Let \(\mathbb{C}^+\) denote the upper half-plane and, for \(p>1\), let \(H^+_p:=H^p(\mathbb{C}^{+})\) be the Hardy space of \(\mathbb{C}^{+}\) and let \(H_p^-:=\overline{H^+_p}\). For a function \(g\), measurable on \(\mathbb{R}\), the Toeplitz operator \(T_g\) with symbol \(g\) is defined on the domain \[D(T_g)=\{f\in H^{+}_p:\, gf\in L^p(\mathbb{R})\}\] by the formula \[T_g f=P^+( gf),\] where \(P^+\) is the Riesz projection from \(L^p(\mathbb{R})\) onto \(H^{+}_p\). Let \(\sigma_p\) be the set of all measurable functions \(g\) on \(\mathbb{R}\) for which \(D(T_g)\not=\{0\}\).
The authors study kernels of unbounded Toeplitz operators in \(H^{+}_p\) with symbols in \(\sigma_p\). The authors give a necessary condition for the kernel of the Toeplitz operator to be non-trivial. The authors examine if, for a given function from \(H^{+}_p\), there exists a minimal Toeplitz kernel containing that function.
Denote \(\lambda_{\pm}(\xi)=\xi\pm i\) and \(r(\xi)=\frac{\xi-i}{\xi+i}\) for \(\xi\in\mathbb{R}\). For a symbol \(g\) admitting a \((q,p')\)-factorization, (\(1<p\leq q<\infty\), \(1/p+1/p'=1\)), that is, \[g=g_{-}r^kg_{+} \text{with } k\in\mathbb{Z},\] where \[g_{-}\in \lambda_{-}H_q^{-},\ g_{-}^{-1}\in \lambda_{-}H_{p'}^{-},\ g_{+}\in \lambda_{+}H_{p'}^{+},\ g_{+}^{-1}\in \lambda_{+}H_{q}^{+},\] the authors describe the kernel of the Toeplitz operator \(T_g\). If, additionally, the symbol \(g\) belongs to \(L^\infty\), they also describe the kernel of the adjoint \(T_g^*\).
To describe kernels of Toeplitz operators with piecewise continuous symbols, the authors study the “relation between kernels of the Toeplitz operators whose symbols differ by a factor which” is a non-integer power of \(r\).