Search Results (7)

The single generalized weighted composition operator $D_{u,\psi }^n$ on various spaces of analytic functions has been investigated for decades, i.e., $D_{u,\psi }^{n}f=u·(f^{\left(n\right)}\circ \psi )$, where $f\in H\left(\mathbb{D}\right)$. However, the study of the finite sum of generalized weighted composition operators with different orders, i.e., $P_{U,\psi }^{k}f=u_0·f\circ \psi +u_1·f^{\prime }\circ \psi +\cdots +u_k·f^{\left(k\right)}\circ \psi ,$ is far from complete. The boundedness, compactness and essential norm of sums of generalized weighted composition operators from weighted Bergman spaces with doubling weights into Bloch-type spaces are investigated. We show a rigidity property of $P_{U,\psi }^k$. Specifically, the boundedness and compactness of the sum $P_{U,\psi }^k$ is equivalent to those of each $D_{u_n,\psi }^n$, $0\le n\le k.$ Full article

Let $n\in {\mathbb{N}}_0$, $\psi$ be an analytic self-map on $\mathbb{D}$ and u be an analytic function on $\mathbb{D}$. The single operator $D_{u,\psi }^n$ acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as $(D_{u,\psi }^{n}f)(z)=u(z)f^{(n)}(\psi (z)),f\in H(\mathbb{D})$. However, the study of the operator $P_{\overrightarrow{u},\psi }^k$, which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator $P_{\overrightarrow{u},\psi }^k$ on the Bloch space are investigated in this paper, and several characterizations for these properties are provided. Full article

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we discuss the bound of the Bloch constant and the norm of the pre-Schwarzian derivative for the classes. Full article

In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates ${\gamma }_k\left(t\right)$ (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs. iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems. Full article

In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied. Special functions significant in both analytic $T(p,q,m,s;\mathrm{\Psi })$ norms and analytic $\mathrm{\Psi }$-Bloch norms serve as a framework for introducing new families of analytic classes. An application in operator theory is provided by establishing important properties of the composition-type operator $C_{\varphi }$ such as the boundedness and compactness with the help of the defined new classes. Full article