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In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

For entire functions f and g we study the problem of when the Hankel product H f ‾ ⁎ H g ‾ is bounded or compact on the Fock space F α 2 . This is a companion to Sarason's Toeplitz product problem which was completely solved for the Fock space by Cho–Park–Zhu in 2014. Surprisingly, our results here are different from the natural conjectures motivated by the Toeplitz product problem.

In this paper, we obtain a sufficient condition for Toeplitz operators with locally integrable symbols to be bounded and compact on the Bergman space $L_{a}^{p}$ with $1\lt p\lt \infty$ in terms of generalized Carleson squares.

In this paper, we study the spectrum of Toeplitz operators on the Dirichlet space D with bounded conjugate analytic symbols and give a characterization of the kernels of Toeplitz operators with harmonic symbols in P ‾ + M ( D ) . Using this characterization, we show that the spectrum of a Toeplitz operator with symbol, the sum of an analytic polynomial with degree at most two and the conjugate of linear polynomial is connected, but there are Toeplitz operators with general harmonic polynomial symbols having disconnected spectrum.

In this paper, we study the asymptotic behavior of the determinants of Bergman–Toeplitz matrices with symbols in $$H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})$$ . We establish a criterion of the asymptotic invertibility and an asymptotic inversion formula for Bergman–Toeplitz operators. These results are applied to obtain two versions of the first Szego theorem for Bergman–Toeplitz matrices. Moreover, we describe the asymptotic distribution of singular values of Bergman–Toeplitz matrices with symbols in $$\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}}) \big )\cap \overline{\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})\big )}$$ .

In this paper, we characterize bounded Toeplitz product $$T_fT_{g}$$ and Hankel product $$H_f^{*}H_g$$ on Fock space $$F_{\alpha }^2$$ for two polynomials f and g in z and $${\overline{z}}$$ . As a consequence, we obtain when Toeplitz operator $$T_f$$ or Hankel operator $$H_f$$ with the polynomial symbol f in z and $${\overline{z}}$$ is bounded.

In this paper, we obtain sufficient and necessary conditions for big truncated Hankel operators on model spaces to be zero, or of finite rank or compact. Our main tools are the properties of Hardy Hankel operators and function algebras.

In this paper, we study the product problem of Toeplitz operators on the Bergman space of the unit disk. We characterize when the product of two Toeplitz operators T f T g is a finite rank perturbation of another Toeplitz operator T h , with f , g bounded harmonic and h in C 2 class with invariant Laplacian in L 1 . As a consequence, we show that there is no nontrivial rank one perturbation. However, in the case rank m ≥ 2 , we construct an example that shows there are bounded harmonic functions f , g and h such that T f T g − T h has rank exactly m .

A necessary and sufficient condition is found for a truncated Toeplitz operator with bounded symbol to be compact on the model space.

The system realizes the setting of assembly plan, the establishment of assembly process files, the feedback of equipment status and the reconfiguration of assembly tasks. It uses industrial Ethernet for data exchange to achieve the integration of management and control through the OPC interface. All the equipments based on logic control of PLC are fully automated. The client running on IPC is programmed with C#. All the device workstations are configured RFID. On the one hand, assembly data can be uploaded to management system together with the code number of the transmission read by RFID to trace the information of assembly process. On the other hand, assembly process parameters generated automatically by management system can be transferred to the equipments to achieve the reconfiguration of technological process.

In this paper, we study the spectrum and essential spectrum of Toeplitz operators on the Dirichlet space. Also a criterion of the asymptotic invertibility of Dirichlet Toeplitz operators and the Szego type theorems of Dirichlet Toeplitz determinants are established.

A200 an important industrial equipment in the field of industrial production, the problem of operating efficiency of asynchronous motor has drawn wide attention with the increasing severity of environmental problems in recent years. Intended to this problem, this paper takes the retarder test bed motor control system as the research object, deduces the motor efficiency optimization control algorithm, and uses the Simulilnk module in Matlab to establish the asynchronous motor loss model. Based on the vector control of asynchronous motor, a simulation model of efficiency optimization control based on loss model is established, and the efficiency optimization control of induction motor based on loss model is realized. At the same time, this paper compares the efficiency optimization control algorithm with the traditional vector control to test the effect of the efficiency optimization algorithm. The experimental results show that the efficiency optimization algorithm used in this paper responds rapidly and the mutation of the load torque fluctuates slightly, but it can be quickly re-stabilized. At the same time, compared with the traditional vector control, efficiency optimization algorithm can significantly improve the test bed operating efficiency. The simulation experiment provides the theoretical support and reference for the future application of the motor control system on the retarder test bench.

In this paper, we show that the spectrum of Toeplitz operators on the Bergman space with harmonic symbols of affine functions of z and $$\bar z$$ equals the image of closed unit disk under the symbol. Surprisingly this does not hold for Toeplitz operators with harmonic symbols of quadratic functions of z and $$\bar z$$ .

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of | z | on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.

