Complex Analysis and Operator Theory最新文献

We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression

in (L^2big ((-1,1); (1-x)^{alpha } (1+x)^{beta } dxbig )), (alpha , beta in {mathbb {R}}). In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general (eta )-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.

In this paper, we analyse properties like nullity, defect, ascent and descent of the powers of regular linear relations. We improve some results related to closure and regularity of powers of linear relations in normed spaces. Further, the obtained results are applied to investigate the descent and essential descent spectrum and to give some stability results.

The role of de Branges–Pontryagin spaces as functional models for entire symmetric operators with finite equal deficiency indices and proper gauges in Pontryagin spaces is reviewed and then extended to symmetric operators that are not entire. These results are used to derive an operator representation for generalized Carathéodory functions. Enroute, boundary mappings and the characteristic function of S are defined. Generalized resolvents of symmetric operators S with non dense domains corresponding to single-valued representing extensions ({{widetilde{S}} }) are characterized in terms of the characteristic function of S. These results are applied to obtain a description of the set of solutions of an indefinite truncated matrix moment problem.

In this paper we study special systems of orthogonal polynomials on the unit circle. More precisely, with a view to the recurrence relations fulfilled by these orthogonal systems, we analyze a link of non-negative arithmetic to harmonic sequences as a main subject. Here, arithmetic sequences appear as coefficients of orthogonal polynomials and harmonic sequences as corresponding Szegő parameters.

The Jordan curve theorem states that any simple closed curve in 3D space divides the space into two regions, an interior and an exterior. In this article, we prove the Jordan curve theorem on the boundary of a 3D ball that is inserted in a complex plane bundle. To do so, we make use of the Brownian motion principle, which is a continuous-time and continuous-state stochastic process. We begin by selecting a random point on an arbitrarily chosen complex plane within a bundle G and on the boundary of the 3D ball considered. Using the two-step random process developed on complex planes earlier by Srinivasa Rao (Multilevel contours on bundles of complex planes, 2022), we draw a contour from the initial point to the next point on this plane. We then continue this process until we finish the Jordan curve that connects points on the boundary of a ball within G.

In the setting of the lattice (mathbb {Z}^n) we consider a pseudo-differential operator A whose symbol belongs to a class defined on (mathbb {Z}^ntimes mathbb {T}^n), where (mathbb {T}^n) is the n-torus. We realize A as an operator acting between the discrete Sobolev spaces (H^{s_j}(mathbb {Z}^n)), (s_jin mathbb {R}), (j=1,2), with the discrete Schwartz space serving as the domain of A. We provide a sufficient condition for the essential adjointness of the pair ((A,,A^{dagger })), where (A^{dagger }) is the formal adjoint of A.

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space (mathcal {H}). Among other results, we show that if (mathcal {H}) is a finite-dimensional Hilbert space and (Tin mathfrak {B}(mathcal {H})), then T is self-adjoint if and only if there exists (p>0) such that (|T|^ple |textrm{Re},(T)|^p). If in addition, T and (textrm{Re},T) are invertible, then T is self-adjoint if and only if (log ,|T|le log ,|textrm{Re},(T)|). Considering the polar decomposition (T=U|T|) of (Tin mathfrak {B}(mathcal {H})), we show that T is self-adjoint if and only if T is p-hyponormal (log-hyponormal) and U is self-adjoint. Also, if (T=U|T|in mathfrak {B}({mathcal {H}})) is a log-hyponormal operator and the spectrum of U is contained within the set of vertices of a regular polygon, then T is necessarily normal.

In this paper, we introduce the strong b-suprametric spaces in which we prove the fixed point principles of Banach and Edelstein. Moreover, we prove a variational principle of Ekeland and deduce a Caristi fixed point theorem. Furthermore, we introduce the strong b-supranormed linear spaces in which we establish the fixed point principles of Brouwer and Schauder. As applications, we study the existence of solutions to an integral equation and to a third-order boundary value problem.

This paper extends the Rubio de Francia extrapolation method to the grand Morrey spaces on Euclidean spaces. By using this extended extrapolation method, we obtain the boundedness of the rough singular integral operators, the spherical maximal functions and the maximal Bochner-Riesz operators on the grand Morrey spaces on Euclidean spaces.

Aim

The aim of this study is to introduce definitions and explore properties of moment problems for sequences of generalized hybrid numbers satisfying a linear recursive equation.

Methods

We analyze complex measures derived from the linear recurrence of hybrid numbers and generalized hybrid numbers sequences.

Results

We present results pertaining to the moments of these complex measures.

Conclusions

This study contributes to the understanding of moment problems in the context of generalized hybrid number sequences.