Complex Analysis and Operator Theory最新文献

An operator T on a complex separable Hilbert space ({mathcal {H}}) is called a real operator if T can be represented as a real matrix relative to some orthonormal basis of ({mathcal {H}}). In this paper, we provide descriptions of concrete real operators, such as real normal operators, real partial isometries, and real Toeplitz operators, among others. Furthermore, we present several structure theorems of real operators, including the polar decomposition, the Riesz decomposition and the block structure.

In this paper, our primary focus is to study the boundedness and compactness of Volterra type operators and multiplication operators on minimal (alpha )-Möbius invariant function spaces. Additionally, we also present a characterization of the boundedness and compactness of Volterra type and multiplication operators from minimal (alpha )-Möbius invariant function spaces to Besov spaces.

Self-intersections and local singular points of the curves play an important role in algebraic geometry and many other areas. In the present paper, we study the self-intersection and local singular points of the n-member chains. For this purpose, we derive and use several new results on trigonometric formulas. A unified approach for calculating self-intersection and local singular points for a wide class of curves is presented. An application to the spectral theory of integro-differential operators with difference kernels is given as well.

A story of the joint research that led to the setting and solution of the Abstract Interpolation Problem.

In this paper, we define the generalized Toeplitz determinants whose entries are the coefficients of holomorphic functions on the unit disk (mathbb {U}) with k-fold symmetric, and then we establish the sharp bounds of the generalized determinants formed over the related terms of homogeneous expansion of a class of holomorphic mappings defined on the unit ball of a complex Banach space. The results presented here would generalize the corresponding results given by Giri and Kumar (Complex Anal Oper Theory 17(6):86, 2023).

Let (T=(T_{1}, T_{2},ldots , T_{n})) be a commuting (n-)tuple of operators on a complex Hilbert space H. We define the extended joint numerical radius of T by

where N is any norm on B(H),

and (overline{B_{n}}(0, 1)) denotes the closure of the unit ball in (mathbb {C}^{n}) with respect to the euclidean norm, i.e.

In this paper, we prove several inequalities for the extended joint numerical radius involving the spherical Aluthge transform in the case where N is the operator norm of B(H) or the numerical radius.

In this work the characteristic properties of image and kernel representations of passive and conservative state/signal systems are presented. Earlier these representations were introduced in joint papers with Olof J. Staffans on theory of linear state/signal systems for much wider class, namely closed state/signal systems. In the case of passive and conservative state/signal systems the central role in our theory play scattering representations instead of these representations. In this paper the connections between image and scattering representations of a passive state/signal system are established, too. Main notions and results of passive s/s theory are connected with known notions and results from Krein spaces that are intensively used here. At the end an example of passive and conservative state/signal system is demonstrate on a simple quantum graph.

We make a progress towards describing the semi-commutants of Toeplitz operators on the Fock–Sobolev space of nonnegative orders. We generalize the results in Bauer et al. (J Funct Anal 268:3017, 2015) and Qin (Bull Sci Math 179:103156, 2022). For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial case, which is different from what is known for the classical Fock space. As an application, we consider the conjecture which was shown to be false for the Fock space in Ma et al. (J Funct Anal 277:2644–2663, 2019). The main result of this paper says that there is a fundamental difference between the geometries of the Fock and Fock–Sobolev space.

Within this article, the maximal function method is used to establish the Calderón-Zygmund estimates for the weak solutions of a class of non-uniformly elliptic equations

where (A(x,Du)approx |Du|^{p_1-2}+mu (x)|Du|^{p_2-2}), (F(x,f)approx |f|^{p_1-2}+mu (x)|f|^{p_2-2}) and (1<p_1<p_2), (0le mu (cdot )in C^{0,alpha }({mathbb {R}}^n),;alpha in (0,1]). The aforementioned problems arise as Euler-Lagrange equations for variational functionals that were originally presented and studied within the context of Homogenization and the Lavrentiev phenomenon by Marcellini (Arch Ration Mech Anal 105:267–284, 1989. https://doi.org/10.1007/BF00251503) and Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33–66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958). They are distinctive in that they exhibit that the growth and ellipticity change between two distinct types of polynomial depending on the position. This feature is characteristic of strongly anisotropic materials. The contribution of this paper is closely tied to the significant advancements made by Colombo and Mingione (J Funct Anal 270:1416–1478, 2016. https://doi.org/10.1016/j.jfa.2015.06.022) in the qualitative analysis of double phase problems, as well as the related techniques used by Zhang et al. (Ann Polon Math, 114:45–65, 2015. https://doi.org/10.4064/ap114-1-4).

Let (alpha >-1) and assume that f is (alpha )-harmonic mapping defined in the unit disk that belongs to the Hardy class (h^p) with (pgeqslant 1). We obtain some sharp estimates of the type (|f(z)|le g(|r|) Vert f^*Vert _p) and (|Df(z)|le h(|r|)Vert f^*Vert _p). We also prove a Schwarz type lemma for the class of (alpha )-harmonic mappings of the unit disk onto itself fixing the origin.