We answer Bauer and Le's question on zero products of Toeplitz operators on the Fock space $F^2(\mathbb C^n)$[JFA, 261 (2011), 9, 2617--2640]. For $n\ge2$, we construct two bounded nonradial Schwartz symbols on $\mathbb C^n$ whose Toeplitz operators are nonzero and have zero product on $F^2(\mathbb C^n)$. For $n=1$ and each $c\in(1/2,1)$, we construct two smooth nonradial symbols of growth at most $Ce^{c|z|^2}$ for some constant $C>0$. Their extended Toeplitz operators in $F^2(\mathbb C)$ are nonzero and have zero product on all holomorphic polynomials. Moreover, the second symbol is bounded when $c\geq3/4$. Our proofs use Gaussian kernel calculations, matrix identities, theta functions and Fourier transform.
We introduce and study a two-parameter family of Toeplitz-type operators $T^{(s,t)}_\varphi$ on the Fock space $F^p(C)$ for $0<p\leq\infty$. For all $0<p, q \leq\infty$ we obtain sharp conditions for $\varphi$ such that $T^{(s,t)}_\varphi: F^p(C)\to F^q(C)$ is bounded or compact. In the Hilbert space case $T^{(s,t)}_\v...
We study absolutely $r$-summing Toeplitz operators $T_f:F_\varphi^p\to F_\varphi^q$ with complex symbols. The weight has a uniformly positive and bounded real Hessian. The main difficulty is that local distance to holomorphic functions does not detect an entire part of the symbol, while the Toeplitz operator does. Usin...
We study Toeplitz operators acting on radial weighted Fock spaces. We use tools from representation theory to construct commutative families of $C^*$-algebras that are generated by Toeplitz operators whose symbols are invariant under the action of $\U(n)$. For a partition $m=(b_1,...,b_k)$ of an integer $n$, we realize...
In this paper, we focus on the boundedness and compactness of the Toeplitz operators and the Hankel operators on the $\mathcal{D}$-weighted Bergman spaces $A_{\omega}^p$ ($1\leq p<\infty$). In particular, the boundedness of Toeplitz operators with $\mathrm{BMO}_\omega^p$-symbols on $A_{\omega}^p$ is characterized in te...
Yong-Jiang Duan, K. Guo, Jun-Han Hong et al.· 1 citation
Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb B$ of a complex Banach space $X$ or on the unit polydisk $\mathbb U^n$ of $\mathbb{C}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fr\'echet derivatives of $F$ when $F$...
S. Panja, Sujoy Majumder, Abhijit Banerjee· 0 citations
We study commutativity and zero products of slant Toeplitz operators on the Bergman space of the unit disk. For bounded polyanalytic symbols and for bounded harmonic symbols with boundary values in $W^{1,2}(\mathbb T)$, we prove that two such operators commute if and only if their symbols are linearly dependent over $\...
Pu-Yu Cui, Ran Li, Yu-Feng Lu· 0 citations
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