In view of the docking assembly difficulties of spacecraft large parts, resulting from the irregular contour and the inadequacy of process clamping points, this paper puts forward a method fitting the attitude relations of the docking parts based on the coordinates of the feature points, obtaining the target attitude of the docking parts based on the process constraints, and controlling 6-axis industrial robot achieves precise docking of large parts. Firstly, the composition of the large parts docking system is introduced, and the docking model is described by the selection of the measurement feature points and the establishment of the coordinate system. After that, a docking control algorithm based on the attitude constraints of large parts is established. Finally, In the RobotStudio and Matlab co-simulation environment, the attitude adjustment process of the large parts is simulated. The simulation results show that this study provides an effective method for precision docking of spacecraft engine large parts based on feature points, which is completed by 6-axis industrial robot.

This paper introduces a kind of management and control system of machining production line for aero-engine nozzle. The main logistics is based on the logic control of PLC, which can dispatch a robot to realize the automatic transfer of the parts between workstations, automatic feeding and unloading of processing and testing equipment. Workstations are equipped with RFID to detect the identification of each nozzle. The system can reconstruct the process route according to the nozzle type, associate processing and test data of the nozzle with their identification number to establish the processing file. It realizes data exchanges between management and control systems by OPC interface, achieves centralized monitoring and control of the equipment of the whole line and improves the informatization level of workshop.

We use the Weierstrass σ-function associated with a lattice in the complex plane to construct finite dimensional zero-based subspaces and quasi-invariant subspaces of given index in the Bargmann-Fock space.

In this paper, we study the Rudin orthogonality problem on the Bergman space, which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the Bergman space of the unit disk. We completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk. As a consequence, it is shown that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n-th power of a Möbius transform, which was obtained via the Hardy space theory of the bidisk in Sun et al. (2008) [10].

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

In this paper we show that there are Toeplitz op- erators on the Bergman space with bounded harmonic symbols having disconnected spectrum and essential spectrum.

In this paper, by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk, we give a proof of the Beurling type theorem on the Bergman space of Aleman, Richter and Sundberg (1996) via the Hardy space of the bidisk.

We consider the question for which square integrable analytic functions f and g on the unit ball the densely defined products T f T g ¯ are bounded on the weighted Bergman spaces. We prove results analogous to those we obtained in the setting of the unit disk and the polydisk.

Motivated by recent works of Ahern and $$\breve{\rm C}$$ u $$\breve{\rm C}$$ kovic on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case.

$m$-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the $m$-Berezin transform as a linear operator from the space of bounded operators to $L^{\infty}$ is found. We show that the $m$-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the $m$-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

We give a necessary and sufficient condition for a composition operator C ϕ C_{\phi } on the Bloch space to have closed range. We show that when ϕ \phi is univalent, it is sufficient to consider the action of C ϕ C_{\phi } on the set of Möbius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

In this paper we use the beautiful formula of Faa di Brune for the nth derivative of composition of two functions to obtain the generalized Scliwarz-Pick estimates. By means of those estimates we show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric.

We prove that a bounded operator S on L for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.

We consider Hankel operators on the Segal-Bargmann space H 2 (C n ,dμ). Our main result is a necessary and sufficient condition for the simultaneous membership of H f and H f in the Schatten class Cp, 1 ≤ p < ∞. We will explain that, since this condition is valid in the case 1 ≤ p ≤ 2 as well as in the case 2 ≤ p < oo, this result reflects the structural difference between the Segal-Bargmann space and other reproducing-kernel spaces such as the Bergman space L 2 a (B n ,dv).

The authors obtain function theoretic characterizations of the compactness on the standard weighted Bergman spaces of the two operators formed by multiplying a composition operator with the adjoint of another composition operator.

We characterize when a Hankel operator and a Toeplitz operator have a compact commutator.

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.

In this paper, using the theory of Hilbert modules we study invariant subspaces of the Bergman spaces on bounded symmetric domains and quasi-invariant subspaces of the Segal–Bargmann spaces. We completely characterize small Hankel operators with finite rank on these spaces.

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

In this note we investigate conditions under which a holomorphic self-map of the unit disk induces a composition operator with closed range on the Bloch space.

This paper studies algebras of functions on the unit disk generated byH ∞(D) and bounded harmonic functions. Using these algebras, we characterize compact semicommutators and commutators of Toeplitz operators with harmonic symbols on the Bergman space.

We consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products TfTg are bounded on the Bergman space. We prove results analogous to those obtained by the second author [17] for such Toeplitz products on the Hardy space. We furthermore obtain similar results for Hankel products HfH*g, where f and g are square integrable on the unit disk, and for the mixed Haplitz products HfTg and TgH*f, where f and g are square integrable on the unit disk and g is analytic.

By making use of M \mathcal M -harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in n n -dimensional complex space.

A sufficient condition is found for the product of two Toeplitz operators on the Hardy space of the unit sphere to be a compact perturbation of a Toeplitz operator. The condition leads to a criterion for a Hankel operator to be compact.

This paper studies products of Hankel operators on the Hardy space. We show thatHfσ(1)*Hfσ(2)Hfσ(3)*=0 for all permutation σ if and only if eitherHf1 orHf2 orHf3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that $$H^\infty [f] \cap H^\infty [g] \cap H^\infty [h] \subset H^\infty + C$$ is a sufficient condition forHf*HgHh*,Hg*HfHh*, andHf*HhHg* to be compact.

In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted Lp-spaces are obtained. Invariant A8 weights are obtained. Several characterizations of invariant A8 weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators of Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.