operators on the Fock space \(\mathcal F_α^{2}\), without prescribing a conjugation a priori. Our approach uses reproducing kernels and two generating functions. The Taylor coefficients of these functions are eigenvectors of the operator and its adjoint, respectively. The conjugations arising from our construction are anti-linear Gaussian integral operators. Some of them are weighted composition conjugations, whereas others are not.
We investigate the commutativity and semi-commutativity of generalized singular integral operators of the form $P_{+} f P_{+} + P_{-} g P_{+} + P_{+} u P_{-} + P_{-} v P_{-}$ on $L^{2}$, where $P_{+}$ denotes the Riesz projection and $P_{-}=I-P_{+}$. Building on this analysis, we develop a unified approach to studying the algebraic properties of operator classes on $L^{2}$ generated by multiplication operators together with the Riesz projection. These classes include, but are not limited to, Toeplitz+Hankel operators, singular integral operators, Foguel--Hankel operators, and asymmetric dual truncated Toeplitz operators. We provide complete characterizations of (i) the quasinormality of singular integral operators, and (ii) the necessary and sufficient conditions under which the product of two asymmetric dual truncated Toeplitz operators is again an asymmetric dual truncated Toeplitz operator. In addition, our methods provide new proofs of several known results, including the classical Brown-Halmos theorems and the commutativity of Hankel operators, singular integral operators, and dual truncated Toeplitz operators. We also improve the conditions for the normality of singular integral operators.
For bounded Lebesgue measurable functions $f,g,ϕ$ and $ψ$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}ϕP_{-}+P_{-}ψP_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or quasi-continuous symbols. As a consequence we obtain the spectra of various classes of GSIOs, the spectral inclusion theorem and comput the Fredholm index of GSIOs. Moreover, we gave the necessary and sufficient conditions for invertibility(Fredholmness) of GSIOs via Winer-Hopf factorization.
Motivated by Barr{\'ı}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. This provides a class of nontrivial projections in Toeplitz algebra.
Compared with harmonic Bergman spaces, this paper introduces a new function space which is called the pluriharmonic Hardy space $h^{2}(\mathbb{T}^{2})$. We character (semi-) commuting Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ with bounded pluriharmonic symbols. Interestingly, these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces, Bergman spaces and harmonic Bergman spaces. Our method for Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ gives new insight into the study of commuting Toeplitz operators on harmonic Bergman spaces.